4. The Quantum State¶
PDF pages 107–150
4.1.1 Example¶
Chapter 4 The Quantum State 4.1 Density Operator Traditionally, the state vector |\psi\rangle represents the state of a quantum system. However, we will need a more general object to represent the quantum state for the purposes of studying light-matter interactions. The density operator represents the state of a quantum system in a more general way than the state vector, and equivalently represents an observer’s state of knowledge of a system. It is particularly important to use the density operator in the quantum theory of open systems, where a quantum system interacts with an external system whose evolution is unknown, and in the quantum theory of measurement and information. When a quantum state can be represented by a state vector |\psi\rangle , the density operator is defined as the product
(4.1) (density operator, pure state) In this case, it is obvious that the information content of the density operator is equivalent to that of the state vector (except for the overall phase, which is not of physical significance). The state vector can represent states of coherent superposition. The power of the density operator lies in the fact that it can represent incoherent superpositions as well. For example, let |\psi\alpha\rangle be a set of states (without any particular restrictions). Then the density operator
X \alpha
(4.2) (density operator, general) models the fact that we don’t know which of the states |\psi\alpha\rangle the system is in, but we assign a probability or weight P\alpha to the quantum state |\psi\alpha\rangle in the mixture defined by \rho. Note that the weights obey X \alpha
(4.3) for proper normalization of the density operator. Another way to say it is this: the state vector |\psi\rangle represents a certain intrinsic uncertainty with respect to quantum observables; the density operator can represent uncertainty beyond the minimum required by quantum mechanics. Equivalently, the density operator can represent an ensemble of identical systems in possibly different states. A state of the form (4.1) is said to be a pure state. One that cannot be written in this form is said to be mixed. 4.1.1 Example As a simple example, consider a qubit, a two-level system with states |0\rangle and |1\rangle . The density operators corresponding to the eigenstates are |0\rangle \langle 0| and |1\rangle \langle 1|; clearly these are pure states. Another pure state is
4.1.2 Evolution¶
Chapter 4. The Quantum State
\sqrt 2, which has the corresponding density operator \rho = 1
. (4.4) The density operator is the sum of the density operators for the eigenstates, plus two extra terms that indicated the purity of the state or the coherence of the superposition. An example of a mixture of the two eigenstates comes from simply removing these last two terms: \rho = 1
. (4.5)
eigenstates |\psi0\rangle = |0\rangle and |\psi1\rangle = |1\rangle . However, we can equally well regard the same mixed state as a different mixture. That is, defining the mixed state
, (4.6) where
\sqrt
. (4.7) it is not hard to see that \rho = \rho′. Thus we see that we have to be a bit careful with our above statement, where we said that a mixed state can be regarded as an association of classical probabilities with being in different pure quantum states. Just given a particular density operator, it is not not in general possible to uniquely define a pure-state decomposition of the form (4.2). Thus stating that the state is really in a pure state, but we don’t quite know which one it’s in, implies some extra information that is not contained in the density operator. 4.1.2 Evolution Differentiating the density operator and employing the Schrödinger equation i¯h\partial t|\psi\rangle = H|\psi\rangle , we can write down the equation of motion for the density operator:
= −i
¯h\rhoH = −i ¯h[H, \rho]. (4.8) (Schrödinger–von Neumann equation) This is referred to as the Schrödinger–von Neumann equation. The derivation here assumed a pure state but carries through in the obvious way for arbitrary density operators. Of course, the point is that using the density operator allows us to write down more general evolution equations than those implied by state-vector dynamics. The more general forms are referred to as Liouville–von Neumann equations or master equations, which we can write in the form
(4.9) (master equation, generic form) Here, L is the Liouvillian superoperator. We use the term ‘‘superoperator’’ because the Liouvillian represents a higher-dimensional object, since it must represent the commutator above (i.e., it ‘‘operates from both sides’’). Thinking of the density operator as a two-dimensional matrix as we discuss below, the Liouvillian is effectively a rank-4 tensor.
4.1.4 The Density Matrix¶
4.1 Density Operator 4.1.3 Expectation Values We can compute expectation values with respect to the density operator via the trace operation. The trace of an operator A is simply the sum over the diagonal matrix elements with respect to any complete, orthonormal set of states |\beta\rangle : Tr[A] := X \beta
(4.10) An important property of the trace is that the trace of a product is invariant under cyclic permutations of the product. For example, for three operators, Tr[ABC] = Tr[BCA] = Tr[CAB]. (4.11) (cyclic permutation invariance) This amounts to simply an interchange in the order of summations. For example, for two operators, working in the position representation, Tr[AB] = Z
= Z dx Z
= Z dx′ Z
= Z
= Tr[BA]. (4.12) Note that this argument assumes sufficiently ‘‘nice’’ operators (it fails, for example, for Tr[xp]). More general permutations [e.g., of the form (4.11)] are obtained by replacements of the form B −\rightarrow BC. Using this property, we can obviously write the expectation value with respect to a pure state as
(4.13) (expectation value, pure state) This obviously extends to the more general form (4.2) of the density operator. Taking an additional average over the ensemble of pure states,
X \alpha
(4.14) (expectation value, ensemble) where the double angle brackets \langle \langle \rangle \rangle denote the ensemble average over expectation values. For simplicity we will drop the extra brackets and simply use single brackets for expectation values with respect to either a pure state or an ensemble (\langle \langle \rangle \rangle −\rightarrow \langle \rangle ). 4.1.4 The Density Matrix The physical content of the density operator is more apparent when we compute the elements \rho\alpha\alpha′ of the density matrix with respect to a complete, orthonormal basis. The density matrix elements are given by
(4.15) (density matrix) To analyze these matrix elements, we will assume the simple form \rho = |\psi\rangle \langle \psi| of the density operator, though the arguments generalize easily to arbitrary density operators. The diagonal elements \rho\alpha\alpha are referred to as populations, and give the measurement probability of the system in the state |\alpha\rangle :
(4.16)
4.1.5 Purity¶
Chapter 4. The Quantum State
the relative phase of different components of the superposition. For example, if we write the state vector as a superposition with explicit phases,
X \alpha
(4.17) then the coherences are
(4.18) Notice that for a density operator not corresponding to a pure state, the coherences in general will be the sum of complex numbers corresponding to different states in the incoherent sum. The phases will not in
a generic mixed state. 4.1.5 Purity How can we tell a pure state from a mixed one in general? Notice that the diagonal elements of the density matrix form a probability distribution. Proper normalization thus requires
X \alpha
(4.19) (normalization) We can do the same computation for \rho2, and we will define the purity to be Tr[\rho2]. For a pure state, the
(4.20) (purity for pure state)
the density operator in (4.2),
X \alpha P 2 \alpha , (4.21) if we assume the states |\psi\alpha\rangle to be orthonormal. For equal probability of being in N such states, Tr[\rho2] = 1/N. Intuitively, then we can see that Tr[\rho2] drops to zero as the state becomes more mixed—that is, as it becomes an incoherent superposition of more and more orthogonal states.
may be diagonalized by a unitary transformation, so we may write
(4.22) where \rho′ is diagonal and S−1 = S\dagger. It is easy to verify that the trace is invariant under unitary transforma- tions, so
X \alpha (\rho′
X \alpha \rho′
(4.23) where the inequality comes from noting that 0 \le \rho′
\alpha\alpha. A diagonal pure state has only a single nonzero diagonal element, while a diagonal mixed state necessarily has more than one nonzero diagonal element. Hence, for a mixed state, Tr[\rho2] < 1. This follows since the diagonal matrix elements are positive,
X k
(4.24) and so the equality occurs only for a single term in the sum.
4.2.1 Unitary Time-Evolution Operator¶
4.2 Pictures 4.2 Pictures 4.2.1 Unitary Time-Evolution Operator The Schrödinger equation generates the time evolution of the state vector |\psi\rangle . It is convenient to represent time evolution in the form of an operator:
(time-evolution operator, definition) (4.25) Here, U(t, t0) is the unitary time-evolution operator1 that evolves the state from time t0 to t. Note that the operator must be unitary to preserve the norm of the state vector. Since
(4.26) if we require this to be equal to \langle \psi(t0)|\psi(t0)\rangle for any initial state |\psi(t0)\rangle , then it follows that
(4.27) (unitary condition) and thus U(t, t0) is unitary. In other words, unitarity of the evolution is required to conserve probability. The time-evolution operator also must have the composition property
(4.28) (composition property) which is sensible for the representation of time evolution. In this relation, note that with the time ordering t0 < t1 < t2, the earliest time appears to the right, since it is that one that operates ‘‘first’’ on the state vector. Finally, we must have the inversion property
(4.29) (inversion property) so that the inverse of an evolution operator corresponds to backwards-time evolution. 4.2.1.1 Infinitesimal Form Again, the Schrödinger equation
(4.30) can be rewritten in differential form as
(4.31) and thus generates the evolution over an interval dt according to
1 −i ¯hH dt
(4.32) Thus, the infinitesimal time-evolution operator is given by
¯hH dt. (4.33) (infinitesimal time-evolution operator) We can verify the above properties for this form of the evolution operator. For example,
1 + i ¯hH dt 1 −i ¯hH dt
(4.34) A similar argument works for the composition property, which gives the form of U(t + 2 dt, t). 1For further reading, see J. J. Sakurai, Modern Quantum Mechanics 2nd ed. (Addison Wesley, 1993), chapter 2, p. 68.
Chapter 4. The Quantum State 4.2.1.2 Differential Equation for the Evolution Operator Now using the composition property,
1 −i ¯hH dt U(t, t0). (4.35) Thus, U(t, t0), regarded as a function of t, undergoes time translation in the same way as the state vector |\psi(t)\rangle . In particular, then, we can write
¯hHU(t, t0), (Schrödinger equation for evolution operator) (4.36) and thus we see that the evolution operator satisfies the Schrödinger equation. 4.2.1.3 General Form Noting again that dt2 = 0 for infinitesimal time increments, we can write
¯hH dt = e−iH dt/¯h. (4.37) Then the composition property extends to give the general form of the evolution operator over finite time intervals,
tY t0
(4.38) where the product is over all infinitesimal time intervals dt\alpha between t0 and t. The product is ordered such
we can combine the elements of the product into a single exponential, using the relation eAeB = eA+B, (4.39) which holds when [A, B] = 0. Then
" −i ¯h t X t0
= exp −i ¯h Z t t0 H(t′) dt′ . (4.40)¶
If the Hamiltonian does not commute with itself at different times, then we can’t use this form, and we must use the form (4.38). We can use the shorthand for this notation2
−i ¯h Z t t0 H(t′) dt′ , (4.41) (general form) where T is the chronological operator, which indicates that the exponential is really a time-ordered product of infinitesimal time-evolution operators. On the other hand, if the Hamiltonian is time-independent,
−i ¯hH(t −t0) , (4.42) (general form, time-independent H) and we see that the time-evolution operator simplifies considerably, being essentially just the exponentiated Hamiltonian operator. 2V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Relativistic Quantum Theory, (Pergamon Press, 1971).
4.2.2 Schrödinger vs. Heisenberg Picture¶
4.2 Pictures 4.2.2 Schrödinger vs. Heisenberg Picture The Schrödinger picture is the usual scheme that you probably learned when you first studied quantum mechanics. In the Schrödinger picture, the state vector |\psi(t)\rangle evolves according to the Schrödinger equation. The operators, on the other hand, are time-independent, so that time-dependent expectation values are computed by
(4.43) (Schrödinger picture) An alternate scheme, the Heisenberg picture, is formulated differently. In the Heisenberg picture, the time dependence is carried by the operators, not the state vector. The state vectors are time-independent here. Thus, the expectation value in the Heisenberg picture is given as
(4.44) (Heisenberg picture) How do we transform between the two pictures? We will use the subscripts ‘‘S’’ and ‘‘H’’ for the Schrödinger and Heisenberg pictures, respectively, so that AS is the Schrödinger operator, and AH(t) is the Heisenberg operator. Then we can use the time-evolution operator to write
(4.45) where we have identified the transformation between pictures as
(4.46) (operator transformation) We also identify the Heisenberg-picture state vector as the initial state vector:
(4.47) (state transformation) Note that the density operator \rho is a Schrödinger-picture operator, since it is equivalent to the state vector. Thus, the density operator transforms as
(4.48) (operator transformation) That is, in the Schrödinger picture, the density operator is time-dependent, while it is time-independent in the Heisenberg picture, which is opposite to the behavior of operators for observables. 4.2.2.1 Heisenberg Equation of Motion In the Heisenberg picture, the operators evolve in time, so we must derive an equation of motion for Heisen- berg operators. Differentiating a Heisenberg operator A and using U as shorthand for U(t, 0),
U \daggerASU =
= i ¯hU \daggerHS(t)ASU −i ¯hU \daggerASHS(t)U = i ¯hU \daggerHS(t)UU \daggerASU −i ¯hU \daggerASUU \daggerHS(t)U, (4.49)
4.2.3 Interaction Picture¶
Chapter 4. The Quantum State and finally, we can write
¯h AH, U \daggerHS(t)U . (4.50) (Heisenberg-picture evolution) Note that we assumed AS to be time-independent—we are still not treating any operators with explicit time dependence. Recall that the Hamiltonian generates time evolution, and so the time dependence is externally imposed. We will thus not speak of the Heisenberg-picture Hamiltonian, although we use the ‘‘S’’ subscript to denote that we introduced it in the Schrödinger picture. Note that for a time-independent Hamiltonian
¯h [AH, H] . (Heisenberg evolution, time-independent H) (4.51) This evolution equation has a similar form to the Schrödinger–von Neumann equation (4.8) for the density operator (differing only by a minus sign). 4.2.3 Interaction Picture The interaction picture is a hybrid of the Schrödinger and Heisenberg pictures. Suppose that the Hamil- tonian can be decomposed as
(4.52) where V is the ‘‘interaction Hamiltonian.’’ Then the interaction picture is essentially the Schrödinger picture with respect to V , but the Heisenberg picture with respect to H0. That is, the state vector carries the time dependence due to V , while the operators carry the time dependence due to H0. For concreteness and simplicity, let’s assume that H0 is time-independent. Thus, the transformation of the state vector to the interaction picture is
(4.53) (interaction-picture state) and the transformation for the density operator follows similarly:
(4.54) (interaction-picture state) The operator transforms according to
(4.55) (interaction-picture operator) Then the background Hamiltonian causes the operator to evolve,
¯h [AI, H0] , (4.56) (interaction-picture evolution) while the state evolves according to the interaction Hamiltonian
¯h[VI(t), \rhoI], (4.57) (interaction-picture evolution) where VI(t) is the interaction-picture form of the Schrödinger-picture operator V (t) in the sense of Eq. (4.55). The interaction picture is useful in perturbation theory, where the evolution due to H0 is already known. It is thus convenient to bury this evolution in the operators, so that it is possible to focus on the perturbation Hamiltonian V .
4.3.1 Marginal Distributions¶
4.3 Wigner Distribution 4.3 Wigner Distribution One important tool for this discussion is the Wigner function (or distribution), which facilitates the de- scription of quantum dynamics in phase space. This is particularly important in comparing quantum and classical mechanics, where the analogous classical object is the phase-space (Liouville) distribution for an ensemble in phase space. The Wigner representation allows us to work with phase-space functions, rather than state vectors and operators. This allows for tremendous insight and simplification for certain aspects of quantum dynamics, and quite a bit of obfuscation for some others. What do we mean by phase space? For a single-particle, classical Hamiltonian of the form
2m + V (x), (4.58) the canonical coordinate pair (x, p) is sufficient to completely determine the state of the particle. Thus, for this system, we can define the (x, p) plane to be the phase space. In general, the classical phase space is the space of all generalized coordinates needed to completely specify the state of the system. For a classical Hamiltonian system of N degrees of freedom, we can generally take the phase space to be the 2N-tuple (x1, . . . , xN, p1, . . . , pN) of canonical coordinates. The Wigner function is defined in terms of the density operator as3
2\pi¯h Z \infty −\infty
(4.59) (Wigner distribution)
r(x, −x′), it is clear that the Wigner function is real-valued. Note that for a pure state, the above formula reduces to
2\pi¯h Z \infty −\infty
(4.60) (Wigner distribution) Obviously, the information content of the Wigner function is equivalent to that of the density operator, since the Wigner transform (4.59) can be inverted: the inverse Fourier transform of W(x, p) over p gives the rotated density operator, which is related by a simple coordinate rotation to the usual density operator. Note that for simplicity here we are sticking to systems of one degree of freedom; the generalizations to more dimensions is reasonably straightforward. 4.3.1 Marginal Distributions The Wigner function is not the only possible quantum phase-space distribution, but it has several features that make it preferable to other distributions. One of its most appealing properties is that each marginal distribution of the Wigner function, where one of the variables is integrated out, results in the probability distribution corresponding to the other variable. The Wigner function itself, however, is not a joint proba- bility distribution, since it can take on negative values, which as we will see below represent the interferences or coherences of the quantum state. To formalize this notion, we can show that when integrated over the variable p, the Wigner function yields the correct spatial probability density: Z \infty −\infty
(4.61) (x marginal distribution) 3E. Wigner, ‘‘On the Quantum Correction For Thermodynamic Equilibrium,’’ Physical Review 40, 749 (1932) (doi: 10.1103/PhysRev.40.749). For good reviews, see also Wolfgang P. Schleich, Quantum Optics in Phase Space (Wiley, 2001); M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, ‘‘Distribution Functions in Physics: Fundamentals,’’ Physics Reports 106, 121 (1984) (doi: 10.1016/0370-1573(84)90160-1); and V. I. Tatarskii, ‘‘The Wigner representation of quantum mechanics,’’ Soviet Physics Uspekhi 26, 311 (1983) (doi: 10.1070/PU1983v026n04ABEH004345).
4.3.2 Overlap¶
Chapter 4. The Quantum State To see this, just use the definition (4.59) and the integral representation of the delta function: Z \infty −\infty
2\pi¯h Z \infty −\infty
Z \infty −\infty dp e−ipx′/¯h = Z \infty −\infty
(4.62) Similarly, we can show that integration over x gives the momentum probability density: Z \infty −\infty
(4.63) (p marginal distribution) To see this, we perform the following steps: insert the definition (4.59), let x −\rightarrow x + x′/2, let x′ −\rightarrow x′ −x, use the expressions
\sqrt 2\pi¯h e−ipx′/¯h,
\sqrt 2\pi¯h eipx/¯h, (4.64) and finally use the completeness relation: Z \infty −\infty
2\pi¯h Z \infty −\infty dx Z \infty −\infty
= 2\pi¯h Z \infty −\infty dx Z \infty −\infty
= 2\pi¯h Z \infty −\infty dx Z \infty −\infty
= Z \infty −\infty dx Z \infty −\infty
(4.65) From either of these arguments it is obvious that the Wigner distribution is normalized such that Z \infty −\infty dx Z \infty −\infty
(4.66) (normalization) The marginal property works along other axes in phase space. This follows from the property that we will show below, that the Wigner-distribution evolution in a harmonic potential is simply a rotation, just as it is for a classical phase-space distribution. Thus, these same arguments can be applied after any phase- space rotation. In fact, this acts as one method for performing a tomographic reconstruction of the Wigner function, since the marginal distributions can be experimentally measured along many axes, then converted to the Wigner function via the (inverse) Radon transform.4 4.3.2 Overlap Another intuitively appealing feature of the Wigner function is that the overlap integral of two Wigner functions yields the overlap of the two corresponding density operators. We can see this for the overlap 4M. G. Raymer, M. Beck, and D. F. McAlister, ‘‘Complex Wave-Field Reconstruction Using Phase-Space Tomography,’’ Physical Review Letters 72, 1137 (1994) (doi: 10.1103/PhysRevLett.72.1137).
4.3.3 Area¶
4.3 Wigner Distribution of two states represented by W1(x, p) and W2(x, p), by again using the integral representation of the delta function: Z \infty −\infty dx Z \infty −\infty dp W1(x, p)W2(x, p) = (2\pi¯h)2 Z \infty −\infty dx Z \infty −\infty dp Z \infty −\infty dx′ Z \infty −\infty
= 2\pi¯h Z \infty −\infty dx Z \infty −\infty dx′ Z \infty −\infty
= 2\pi¯h Z \infty −\infty dx Z \infty −\infty
(4.67) Again letting x −\rightarrow x + x′/2 and then x′ −\rightarrow x′ −x, Z \infty −\infty dx Z \infty −\infty
2\pi¯h Z \infty −\infty dx Z \infty −\infty
= 2\pi¯h Z \infty −\infty dx Z \infty −\infty
= 2\pi¯h Z \infty −\infty
=
(4.68) Recall that for pure states,
(4.69) where the latter expression is known as the fidelity of two pure states. Thus, Tr[\rho1\rho2] is a generalized overlap for mixed states, and 2\pi¯h Z \infty −\infty dx Z \infty −\infty
(4.70) (overlap integral) represents the same overlap in terms of the Wigner functions. Note that Tr[\rho1\rho2] also represents the more
max |\langle 1|2\rangle |2, where the maximum is taken over all ‘‘purifications’’ (Section 4.4.5) |1\rangle of \rho1 and |2\rangle of \rho2.5
4.3.3 Area Recall that for the density operator, we showed that Tr[\rho2] \le 1, with the equality holding for pure states. Using Eq. (4.70),
Z \infty −\infty dx Z \infty −\infty
(4.71) Inverting this relation, we find a sort of generalized uncertainty relation, Z \infty −\infty dx Z \infty −\infty dp W 2(x, p) −1 \ge 2\pi¯h, (4.72) (area theorem) 5The important concept of fidelity for mixed states was introduced by Richard Jozsa, ‘‘Fidelity for Mixed Quantum States,’’ Journal of Modern Optics 41, 2315 (1994) (doi: 10.1080/09500349414552171); see also Benjamin Schumacher, ‘‘Sending entan- glement through noisy quantum channels,’’ Physical Review A 54, 2614 (1996) (doi: 10.1103/PhysRevA.54.2614). 6Richard Jozsa, op. cit.
4.3.4 Sample Wigner Distributions¶
Chapter 4. The Quantum State again with the equality holding for pure states. Recalling that the Wigner function is a normalized distribu- tion, along with the definition for the width \delta\omega for the normalized spectral function s(\omega) [Eq. (2.59)], we see that the quantity on the left-hand side is the two-dimensional generalization of the frequency width. Thus, the quantity on the left-hand side represents the area of the Wigner function, and is sometimes referred to as the Süßmann measure7 for the Wigner function. Again, this measure of area is more ‘‘robust’’ for certain distributions than corresponding measures based on rms widths. Eq. (4.72) therefore has the nice interpretation that the Wigner function for a pure state occupies an area of h in phase space, with mixed states occupying more area. 4.3.4 Sample Wigner Distributions To gain more intuition, since our discussion has so far been rather abstract, we will consider a few specific examples of Wigner functions. 4.3.4.1 Gaussian State For a Gaussian state, the definition (4.59) amounts to a Fourier transform of a Gaussian function. In this case, the Wigner function is clearly Gaussian. The most general Gaussian distribution we can write down for two dimensions (centered about the origin) is8
2\pi p
exp −1 2z\alpha S−1
, (4.73) (Gaussian state)
Vx Cxp Cxp Vp (4.74) (covariance matrix) is the covariance matrix, whose inverse is S−1
VxVp −C 2 xp Vp −Cxp −Cxp Vx . (4.75) Here, [a, b]+ = ab + ba is the anticommutator bracket, Vx =
x2 is the position variance, Vp =
p2 is the momentum variance, and Cxp =\langle xp + px\rangle /2 is the covariance. This form can be derived as follows. For two independent Gaussian random variables, the joint distribution is just the product of two individual Gaussian distributions. Written in matrix form, this gives the above form for W(x, p) where the covariance matrix is diagonal (recall that we just want the distribution with the correct marginals, so in this case classical arguments suffice.) Applying a rotation (or more generally, any linear, symplectic transformation) then gives the form where the covariance matrix is not diagonal. It is then easy to verify that this distribution is normalized and has the correct variances: Z \infty −\infty dx Z \infty −\infty
x2 Z \infty −\infty dx Z \infty −\infty
p2 Z \infty −\infty dx Z \infty −\infty
(4.76) (Gaussian moments) Note that in the last case, the covariance is associated with the symmetrically ordered expectation value, a point we will return to below. 7Wolfgang P. Schleich, Quantum Optics in Phase Space (Wiley, 2001). 8Note the slightly different form from Eq. (2.85), which is for complex variables.
4.3 Wigner Distribution x p For the Gaussian state, we can evaluate the integral Z \infty −\infty dx Z \infty −\infty
Z \infty −\infty dx Z \infty −\infty dp
h −z\alpha S−1
i = 4\pi p
= 4\pi q VxVp −C 2 xp , (4.77) which follows from the fact that the form of W(x, p) is normalized. From Eq. (4.72), we thus find the inequality VxVp −C 2 xp \ge ¯h2 4 , (4.78) (generalized uncertainty relation) which acts as a generalized uncertainty relation for Gaussian states, and is stronger than the usual uncertainty relation VxVp \ge ¯h2/4, since it maintains the equality for pure Gaussian states even if they are rotated in phase space. Again, the equality holds for pure states. From Eq. (4.71), we can then see that Tr \rho2 = ¯h/2 q VxVp −C 2 xp , (4.79) (Gaussian-state purity) and thus that the size of the Gaussian state is simply related to its purity. The Gaussian state is also important in another sense. Hudson’s theorem9 states that the only pure states that do not take on any negative values are Gaussian, at least for systems of one degree of freedom. In this sense the Gaussian pure states are the ‘‘most classical,’’ since they can be given a sensible classical interpretation, at least in terms of a classical probability distribution in phase space. 4.3.4.2 Coherent Superpositions Consider the coherent superposition of two positions \pmx0. To keep things physical, we will consider a superposition of two Gaussian states, each of rms width \sigma:
\sqrt
h
. (4.80) 9R. L. Hudson, ‘‘When is the Wigner Quasi-Probability Density Non-Negative?’’ Reports on Mathematical Physics 6, 249 (1974).
Chapter 4. The Quantum State Putting this into the definition (4.60), we find
i . (4.81) The first two terms in brackets are Gaussian distributions centered at \pmx0, corresponding to each of the first distributions separately. The final term looks like some sort of interference term: it is a Gaussian distribution centered at the origin, but with a sinusoidal modulation in the p-direction with period \pi¯h/x0. The interpretation is that the coherence of the superposition is encoded in this oscillatory structure that lies between the two structures that represent the population. Indeed, had we chosen a different relative phase for the two Gaussian states, we would have simply found a different phase for the sinusoidal modulation. It is also easy to see that for an incoherent superposition of the two Gaussian states (i.e., the density operator being the sum of two Gaussian density operators), the Wigner function is linear in the density operator and thus the oscillatory structure would be missing. x p In the wave function, coherences between components of a superposition are again ‘‘encoded’’ in the phases of the complex numbers. The Wigner function is real-valued, and thus complex phases cannot contain the same information. Instead, coherence between two phase-space regions is encoded as oscillations in the region directly between them. In this sense, negative values of the Wigner distribution are indicators of coherence. We can also see that the oscillatory structure is necessary to recover the proper marginal distributions. The position distribution is given by integrating over p, in which case the oscillatory part vanishes, leaving just the two Gaussian states:
Z \infty −\infty dp W(x, p) = \sqrt 2\pi\sigma2 h
\approx \sqrt 2\pi\sigma2 h
, (4.82) where the last expression follows through when the Gaussians are well resolved, |x0| ≫\sigma, so that the overlap terms are negligible. On the other hand, the coherent superposition leads to interference fringes in
4.3 Wigner Distribution the momentum distribution (as you would expect, for example, for the far-field diffraction pattern of two Gaussian apertures):
Z \infty −\infty
2\sigma \sqrt
(4.83) Thus, the orientation of the modulation is critical: a coherent superposition of two phase-space regions implies an array of ‘‘stripes’’ pointing between the two regions, with the stripes or ridges becoming denser and more numerous as the separation increases. 4.3.4.3 Harmonic Oscillator States For the harmonic oscillator with Hamiltonian
2m + 1 2m\omega2 0x2, (4.84) we will see below that the quantum and classical equations of motion are identical. The classical motion of a single point particle corresponds to a closed, elliptical trajectory in phase space. This is clear from the form of the Hamiltonian, since classical trajectories correspond to surfaces of constant energy. In particular, in rescaled coordinates such that the Hamiltonian has the form
2 + x′2 2 , (4.85) the classical trajectories are circles in phase space, all rotating at the same frequency \omega0. Thus, time evolution in the rescaled harmonic-oscillator phase space is equivalent to rotation at frequency \omega0. Similarly, then, in the proper coordinates, the Wigner function simply rotates in time at frequency \omega0. We can thus infer that in the proper coordinates, the Wigner functions for the harmonic-oscillator eigenstates must be rotationally invariant. Indeed, it can be shown that the nth eigenstate has the form10
\pi¯h e−r2(x,p)/¯hLn 2r2(x, p)/¯h , (harmonic-oscillator eigenstate) (4.86) where
p2 m\omega0 , (4.87) and the Ln(x) are the Laguerre polynomials, given explicitly by
n X j=0 n j (−x)j j! . (4.88) The functions Wn(x, p) are thus called Laguerre–Gaussian functions. Naturally, the marginals of these distributions must reproduce the position and momentum distributions for the harmonic oscillator, given by
2nn! rm\omega0 \pi¯h exp −m\omega0x2 ¯h H 2 n rm\omega0 ¯h x
2nn! r
− p2 m\omega0¯h H 2 n r m\omega0¯h p . (4.89) Here, the Hn(x) are the Hermite polynomials, given explicitly by
x e−x2/2, (4.90) and the wave functions in x and p are the Hermite–Gaussian functions. Clearly, for n > 0, the Wigner functions must take on negative values to reproduce the zeros in the marginal distributions. Below is a plot of the Wigner distribution and the marginals for the n = 1 harmonic-oscillator eigen- state, 10Wolfgang P. Schleich, Quantum Optics in Phase Space (Wiley, 2001).
4.3.5 Weyl Correspondence and Operator Ordering¶
Chapter 4. The Quantum State x p and below is the corresponding plot for the n = 2 harmonic-oscillator eigenstate. x p 4.3.5 Weyl Correspondence and Operator Ordering The Wigner representation casts quantum mechanics in terms of real-valued functions, rather than state vectors and operators. But then, how do we associate functions with operators, for example, if we want to compute expectation values in the Wigner formalism? While simple phase-space functions, such as x and p have obvious associations with the operators ˆx and ˆp, more complicated functions such as x2p2 have ambiguous operator associations due to the ordering problem, where multiple possible operator orderings can correspond to the same classical function. (In this section we explicitly mark operators with hats to distinguish them from real numbers, since this is the whole point, with the exception of the density operator where the notation is already clear.) Indeed, it can be difficult to formulate a prescription for uniquely
4.3 Wigner Distribution associating phase-space functions with operators.11 4.3.5.1 Weyl’s Rule Consider the exponential phase-space function
(4.91) The notation here is intended to suggest that x and p are, in a sense, coordinates with ‘‘conjugate’’ variables \pix and \pip, respectively, because we will be constructing Fourier transforms in this way. We can choose to associate this function with the characteristic operator ˆ
(4.92) (characteristic operator) This association of the exponential function,
(4.93) (Weyl association) is sufficient to uniquely fix the operator association with any phase-space function. Weyl’s prescription12 for the association of an arbitrary phase-space function F(x, p) to an operator proceeds then as follows. The exponential function (4.91) implies the (inverse) Fourier transform relation
2\pi¯h Z \infty −\infty d\pix Z \infty −\infty
(4.94) Converting the exponential factor to the characteristic operator, we find an operator related to the same Fourier-transform function ˜F(\pix, \pip):
2\pi¯h Z \infty −\infty d\pix Z \infty −\infty
(4.95) These relations uniquely define the association F(x, p) −\rightarrow ˆF(ˆx, ˆp). (4.96) For example, we can eliminate the intermediate function in Eq. (4.95),
(2\pi¯h)2 Z \infty −\infty d\pix Z \infty −\infty d\pip Z \infty −\infty dx Z \infty −\infty dp F(x, p) ei[\pix(ˆx−x)+\pip(ˆp−p)]/¯h, (Weyl’s rule) (4.97) to obtain the operator explicitly in terms of the original function. 4.3.5.2 Expectation Values Suppose we now compute the expectation value of Eq. (4.97) with respect to an arbitrary state \rho. Then we find D ˆF(ˆx, ˆp) E = (2\pi¯h)2 Z \infty −\infty d\pix Z \infty −\infty d\pip Z \infty −\infty dx Z \infty −\infty
(4.98) where
D ˆ
E = D
(4.99) (characteristic function) 11John Robert Shewell, ‘‘On the Formation of Quantum-Mechanical Operators,’’ American Journal of Physics 27, 16 (1959) (doi: 10.1119/1.1934740 ). 12H. Weyl, ‘‘Quantenmechanik und Gruppentheorie,’’ Zeitschrift für Physik 46, 1 (1927) (doi: 10.1007/BF02055756); Her- mann Weyl, The Theory of Groups and Quantum Mechanics (Dover, 1950).
Chapter 4. The Quantum State is the characteristic function. Using a special case of the Baker–Campbell–Hausdorff (BCH) ex- pansion,13 which states that
−1 2[A, B]
1 2[A, B] (4.100) if [A, [A, B]] = [B, [A, B]] = 0, we can see that
(4.101) since [x, p] = i¯h. Then we can write the characteristic function as
h
i = Z \infty −\infty
= Z \infty −\infty
(4.102) In the last step, we use the shifting property of the exponential factors:
Z \infty −\infty
= \sqrt 2\pi¯h Z \infty −\infty dp′ e−ip′(x−x0)/¯h|p′\rangle = \sqrt 2\pi¯h Z \infty −\infty dx′ Z \infty −\infty
= 2\pi¯h Z \infty −\infty dx′ Z \infty −\infty dp′ e−ip′(x−x0−x′)/¯h|x′\rangle = Z \infty −\infty
(4.103) Eq. (4.102) has the form of an inverse Fourier transform, which we can happily invert, with the result
2\pi¯h Z \infty −\infty
(4.104) Now applying (2\pi¯h)−1 R d\pip e−i\pipp/¯h to both sides, we find an expression for the Wigner function in terms of the characteristic function:
(2\pi¯h)2 Z \infty −\infty d\pix Z \infty −\infty
(4.105) (alternate definition) That is, the Wigner function is, up to a constant factor, just the Fourier transform of the characteristic func- tion. Essentially, we have just motivated the definition of the Wigner function as the Weyl correspondence of the density operator. Using this expression to simplify Eq. (4.98), we find D ˆF(ˆx, ˆp) E = Z \infty −\infty dx Z \infty −\infty dp F(x, p) W(x, p). (4.106) (operator expectation value) 13R. M. Wilcox, ‘‘Exponential Operators and Parameter Differentiation in Quantum Physics,’’ Journal of Mathematical Physics 8, 962 (1967) (doi: 10.1063/1.1705306).
4.3 Wigner Distribution Thus we find another intuitively appealing result, that the expectation values of operators are given by an overlap integral of the corresponding phase-space function with the Wigner distribution. This relation further cements the analogy of the Wigner distribution with a joint probability density. Note that the expectation value here assumes the particular ordering implied by Weyl’s rule. This turns out to be a symmetrized ordering, which we will explore more carefully below. 4.3.5.3 Weyl Correspondence: Inverse Form In Eq. (4.97), we gave the operator ˆF(ˆx, ˆp) in explicitly in terms of the function F(x, p), but we did not give the inverse relation. We do so here, as a convenient prescription for obtaining the phase-space function from the operator. We start by writing the expectation value as D ˆF(ˆx, ˆp) E = Tr h ˆF(ˆx, ˆp)\rho i = Z \infty −\infty dx Z \infty −\infty dx′ Tr h
i = Z \infty −\infty dx Z \infty −\infty
= Z \infty −\infty dx Z \infty −\infty
(4.107) In the last step, we performed the usual trick of letting x′ −\rightarrow x′ + x and then x −\rightarrow x −x′/2. We can compare this expression with Eq. (4.106), which gives D ˆF(ˆx, ˆp) E = 2\pi¯h Z \infty −\infty dx Z \infty −\infty dp Z \infty −\infty
(4.108) If both relations are to hold for any density operator \rho, then we may identify the two integrands:
2\pi¯h Z \infty −\infty dp F(x, p)e−ipx′/¯h. (4.109) Inverting this relation and letting x′ −\rightarrow −x′, we find
Z \infty −\infty
(4.110) (Weyl correspondence) Thus, we see how an arbitrary operator transforms to a phase-space function, and we see that this transfor- mation also motivates the original form for the Wigner transform (4.59). 4.3.5.4 Weyl Ordering As we noted above, the operator ordering implied by Weyl’s rule is a symmetrized ordering. Again, the association (4.93) gives
(4.111) after applying the BCH formula (4.100). Then noting that
\infty X n=0 n! i
n = \infty X n=0 i ¯h n 1 n! n X k=0 n k (\pixx)k(\pipp)n−k, (4.112) we can match the coefficients of \pi k x \pi n−k p to find the correspondence n! n k xkpn−k −\rightarrow k X m=0 m!(n −k)!(k −m)! ˆx m ˆpn−k ˆx k−m (4.113)
4.3.6 Operator Products and Commutators¶
Chapter 4. The Quantum State after expansion of the operator exponentials. Dividing through by the factorials on the left-hand side, this simplifies to xkpn−k −\rightarrow 1 2k k X m=0 k m ˆxmˆpn−kˆxk−m, (4.114) and letting l = n −k, we find the explicit Weyl ordering xkpl −\rightarrow 1 2k k X m=0 k m ˆxmˆplˆxk−m. (4.115) (Weyl ordering rule) Here we explicitly see the symmetric nature of the ordering. In particular, we have shown that the charac- teristic function gives all the symmetrically ordered moments
(ˆxk ˆpl)W
(4.116) (moment formula) where (ˆxk ˆpl)W denotes the symmetrized Weyl ordering of Eq. (4.115). It is somewhat inconvenient to order any particular function by expanding it, applying the above rule term-by-term, and then resumming it. For general functions, it is easier to use McCoy’s formula,14 which says that if ˆFstd(ˆx, ˆp) is an operator with ‘‘standard’’ ordering, having all ˆx’s written to the left and all ˆp’s written to the right, then the corresponding operator ˆFW(ˆx, ˆp) with Weyl-ordering operator is obtained by
−i¯h \partial 2
ˆFstd(ˆx, ˆp). (4.117) (McCoy’s formula) The orders of the factors must obviously be preserved in the differentiation. 4.3.6 Operator Products and Commutators Consider the operator product ˆA = ˆB ˆC. (4.118) How does the product go over to the Wigner representation? Using the Wigner correspondence in the form (4.110), we find that the operator product implies
¯h 2i \leftarrow − \partial p −\rightarrow \partial x − \leftarrow − \partial x −\rightarrow \partial p C(x, p), (Weyl product correspondence) (4.119) where the arrows on the derivative operators indicate the direction of operation. Mostly, the Wigner function has given us relatively simple and intuitively appealing results. However, as we see now, the complexities we have hidden thus far start to become more obvious when looking at operator products. The proof of this correspondence is left as an exercise (Problem 4.3), but the outline of the derivation is as follows. First, note that Eq. (4.95) gives the operator matrix elements as
2\pi¯h Z \infty −\infty
(4.120) This relation can then be used in the correspondence equation (4.110), and then the derivation carries through in essentially the same way as the others above. 14Neal H. McCoy, ‘‘On the Function in Quantum Mechanics which Corresponds to a Given Function in Classical Mechanics,’’ Proceedings of the National Academy of Sciences (18), 674 (1932). See also John Robert Shewell, ‘‘On the Formation of Quantum-Mechanical Operators,’’ American Journal of Physics 27, 16 (1959) (doi: 10.1119/1.1934740).
4.3.7 Moyal Bracket¶
4.3 Wigner Distribution An alternate form of the operator product is¶
x −¯h
2i\partial x
x + ¯h 2i\partial p, p −¯h 2i\partial x B(x, p). (Weyl product correspondence) (4.121) These expressions follow from an argument similar to the one for the original form (4.119) (Problem 4.3). These expressions for the product then give the following correspondence for the commutator: [ ˆA, ˆB] −\rightarrow 2 i A(x, p) sin ¯h \leftarrow − \partial p −\rightarrow \partial x − \leftarrow − \partial x −\rightarrow \partial p B(x, p). (Weyl commutator correspondence) (4.122) With the alternate forms (4.121), we can also write [ ˆA, ˆB] −\rightarrow A x −¯h
2i\partial x −A x + ¯h 2i\partial p, p −¯h 2i\partial x B(x, p) (Weyl commutator correspondence) (4.123) as an alternate correspondence. While the other properties of the Wigner function make it intuitively appealing, we can see that when operator products are involved, the situation becomes substantially more complicated. 4.3.7 Moyal Bracket Finally, based on what we now know, it is straightforward to obtain the equation of motion for the Wigner function. Recalling the Schrödinger–von Neumann equation (4.8), we note by comparing the definition (4.59) to the Weyl correspondence (4.110), we see that the Weyl correspondence for the density operator reads
(4.124) (Weyl quantum-state correspondence) Using the commutator correspondence (4.122), we find that the Schrödinger–von Neumann equation (4.8) becomes
¯hH(x, p) sin ¯h \leftarrow − \partial p −\rightarrow \partial x − \leftarrow − \partial x −\rightarrow \partial p W(x, p) =: {H, W}M. (4.125) (Moyal bracket) The final abbreviation {A, B}M is the Moyal bracket,15 so named to emphasize the connection to the classical Poisson bracket
(4.126) (Poisson bracket)
the Poisson bracket plus quantum ‘‘correction’’ terms:
\infty X n=1 (−1)n¯h2n
x
p W). (Moyal bracket for particle Hamiltonian) (4.127) This equation is especially suitable for comparing the quantum evolution with the evolution of a classical (‘‘Liouville’’) distribution \rhoL,
(4.128) (Liouville equation) 15J. E. Moyal, ‘‘Quantum Mechanics as a Statistical Theory,’’ Proceedings of the Cambridge Philosophical Society 45, 99 (1949).
4.3.8 Summary: Defining Properties¶
Chapter 4. The Quantum State which is described only by the Poisson bracket. This equation of motion follows from the classical expansion for a general phase-space function, df(x, p, t) dt
\partial x dx
\partial p dp
\partial t
\partial x \partial H
\partial p \partial H
\partial t
\partial t , (4.129)
classical trajectory. Thus we see that the usual analogy between quantum and classical mechanics16 of i¯h[ ˆA, ˆB] −\rightarrow {A, B}P, (4.130) seen by comparing the Liouville equation (4.128) to the Heisenberg equation of motion (4.51), is made explicit via the Moyal bracket.
spondence seems easy in this formulation; however, it must be emphasized that taking the limit ¯h \rightarrow 0 for a quantum system is not trivial and is not, in general, well defined due to the singular nature of the limit. It is immediately clear from the form of the Moyal bracket (4.127) that quantum-classical correspon- dence is particularly simple for linear systems, such as the free particle and the harmonic oscillator, because the quantum-correction terms vanish. This yields identical quantum and classical evolution equations for the harmonic oscillator. This point was recognized early on by Schrödinger, when he constructed the coher- ent states of the harmonic oscillator that mimic the classical oscillating trajectories.17 This is a critically important point, and so I’ll repeat it: for harmonic oscillators, the quantum and classical evolution equations are equivalent. Thus, all quantum effects in the harmonic oscillator are only in the initial condition. It is only in nonlinear potentials that the dynamical evolution generates quantum effects. 4.3.8 Summary: Defining Properties We conclude our discussion of the Wigner distribution with a summary of the properties that define it. It turns out that the following five properties are sufficient to uniquely define the Wigner function:18
ˆW is a Hermitian operator depending on x and p. This implies W(x, p) is real. 2. W(x, p) is normalized and produces the correct marginal probability densities of x and p. 3. The definition of W(x, p) exhibits Galilean invariance: the replacement \psi(x) −\rightarrow \psi(x + x0) implies W(x, p) −\rightarrow W(x + x0, p) for translations, and \psi(x) −\rightarrow eip0x/¯h\psi(x) implies W(x, p) −\rightarrow W(x, p + p0) for boosts. 4. The definition of W(x, p) is invariant under the reflections x −\rightarrow −x and t −\rightarrow −t. Mathematically, this means that \psi(x) −\rightarrow \psi(−x) implies W(x, p) −\rightarrow W(−x, p) for space reflections, and \psi(x) −\rightarrow \psi∗(x) implies W(x, p) −\rightarrow W(x, −p) for time reflections. 5. The equation of motion for W(x, p) is the classical one in the case of the free particle. We have discussed most of these properties already, and the others are easy to see. The extra condition 16See P. A. M. Dirac, The Principles of Quantum Mechanics, 4th revised ed. (Oxford, 1967), § 21, p. 84. 17E. Schrödinger, ‘‘Der stetige Übergang von der Mikro- zur Makromechanik,’’ Naturwissenschaften 14, 664 (1926) (doi: 10.1007/BF01507634). 18M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, ‘‘Distribution Functions in Physics: Fundamentals,’’ Physics Reports 106, 121 (1984) (doi: 10.1016/0370-1573(84)90160-1), and references therein.
4.3.9 Other Representations¶
4.3 Wigner Distribution 6. The squared modulus of the overlap of two states is given by 2\pi¯h times the phase-space overlap integral of the corresponding Wigner functions, as in Eq. (4.70). gives an alternate set of conditions sufficient to fix the form of the Wigner function, obtained by replacing condition 5 with this last condition. 4.3.9 Other Representations Naturally, other choices regarding the desirable properties of quantum phase-space distributions lead to other distributions. We will consider a few distributions, some of which have some utility in quantum optics, but we will consider them only briefly. 4.3.9.1 Husimi or Q Function Consider the following Wigner function for a pure Gaussian state, centered at (0, 0), from Eq. (4.73):
2\pi p VxVp exp − x2 2Vx + p2 2Vp . (4.131) Note that here we have assumed a covariance Cxp = 0 for simplicity, and thus VxVp = ¯h2/4. The Husimi Distribution19 for an arbitrary state \rho with Wigner distribution W(x, p) is then given by the convolution of the Wigner function corresponding to \rho with the Gaussian Wigner function:
W ∗WVxVp (x, p) = Z \infty −\infty dx′ Z \infty −\infty dp′ W(x′, p′) WVx,Vp(x −x′, p −p′). (4.132) (Husimi distribution) From the overlap relation (4.70), we can see that the Husimi distribution has the following interpretation: WH(x′, p′) represents the projection (up to a factor of 2\pi¯h) of \rho into the Gaussian state represented by WVx,Vp(x −x′, p −p′) [i.e., the Gaussian state centered at (x, p)]. Thus, the Husimi distribution represents a projection into an overcomplete basis of displaced Gaussian states. Because of the above projection interpretation, it follows that the Husimi distribution is everywhere positive. It also is a ‘‘smoothed’’ version of the Wigner function, and can be useful for visualizing quantum states in phase space, especially in cases where interferences cause complicated oscillations in the Wigner distribution. The flip side of this is that the Husimi distribution tends to ‘‘hide’’ the quantum nature of certain states; this argument has been used both for and against it for comparing quantum states to classical distributions. The Husimi function also has neither the correct marginals nor a simple equation of motion. In terms of ordering, the Husimi function corresponds to anti-normal ordering: that is, if ˆx and ˆp are decomposed into the creation and annihilation operators for the harmonic oscillator (ˆa\dagger and ˆa, respectively), then the ordering is such that all annihilation operators are to the left of the creation operators. In the case of a harmonic oscillator, and the variances Vx and Vp of the convolution kernel are chosen to match the ground state (i.e., WVx,Vp(x′ −x, p′ −p) is a displaced ground state or a coherent state) the Husimi distribution reduces to the Q function. In this case, the distribution is connected with the operator association20
(4.133) in place of Eq. (4.93) for the Wigner distribution. Here, \alpha is the eigenvalue of ˆa, since it turns out that the coherent states are eigenstates of the annihilation operator. 19K. Husimi, Proceedings of the Physico-Mathematical Society of Japan 22, 264 (1940). 20Hai-Woong Lee, ‘‘Theory and Application of the Quantum Phase-Space Distribution Functions,’’ Physics Reports 259, 147
4.4.1 Merging Hilbert Spaces¶
Chapter 4. The Quantum State 4.3.9.2 P Function The Glauber–Sudarshan P function, on the other hand, corresponds to a normal ordering, where all the annihilation operators are written to the right. The formal association rule is thus
(4.134) This correspondence is sufficient to define the function as well as the operator associations, as in Eqs. (4.94)
thus perform the integration. 4.3.9.3 Standard-Ordered Distribution The standard ordering gives the exponential association
(4.135) which gives rise to the standard-ordered distribution. It turns out that the corresponding distribution can be written as21
2\pi¯h Z \infty −\infty
(4.136) which follows from an argument similar to the one embodied by Eq. (4.102), in analogy with the definition (4.59) of the Wigner distribution. 4.3.9.4 Antistandard-Ordered Distribution Finally, the antistandard ordering corresponds to
(4.137) giving rise to the antistandard-ordered distribution, also known as the the Kirkwood distribution or the Rihaczek distribution. It turns out that this distribution can be written as22
2\pi¯h Z \infty −\infty
(4.138) This distribution is the complex conjugate of the standard-ordered distribution. 4.4 Multiple Degrees of Freedom 4.4.1 Merging Hilbert Spaces Suppose two degrees of freedom are prepared in two quantum states completely independently of each other. This could happen, say, for two particles prepared in separate, distant galaxies. We will refer to the two degrees of freedom as ‘‘particles,’’ even though they could correspond to different degrees of freedom of the same system, such as the spin and center-of-mass position of an atom, or the spin and spatial profile of a photon. Labeling the two particles as A and B, if the individual states of the particles are |\psi\rangle A and |\psi\rangle B, then we can write the composite state as
(4.139) where \otimesdenotes the tensor product (or direct product). Often, this is product is written without an explicit tensor-product symbol:
(4.140) 21Ibid. 22Ibid.
4.4.2 Entanglement¶
4.4 Multiple Degrees of Freedom The particle labels can even be dropped, since the ordering determines which state applies to which particle. We can also see the meaning of the tensor product in component form. Let each separate state be expressed in an orthonormal basis as
X \alpha c(A)
X \beta c(B) \beta
(4.141) Then we can express the composite state as
X \alpha\beta
(4.142) where
\alpha c(B) \beta . (4.143) Note that c\alpha\beta is still understood to be a vector-like object, with a single index. Thus, there is an implicit (bijective) mapping of the ordered index pair (\alpha, \beta) to a single index, which we simply denote as \alpha\beta. Similarly, we can write a density operator for two independent particles by the same tensor product:
(4.144) We can also write this in component form for the density matrices as
µ\nu , (4.145) where again \alphaµ and \beta\nu are to be taken as composite indices. The same tensor-product notation applies to Hilbert spaces. That is, we can write
(4.146)
4.4.2 Entanglement The above composite states, described by tensor products of separated states, are called separable states. However, not all states are separable, and those that are not separable are called entangled. In some sense, entanglement is the ‘‘most quantum’’ of all quantum effects. Thus, we can see that a composite state |\psi\rangle is entangled if and only if it cannot be written in the separable form
(4.147) The definition for density operators is somewhat more general: a density operator for a composite system is separable if and only if it can be written in the form
X \alpha
\alpha
\alpha . (4.148) Unfortunately, given an arbitrary mixed density operator, it is difficult to tell if it corresponds to an entangled state (in fact, this turns out to be an NP-hard problem). The point is that two entangled systems do not have local states that can be treated independently. This is in conflict with the apparently reasonable assumption of local realism, which states that distant systems should have independent, observer-independent realities (in particular, they should not directly influence each other). Herein lies the importance of the famous Bell-type inequalities and the experimental verification that quantum mechanics violates the inequalities: local realism contradicts quantum mechanics. Specifically, quantum mechanics is inherently nonlocal because of entanglement.
4.4.3 Peres-Horodecki Criterion¶
Chapter 4. The Quantum State 4.4.2.1 Cloning With the language of entanglement, it is relatively simple to demonstrate the no-cloning theorem,23 which says that the state of a single quantum system cannot be copied to another particle. This turns out to be a simple consequence of unitary evolution. Let’s examine just a simple case. Suppose that cloning is possible on a two-state system from particle A to particle B. Particle B must be in a particular state to begin with, and without loss of generality we may take this to be the ‘‘0’’ state. Then to copy the eigenstates of A, we see that there must be a unitary transformation U such that
(4.149) However, if particle A is in the superposition state
\sqrt
, (4.150) then we see that the cloning operator gives
\sqrt
, (4.151) which is an entangled state (a Bell state). However, what we wanted for cloning to work properly is the separable state
. (4.152) We can see that the problem in this particular example is that U acts nonlocally, and thus induces entangle- ment between the two particles. In fact, the controlled-NOT (CNOT) gate is a quantum operation that effects the transformations in Eqs. (4.149) (if A and B are in eigenstates, the CNOT flips the state of system B if and only if system A is in state 1). Of course, it is possible to clone a state if you already know everything about it (i.e., you have classical knowledge of the state), or if you have an infinite ensemble of copies. (Copying a state is possible to within some fidelity tolerance for a finite ensemble of copies.) In this case, enough measurements may be made to reconstruct the state of the system arbitrarily well, and of course this procedure does not correspond to a unitary transformation. The problem with the single system is that in general, a measurement of the system destroys its state, and a single measurement is not enough to determine the state of the system. Of course, there is no problem with the cloning of the basis states, as in Eqs. (4.149); the problem is in cloning general states that are not orthogonal to the basis states. In particular this means that with a bit of extra information beyond what is contained in the quantum state (e.g., the state of particle A is either |0\rangle A or |1\rangle A, but not any coherent superposition of the two), cloning may in fact be possible. 4.4.3 Peres–Horodecki Criterion Given an arbitrary density operator, how do we tell if it corresponds to an entangled or separable state? As we mentioned above, this is a very difficult problem in general. Nevertheless, we can briefly discuss one important criterion for separability. The Peres–Horodecki or positive partial transpose (PPT) criterion24 starts with the following observation. Suppose that we have a separable, composite state for two systems of the form
(4.153) 23W. K. Wootters and W. H. Zurek, ‘‘A single quantum cannot be cloned,’’ Nature 299, 802 (1982); D. Dieks, ‘‘Communication by EPR devices,’’ Physics Letters A 92, 271 (1982). 24Asher Peres, ‘‘Separability Criterion for Density Matrices,’’ Physical Review Letters 77, 1413 (1996). Michal Horodecki, Pawel Horodecki, and Ryszard Horodecki, ‘‘Separability of Mixed States: Necessary and Sufficient Conditions,’’ Physics Letters A 223, 1 (1996).
4.4 Multiple Degrees of Freedom Now suppose that we take the transpose of one subsystem, say B, to obtain a new ‘‘density operator:’’
\rho(B)T . (4.154) Note here that for the transposition operation, \rho(B)T ≡ \rho(B)∗. The question now is, is ˜\rho still a valid density operator? In general, a Hermitian operator \rho is a valid density operator if it has unit trace and is positive semidefinite; that is, if all its eigenvalues are nonnegative (equivalently, every diagonal matrix element of the density operator is nonnegative in every basis). In this sense, it represents a sensible quantum- mechanical probability distribution. This property often goes by the name of rho-positivity. The transpose operation does not affect the eigenvalue spectrum, so the transpose of \rho(B) is still a valid density operator, and it follows that ˜\rho is also a valid density operator. Clearly, this argument also holds for the more general separable state (4.148). Thus, we have established a necessary condition for the composite density operator to be separable. In component form, we can again write the density operator as
µ\nu , (4.155) in which case the PPT is given by interchanging µ and \nu:
\nuµ . (4.156) Thus, for an arbitrary density operator, the PPT test consists of computing the PPT of the operator, and then testing ˜\rho for positivity. The positivity of the transposed density operator,
(4.157) (positive partial transpose criterion) is a necessary (and sufficient in some cases, as we will discuss below) criterion for separability. Fine, so let’s take it for a test drive. Consider the entangled Schrödinger-cat state
\sqrt
. (4.158) The density operator is \rho = 1
, (4.159) so that the PPT operation on subsystem B gives
. (4.160) With the index ordering (00, 01, 10, 11), this density operator corresponds to the density matrix
, (4.161) which has eigenvalues 1/2 and −1/2 with multiplicities 3 and 1, respectively. (We can also see that there is a problem since it is easy to see that the determinant is −1/24, and so at least one eigenvalue must be negative.) Hence this is not a proper density matrix, and the state is not separable according to the PPT criterion. Horodecki3 have shown that the PPT criterion is also sufficient for density operators in H2 \otimesH2 and H2 \otimesH3. That is sufficiency holds if the subsystems are both qubits (two-state quantum systems) or there is one qubit and one qutrit (three-state quantum system). For larger Hilbert spaces, the PPT criterion breaks down as a sufficient condition, but remains a useful tool. More general and powerful criteria than the PPT can be devised,25 though of course the general problem of distinguishing separable from entangled states is still computationally inefficient. 25Andrew C. Doherty, Pablo A. Parrilo, Federico M. Spedalieri, ‘‘Distinguishing separable and entangled states,’’ Physical Review Letters 88, 187904 (2002).
Chapter 4. The Quantum State 4.4.3.1 Wigner Representation The PPT condition has an interesting interpretation in the Wigner representation. As we noted above, the transpose of the state is merely the complex conjugation operation. But as we discussed for the Wigner function, complex conjugation corresponds to time-reversal, and thus under the transformation \psi(x) −\rightarrow \psi∗(x), the Wigner function undergoes the corresponding transformation W(x, p) −\rightarrow W(x, −p). Now consider the Wigner function generalized to N degrees of freedom:
(2\pi¯h)N Z \infty −\infty dx′
Z \infty −\infty dx′ N e−ip\betax′
(4.162) Thus, in the Wigner representation the PPT criterion reads as follows. For the bipartite Wigner function W(x1, x2, p1, p2), a necessary condition for separability is that
(4.163) (positive partial transpose) be a valid Wigner function. The inversion of one momentum here looks like a physical transformation—it is for a system of one degree of freedom—but for a bipartite system it is not, because it maps some physical states to unphysical ones. This test is not necessarily easier than the original one for the density operator. However, Simon26 showed that this form of the Peres–Horodecki criterion is also sufficient for Gaussian bipartite states. We can see from the definition (4.162) of the Wigner function that the area theorem generalizes to Z dNx Z
−1
(4.164) (area theorem) We can write the Gaussian Wigner function as
(2\pi)Np
exp −1 2z\alpha S−1
, (4.165) (Gaussian state) where we are using the generalized coordinate ordering
(4.166) and the covariance matrix is thus
(4.167)
anticommutator bracket. Thus, the area theorem again implies the generalized ‘‘uncertainty relation’’ for Gaussian states
¯h2 N , (4.168) (area theorem) where again equality holds only for pure states. We can use this as one necessary criterion for the validity of a Gaussian Wigner function. However, the PPT operation corresponds to a sign change in the covariance matrix of the form + + + + + + + + + + + + + + + + −\rightarrow + + + − + + + − + + + − − − − + , (4.169) and thus we can see that det(S\alpha\beta) is actually invariant under PPT. Thus, the condition (4.168) is not strong enough to see why something goes wrong with the Wigner function under PPT. 26R. Simon, ‘‘Peres–Horodecki Separability Criterion for Continuous Variable Systems,’’ Physical Review Letters 84, 2726 (2000) (doi: 10.1103/PhysRevA.84.2726).
4.4 Multiple Degrees of Freedom 4.4.3.2 Generalized Uncertainty Relation To express a better uncertainty relation, we first note that the commutators for the canonical coordinates can be written in the compact form
(4.170) (commutation relation) where Ω\alpha\beta is the canonical cosymplectic two-form, defined by
0n In −In 0n , (4.171) (canonical cosymplectic form) with In denoting the n \times n identity matrix and 0n the n \times n null matrix. [Note that the cosymplectic
Hamiltonian mechanics. In particular, the classical Poisson bracket reads
=\Rightarrow
\partial g
, (4.172) and the classical Hamilton equations are (Section 8.2.2.2)
=\Rightarrow
\partial H
(4.173) Essentially, the cosymplectic form mixes positions and momenta, but only if they belong to the same degree of freedom. It also treats them on equal footing, but an exchange of the positions with the momenta is accompanied by a minus sign. The point of all this is that we can use the cosymplectic form to write down a generalized uncertainty relation:27
(4.174) (generalized uncertainty relation)
eigenvalues). So what does that mean? Well, we can try this out for one degree of freedom, in which case we have Vx
Cxp −i¯h/2 Vp \ge 0. (4.175) Recalling positive semidefiniteness (positivity) is also required of the density operator, we note that this means that all the eigenvalues of the above matrix must be nonnegative. Diagonalizing the above matrix, we find that Vx + Vp \pm q (Vx −Vp)2 + 4C 2 xp + ¯h2 \ge 0. (4.176) Only the case of the negative sign is nontrivial, and this simplifies to VxVp −C 2 xp \ge ¯h2 4 , (4.177) which is precisely the uncertainty condition that we saw before in Eq. (4.72) for the Gaussian state [i.e., the one-dimensional version of Eq. (4.168)]. However, we’re trying to establish this as a general criterion, not restricted to Gaussian states. To establish this result in the case of a single degree of freedom, note that this result (4.177) reduced
27R. Simon, N. Mukunda, and Biswadeb Dutta, ‘‘Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,’’ Physical Review A 49 1567 (1994) (doi: 10.1103/PhysRevA.49.1567).
Chapter 4. The Quantum State matrix may be obtained from a diagonal one by a suitable initial choice of diagonal variance matrix and a coordinate rotation in phase space. A coordinate rotation preserves areas in phase space, so in a straightfor- ward way, this establishes the result (4.177) for an arbitrary covariance matrix for a one-degree-of-freedom system. The coordinate rotation here is a special case of a symplectic transformation, which we will now examine more closely. To handle the case of multiple degrees of freedom, we need to be more sophisticated about our co- ordinate transformations. In particular, let’s make a time-independent coordinate transformation from z\alpha
of motion as
\partial H
= \partial ˜zµ
\partial H
, (4.178) where we used Hamilton’s equations in the old coordinate system. Thus we can define
, (4.179) which is the cosymplectic form for the new coordinate system. (Recall that Ω\alpha\beta is the cosymplectic form in canonical coordinates.) Hamilton’s equations in the new coordinates can thus be written
\partial H
= {˜zµ, H}˜ P, (4.180) where the new Poisson bracket is {f, g}˜
\partial ˜zµ ˜Ωµ\nu \partial g
. (4.181) Now we can define a canonical transformation, which is a coordinate transformation that leaves the cosymplectic form unchanged. In this case, the new coordinates ˜z\alpha are canonical. Thus, the transformation from z\alpha to ˜z\alpha is canonical if and only if
⇐\Rightarrow
. (4.182) (canonical transformation) Again, this is because ˜Ω\alpha\beta must have the special form for the new coordinates to be canonical. In particular,
these properties, and thus time evolution represents a canonical transformation. Consider now a linear, canonical coordinate change, or possibly a nonlinear canonical coordinate change only in the neighborhood of a particular point z\alpha, so that we may linearize the transformation. Then we may represent the transformation by a matrix,
(4.183) where Aµ\nu is independent of the coordinates and ˜z\alpha = A\alpha\betaz\beta. Being a canonical transformation, the linear mapping Aµ\nu satisfies
(4.184) or in other words A preserves the canonical cosymplectic form under a similarity transformation: AAT = Ω. (4.185) (symplectic matrix) Any matrix that satisfies this condition is said to be a symplectic matrix.28 28The set of all 2n \times 2n real symplectic matrices forms a group, called the symplectic group, denoted Sp(2n, R). Hermann Weyl coined this term; he originally wanted to call this the ‘‘complex group,’’ but wanted to avoid confusion with complex numbers. So he chose the term ‘‘symplectic’’ as a (rough) Greek translation of the word ‘‘complex.’’ See Hermann Weyl, The Classical Groups: Their Invariants and Representations (Princeton, 1939), p. 165.
4.4 Multiple Degrees of Freedom How does the covariance matrix transform? In general, we can expand the new coordinates as
z\alpha=0
(4.186) In new coordinates,
\partial zµ
zµ=0
z\nu=0 + O(z3). (4.187) Thus, we see that the covariance matrix only goes simply over to a covariance matrix under a linear trans- formation, in which case the coordinate transformation reduces to a symplectic matrix A\alpha\beta, and we can write
(4.188) Thus, we can see that A\alphaµ
2 Ωµ\nu
2 Ωµ\nu, (4.189) and thus the uncertainty relation (4.174) is invariant under linear canonical transformations,
(4.190) as we expect. We have used the fact that the statement M \ge 0 is equivalent to the statement ˜M \ge 0 if M and ˜M are related by an invertible transformation (you should try proving this, it isn’t difficult). This is certainly true here of the symplectic matrices, since we generally assume nonsingular coordinate transformations. Okay, now let’s return to the uncertainty relation for N degrees of freedom. We’ll start again by considering only a diagonal covariance matrix. Then the matrix
(4.191) only has elements along three diagonals: the cosymplectic form again couples positions to momenta, but only if they belong to the same degree of freedom. Thus, this matrix is easy to diagonalize: the matrix essentially decomposes into a set of N 2 \times 2 blocks of the form Vx\alpha i¯h/2 −i¯h/2 Vp\alpha , (4.192) just as we had in the case of one degree of freedom. Then, by the same argument, we can see that the uncertainty relation in this case reduces to the set of uncertainty relations
4 , (4.193) which we know to be true. For the general case, we then rely on the fact that any covariance matrix can be reduced to diagonal form by a linear, symplectic transformation (i.e., via a symplectic matrix). This seems reasonable, since as long as we are doing an effective linearization by only considering the covariance matrix, there should exist a linear, canonical transformation that takes any covariance matrix to any other (with the same ‘‘volume’’ or purity). However, this result can be formalized in Williamson’s theorem29, which guarantees that we can always use a symplectic transformation to obtain the diagonal form. We showed that the uncertainty relation is invariant under linear, canonical transformations, and thus the uncertainty relation (4.174) holds in general. 29Simon et al., ibid.
Chapter 4. The Quantum State 4.4.3.3 Sufficiency for Gaussian States Now we can get back to the original question, what goes wrong with the Wigner function under the partial transpose operation for entangled states? Well, we need to do a little more work with the uncertainty relation (4.174) to see what it means. Let M\alpha\beta be positive semidefinite. This means that c∗
(4.194) for any complex vector c\alpha, and note that we may take M\alpha\beta to be Hermitian. Now let
(4.195) where a\alpha and b\alpha are real vectors. Then the positivity condition reads
(4.196) Since M\alpha\beta is Hermitian,
(4.197) Then with
(4.198) the positivity condition is
(4.199)
uncertainty relation becomes
(4.200) The left-hand side is invariant under the exchange a\alpha \leftarrow \rightarrow b\alpha, but the right-hand side changes sign, giving a second condition. Only one of these two conditions is nontrivial, and so it is appropriate to rewrite the uncertainty condition as
(4.201) to emphasize the nontrivial condition. In the case of two degrees of freedom,
−1 −1 b1 b2 b3 b4 = b3 b4 −b1 −b2 , (4.202) and so
(4.203) Thus, the uncertainty condition becomes
(4.204) Recalling that the partial transpose corresponds to p2 −\rightarrow −p2, we see that the partial transpose is induced
z\alpha −\rightarrow z′
(4.205) Note that \Lambda\alpha\beta does not correspond to a unitary transformation, nor does it represent a canonical transfor- mation, as we can see from the transformation of the cosymplectic form:
J −J , J = 1 −1 . (4.206)
4.4 Multiple Degrees of Freedom This clearly does not have the same form as the original cosymplectic form. Again, the Peres–Horodecki criterion says that under partial transposition, the resulting Wigner function for a separable state must still be a valid state. Thus, we have the new uncertainty relation that must be satisfied for separable states: (S′
(4.207)
2 (Ω′
(4.208) Repeating the above derivation, we see that Ω′
−1 −1 b1 b2 b3 b4 = b3 −b4 −b1 b2 , (4.209) and thus a\alphaΩ′
(4.210) This is the same as the uncertainty relation as before save for a minus sign. Noting that w \ge |u + v| and
uncertainty relation for separable states reads
(4.211) We can see that this is a stronger condition than the uncertainty relation (4.204). Thus we see that separable states obey a stricter uncertainty law than generic states. Now let’s consider an example to try to get a feeling for this. Consider the case
x0 ,
p0 −1 , (4.212) where x0 and p0 are length and momentum scales, respectively, that we introduce to make the units come out right. Then
x0p0 ,
x0p0 , (4.213) and the terms on the left-hand side of the uncertainty relation are
x 2
x 2
(x1 + x2)2
p 2
p 2
(p1 −p2)2 . (4.214) In this case, we can see that the uncertainty relation x 2
(x1 + x2)2 + 1 p 2
(p1 −p2)2 \ge 0, (4.215) which always trivially holds. The separability condition, on the other hand, gives x 2
(x1 + x2)2 + 1 p 2
(p1 −p2)2 \ge 2¯h x0p0 . (4.216)
4.4.4 Indistinguishability¶
4.4 Multiple Degrees of Freedom 4.4.4 Indistinguishability Bosonic and fermionic quantum particles that are identical are furthermore indistinguishable, even in prin- ciple. This induces a structure that looks a lot like entanglement, but isn’t really the same.31 Consider the two-particle state
(4.222) That is, particle A is in state |\psi1\rangle , and particle B is in state |\psi2\rangle . This state is appropriate for distinguishable particles. But if the two particles are indistinguishable, the state must be invariant under exchange of the particle labels,
(4.223) which for this state would imply that |\psi1\rangle = |\psi2\rangle . It is hardly satisfactory for every indistinguishable particle to be in the same state, so we can introduce an explicit symmetrization (antisymmetrization) as follows:
\sqrt
, (4.224) In the case of the minus sign, the particle exchange is accompanied by a factor (−1), amounting to just an overall phase, which is certainly permissible. What happens when we try to superpose these states? We can see that \sqrt
(4.225) and so we end up in a state that’s no good. Evidently, we can’t superpose states corresponding to different symmetrizations. Thus, we must postulate that once a pair of particles obey a certain symmetrization rule, they must always do so. (This is equivalent to the statement that the operator corresponding to the exchange operation commutes with the Hamiltonian.) Of course, the particles corresponding to the + sign in (4.224) are bosons, and those corresponding to the −sign are fermions. Again, Eq. (4.224) is suggestive of an entangled state, but only in a trivial sense, since it is nonsensical to speak of separate identities for the two particles. It is completely, fundamentally impossible to tell them apart. Another way to see this that even with an unsymmetrized state vector, we can always impose the symmetrization via the representation: \sqrt \langle (x1)A (x2)B| \pm \langle (x2)A (x1)B|
\sqrt
(4.226) Here, x1 and x2 represent two different position coordinates, it is the ordering of the arguments that deter- mines which particle is associated with which position. 4.4.4.1 Exchange “Force” One consequence of indistinguishability is an interference effect that looks something like an effective force between indistinguishable particles. This effect, the exchange force, is particularly important in understand- ing atomic and molecular structure, condensed matter systems, quantum degenerate gases, and astrophysics (where ‘‘degeneracy pressure’’ prevents white dwarfs and neutron stars from collapsing). Consider two particles A and B in respective states |\psi1\rangle and |\psi2\rangle , which we assume to be orthonormal. For distinguishable particles, the composite state is
(4.227) The joint spatial probability density is
(distinguishable), (4.228) 31The presentation here follows parts of David J. Griffiths, Introduction to Quantum Mechanics (Prentice–Hall, 1995), Chapter 5, p. 177; Asher Peres, Quantum Theory: Concepts and Methods (Kluwer, 1995), Section 5-4, p. 126; and lecture notes by M. Baranger and J. Negele. See also Daniel F. Styer, ‘‘Common misconceptions regarding quantum mechanics,’’ American Journal of Physics 64, 31 (1996).
Chapter 4. The Quantum State
we can compute the probability of finding one at x and the other at x′,
(distinguishable). (4.229) This expression facilitates comparison with the indistinguishable case. Now consider indistinguishable particles, either bosons or fermions
\sqrt
, (4.230) In this case the joint spatial probability density becomes
= 1
\pm 2Re[\psi∗ 1(xA)\psi∗
(bosons/fermions). (4.231) Again, we must drop the particle labels, so
\pm 2Re[\psi∗ 1(x)\psi∗
(bosons/fermions). (4.232) The final interference term is the exchange term. Note that it is nonvanishing only if the two wave functions \psi1(x) and \psi2(x) overlap. To see the effect of the exchange term, consider the probability density when x = x′. For distinguishable particles, the probability density is simply
(4.233) (distinguishable particles) while in the indistinguishable case,
= (
(bosons) (fermions) (indistinguishable particles) (4.234) Thus, we see that the probability density for being at the same position doubles with respect to the dis- tinguishable case for bosons, and vanishes for fermions. Thus, it is common to speak of the attractive ‘‘exchange force,’’ or exchange interaction between bosons, and the repulsive ‘‘force’’ between fermions. This effect is not a force, however; it is simply an interference effect due to the symmetry properties under exchange. In particular, if two noninteracting particles ‘‘collide’’ and then separate, there is no scattering or net phase shift after the crossing of the particles due to the exchange interaction. However, these would occur if the exchange interaction really could be modeled as a force (instead, the exchange interaction affects the details of the particles spatial distributions only while they are overlapping).32 32W. J. Mullin and G. Blaylock, ‘‘Quantum statistics: Is there an effective fermion repulsion or boson attraction?’’ American Journal of Physics 71, 1223 (2003) (doi: 10.1119/1.1590658).
4.4.5 Open Systems: Church of the Larger Hilbert Space¶
4.4 Multiple Degrees of Freedom 4.4.5 Open Systems: Church of the Larger Hilbert Space One important use of the density operator is in describing open quantum systems—systems interacting with auxiliary systems (environments or reservoirs) that we don’t have access to. We will treat open quantum systems in great detail, but for now let’s examine a simple model for why the density operator is useful. Consider the entangled state
\sqrt
(4.235) between particles (qubits) A and B. Suppose that we have access to particle A, but particle B is locked up in a box, so that we don’t know anything about it. The density operator for the composite system is
(4.236) We can define the reduced density operator that describes only particle A by performing a partial trace over the state of particle B:
X \alpha
. (reduced density operator) (4.237) Thus, we can see that the reduced state of particle A corresponds to a completely incoherent superposition of the two states, even though the composite system carried a completely coherent superposition. This is a simple model for the process of decoherence. A quantum system can start in a local state of coherent superposition. But if it interacts with the environment, the coupling causes entanglement between the system and environment, since the interaction is nonlocal. Because we don’t have access to the state of the environment, we must trace over it, which reduces the purity of the reduced density operator. Note that we can’t keep track of the environment even in principle, since it generally has many degrees of freedom. As the interaction continues, the entanglement progresses, driving the reduced density operator towards a completely incoherent superposition. This is, at a simple level, why classical (macroscopic) things behave classically: coupling to the environment destroys quantum coherence. Conversely, whenever we have a system described by a mixed density operator,
X \alpha
(4.238) we can always think of it as part of a larger system. We can see this as follows. We will introduce a fictitious environment with orthonormal basis states |\alphaE\rangle . Then we can write the state vector for the composite system as
X \alpha p
(4.239) (purification of mixed state) When we compute the total density operator for the composite pure state and trace over the environment, we recover the original density operator (4.238) as the reduced density operator of the larger state. This procedure of switching to a larger pure state is referred to as purification or ‘‘the doctrine of the Church of the larger Hilbert space.’’33 The space of these extra environmental degrees of freedom is often referred to as the ancilla.34 Often, this is a useful picture for thinking about mixed quantum states, especially in quantum-information problems. 33Terminology introduced by John Smolin; see Daniel Gottesman and Hoi-Kwong Lo, ‘‘From Quantum Cheating to Quantum Security,’’ Physics Today 53, no. 11, 22 (2000) (doi: 10.1063/1.1333282). 34Although it just doesn’t sound right to my ear, ‘‘ancilla’’ is a singular noun that comes from latin, where it means ‘‘hand- maiden,’’ but in contemporary English can mean something that helps to achieve something difficult. (The plural form can be ‘‘ancillae’’ or ‘‘ancillas.’’)
4.5.1 Interaction Representation¶
Chapter 4. The Quantum State 4.5 Master Equation A main and obvious advantage of the density-operator formalism is that it provides a method for handling nonunitary evolution of the quantum state. This generally occurs in the treatment of open quantum systems: quantum systems coupled to external systems that we do not directly track.35 We saw the simplest example of this in the previous section: maximal entanglement of two qubits enforces minimum purity in the reduced state of one qubit. Now we will take this idea and study more generally how weak entanglement with an external system leads to a general, nonunitary evolution equation for the density operator. We will thus study the evolution of a quantum system, described by Hamiltonian HS, interacting with a ‘‘reservoir’’ (or ‘‘heat bath’’ or ‘‘environment’’), described by Hamiltonian HR. We will assume the system–reservoir interaction, described by HSR, to be weak, causing slow evolution on the uncoupled time scales of the system and reservoir separately. The evolution of the total system is unitary, given by
¯h[H, \rhoSR], (4.240) where \rhoSR is the combined state of the system and reservoir, and the total Hamiltonian is
(4.241) Our goal is to derive an equation of motion for the state of the system alone, given by a partial trace over the reservoir degrees of freedom:
(4.242) Note that so long as we are interested in operators that act solely on the system’s Hilbert space, this reduced density operator is sufficient to compute any appropriate expectation values. We will derive the master equation with a number of approximations and idealizations, mostly related to the reservoir having many degrees of freedom. The approximations here typically work extremely well in quantum optics, though not necessarily in other areas such as condensed-matter physics where, for example, the weak-coupling idealization may break down. Examples of reservoirs include the quantum electromagnetic field (in a vacuum or thermal state), or the internal degrees of freedom of a composite object. 4.5.1 Interaction Representation The first step is to switch to the interaction representation, in effect hiding the fast dynamics of the uncoupled system and reservoir, and focusing on the slow dynamics induced by HSR. We do this as in Section 4.2.3 via the transformations
(4.243) so that the formerly time-independent interaction becomes explicitly time-dependent. The equation of motion then becomes
¯h[ ˜HSR(t), ˜\rhoSR(t)]. (4.244) Integrating this from t to t + ∆t,
¯h Z t+∆t t dt′ [ ˜HSR(t′), ˜\rhoSR(t′)]. (4.245) Iterating this equation by using it as an expression for ˜\rhoSR(t′),
¯h Z t+∆t t dt′ [ ˜HSR(t′), ˜\rhoSR(t)] −1 ¯h2 Z t+∆t t dt′ Z t′ t dt′′ [ ˜HSR(t′), [ ˜HSR(t′′), ˜\rhoSR(t′′)]]. (4.246) 35For further reading, see William H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973), Chapter 6; Claude Cohen–Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 1992), Chapter IV; and Howard Carmichael, An Open Systems Approach to Quantum Optics (Springer, 1993), Chapter 1.
4.5.2 Born-Markov Approximation¶
4.5 Master Equation Now in taking the trace over the reservoir, we will assume that the first term on the right-hand side vanishes. More specifically, we assume
(4.247) This follows by assuming that the total system–reservoir state always approximately factorizes
(4.248) where ˜\rhoR is the stationary state of the reservoir. This amounts to assuming that the reservoir is large and complex, and weak coupling of the system to the reservoir, so that the perturbation to the reservoir by the system is small. In this case, the time interval ∆t ≫\tauc, where \tauc is the correlation time of the reservoir—the time for reservoir and system–reservoir correlations to decay away. This also amounts to a coarse-graining approximation, which means that we are smoothing out any fast dynamics on time scales of the order of \tauc or shorter. Thus, any correlations that have arisen in past time intervals have decayed away. Of course, new correlations arise due to the coupling in the present time interval, which will give rise to nonunitary terms in the evolution equation for the reduced state. Then the assumption (4.247) amounts to
(4.249) This assumption means essentially that there is no dc component to the system–reservoir coupling—that is, the system–reservoir coupling consists of fluctuations about a zero mean. This can always be arranged by absorbing any nonzero mean into the system Hamiltonian. 4.5.2 Born–Markov Approximation Since the first term vanishes under the partial trace, with the trace Eq. (4.246) becomes
¯h2 Z t+∆t t dt′ Z t′ t dt′′ TrR[ ˜HSR(t′), [ ˜HSR(t′′), ˜\rhoSR(t′′)]], (4.250)
(4.251) In fact there is a pair of approximations at work here. The Born approximation amounts to assuming the factorization in (4.248), which we have justified in terms of a large, complex reservoir with a short coherence time. The Markov approximation amounts to setting \rho(t′′) to \rho(t) in (4.251), which will result in an evolution equation that only depends on \rho(t), and not the past history of the density operator. We can justify this approximation by noting that ∆t is small and HSR induces a weak perturbation, so that
Then this amounts to a lowest-order expansion in ∆t of the right-hand side of Eq. (4.250), which is appropriate in view of the limit ∆t −\rightarrow 0 to obtain a differential equation (though in
Next we change integration variables by setting
(4.252) so that the integration becomes Z t+∆t t dt′ Z t′ t dt′′ = Z ∆t d\tau Z t+∆t
dt′ \approx Z \infty d\tau Z t+∆t t dt′. (4.253) In writing down the final, approximate form for the integrals, we have used the fact that the integrand involves an expectation value of the interaction Hamiltonian taken at times that differ by \tau, as we will explore further shortly. That is, the integrand involves reservoir correlation functions, which decay away on the time scale \tauc.
4.5.3 Interaction¶
Chapter 4. The Quantum State 4.5.3 Interaction Now we make a reasonably general assumption regarding the interaction Hamiltonian; namely, that it can be written as a sum of products over system and reservoir operators:
(4.254) (Recall that repeated indices imply summation.) The interpretation here is that if S\alpha is a Hermitian operator, then it represents an observable that is being effectively (or actually) monitored via coupling to the environment. For example, a position measurement is represented by an interaction of the form HSR = xR (Chapter 19). Alternately, the operators need not be Hermitian. For example, an interaction of the form
would thus represent dissipation or loss of energy to the reservoir. Such interactions occur in spontaneous emission (Chapter 11) and cavity decay (Chapter 12). With the interaction of the form (4.254) and the change of integration in Eqs. (4.253), the change (4.250) in the quantum state becomes
Z \infty d\tau Z t+∆t t dt′ h
i
- h
i
, (4.255) where we have defined the reservoir correlation functions
h
i = D
E R = D
E R , (4.256) which depend only on the time difference because the reservoir is in a stationary state. Now we make the further assumption
(4.257) about the interaction-picture system operators. This is not necessarily a restrictive assumption, since multiple frequencies for a given system operator may be separated in the sum in (4.254). Then Eq. (4.255) becomes
Z \infty d\tau Z t+∆t t dt′ h
i
- h
i
(4.258) Now defining
Z t+∆t t
w+
Z \infty
w−
Z \infty
(4.259) we can write
h
i w+
h
i w− \beta\alpha
(4.260) Under the assumption of fast (uncoupled) system and reservoir dynamics,
(4.261)
Kronecker delta,
(4.262)
4.5 Master Equation Now formally taking the limit of small ∆t,¶
∆t
h
i w+
h
i w− \beta\alpha , (4.263) where again we must keep in mind that this differential equation is coarse-grained in the sense of not representing dynamics on time scales as short as \tauc or (\omega\alpha + \omega\beta)−1 for different frequencies. Now transforming
h
i w+
h
i w− \beta\alpha . (4.264) Now we use the fact that HSR is Hermitian, so terms of the form SR in (4.254) that are not Hermitian must be accompanied by their adjoint terms S\daggerR\dagger. Clearly, terms where S\alpha = S\dagger
can explicitly combine these pairs of terms to write the master equation in terms of only a single sum:
X \alpha h
i w+
h
i w− \alpha . (4.265) Of course, terms of the same form carry through when S\alpha is Hermitian. In the expression above we have also defined the reduced integrals w+
Z \infty
˜R\dagger
E R w−
Z \infty
˜R\dagger
E
\alpha ]∗. (4.266)
be missing here. However, if we end up with terms like S1\rhoS\dagger 2, this can always be absorbed into terms of the form (S1 + S2)\rho(S1 + S2)\dagger, representing interferences in the couplings represented by S1,2. The cross terms are weighted by a cross-correlation function between R1 and R2, representing the cross terms of the coherence. In the absence of cross coherence, only terms of the form S1\rhoS\dagger
2 should appear. Weighted combinations of these terms with (S1 + S2)\rho(S1 + S2)\dagger terms can account for any degree of coherence. (See Section 6.2.4.1 for a discussion of interference contributions of this form in the context of quantum beats in three-level atoms.) Now separating out the real and imaginary parts of the integrals (4.266) in (4.265),
X \alpha 2Re[w+ \alpha ]
\alpha −1 h S\dagger
i −i X \alpha Im[w+ \alpha ] h S\dagger
i . (4.267) Note that the last term has the form of Hamiltonian evolution, while the second term does not; these represent energy shifts and dissipation/diffusion effects, respectively, due to the interaction with the reservoir. Now separating out the real and imaginary parts of the integrals, we have the final result
X \alpha
(Born–Markov master equation) (4.268) where the effective Hamiltonian for the reservoir interaction, leading to a ‘‘generalized Lamb shift,’’ is Heff := ¯h X \alpha Im[w+
(effective Hamiltonian for generalized Lamb shift) (4.269)
Chapter 4. The Quantum State and we have defined the Lindblad superoperator
. (4.270) (Lindblad superoperator) with coefficient
\alpha ]. (4.271) (dissipation/diffusion coupling coefficient) We have thus arrived at the general Lindblad form of the master equation in the Born–Markov approxi- mation, which we return to and justify in the context of measurement in Section 19.1. Again, the system operators S\alpha represent the coupling channel of the system to the reservoir, and thus the channel by which the system may be observed. Thus, for example, if S\alpha −\rightarrow x, then we have the master equation for a position measurement, whereas if S\alpha −\rightarrow a, where a is the annihilation operator for the harmonic oscillator, then we have the master equation for energy loss (and thus damping) of a quantum harmonic oscillator.
4.6 Exercises¶
4.6 Exercises 4.6 Exercises Problem 4.1 (a) Using the expression for the Wigner function in terms of a pure state, argue that the Wigner function corresponds to an overlap integral of two wave functions, and thus derive the upper bound
\pi¯h (4.272) for the magnitude. (b) Derive a similar upper bound for the Husimi distribution WH(x, p). Problem 4.2 For a harmonic oscillator of frequency \omega and mass m, the density operator for a thermal state of temperature T is given by the Boltzmann-type sum
X n=0
(4.273) Carry out the appropriate summation over the Wigner functions for the harmonic-oscillator eigenstates to show that the thermal state is Gaussian.36 What are the variances of the thermal state? Show that your variance expressions are sensible in the limits of low and high temperature. It may help to know that the summation formula \infty X j=0 e−jx (n + j)! j!
(4.274) valid for nonnegative n, is not difficult to prove by induction. (Translation: if you use it, you should prove it by induction.) Problem 4.3 Derive the Weyl correspondences
¯h 2i \leftarrow − \partial p −\rightarrow \partial x − \leftarrow − \partial x −\rightarrow \partial p C(x, p) (4.275) and
x −¯h
2i\partial x
x + ¯h 2i\partial p, p −¯h 2i\partial x B(x, p) (4.276) for the operator product ˆA = ˆB ˆC. Problem 4.4 Let A be a symplectic matrix, so that A satisfies AΩAT = Ω.
(b) Show that AT is symplectic. Problem 4.5 Let A be a symplectic matrix. (a) Show that if \lambda is an eigenvalue of A, then 1/\lambda is also an eigenvalue of A. (b) What is the determinant of A? (c) Give a physical interpretation of the eigenvalues and the determinant of A. 36Note that in the Church of the Larger Hilbert Space, the thermal state corresponds to a two-mode (Gaussian) squeezed state—that is, when you trace over one of the modes, you can choose the variances such that you recover the thermal state for the remaining mode.
Chapter 4. The Quantum State Problem 4.6 Show that under Hamiltonian evolution,
¯h[H, \rho], (4.277) the purity Tr[\rho2] is a constant of the motion. Problem 4.7 (a) Show that for a single particle in the sinusoidal (pendulum) potential
(4.278) the equation of motion for the Wigner function may be written
¯h sin(kx) W x, p + ¯hk −W x, p −¯hk . (4.279) (b) We can make this equation of motion look ‘‘more classical’’ by defining the Wigner effective potential by writing the Liouville-type equation
(4.280) Write down an expression for \partial xVeff, and then show that for the minimum-uncertainty Gaussian state with variance Vx and no covariance,
\pi¯h exp
2Vx
¯h2 , (4.281) the effective potential can be written
−k2Vx sinh 2kVx ¯h
2kVx ¯h
. (4.282) (c) Argue that the last factor in the above expression is negligible for a localized wave packet, and then show that in this limit, the above effective potential is approximately equal to what we will call the Ehrenfest effective potential V (E) eff
(4.283) where it turns out the last equality holds for the Gaussian state in a cosine potential. Problem 4.8 The master equation for a damped harmonic oscillator, coupled to a reservoir in the vacuum state in Lindblad form is
(4.284) where \kappa is the rate of energy decay of the oscillator, and H is the harmonic-oscillator Hamiltonian. Consider the master equation for a damped, anharmonic oscillator. A reasonable guess might be
where V (x) is an anharmonic potential, while keeping the damping terms the same (and assuming that a is still defined as an appropriate linear combination of x and p). Explain why the Born–Markov master-equation formalism does not lead to this master equation, and discuss the most general form for the master equation of a damped, anharmonic oscillator. (Hint: what is the spectrum of a harmonic oscillator? An anharmonic oscillator? What does a for a harmonic oscillator look like in the Heisenberg picture? What would it look like if generalized to an anharmonic oscillator? Make sure to keep track of the assumptions in the Born–Markov derivation.)