Language of Quantum Mechanics System s Hilbert space. Pure and mixed states A quantum-mechanical system 1 is characterized by a complete inner-product space that is, Hilbert space with either finite or countably infinite basis. This space is termed system s Hilbert space. In the system s Hilbert space, each vector ψ (referred to as a vector of state, aka pure state, aka wave function) can be represented as the following finite or countably infinite sum ψ = n c n e n, c n = e n ψ, (1) where { e n } is a certain orthonormal basis (ONB): e n1 e n2 = δ n1,n 2. (2) In the most general case, the state of the system is described by so-called density matrix, ˆρ, a Hermitian operator with non-negative eigenvalues and unity trace. Tr ˆρ e n ˆρ e n ρ nn = 1. (3) n n A special class of states, the pure states, are associated with the (normalized to unity) vectors of states. By definition, the density matrix ˆρ ψ of a pure state ψ has the form ˆρ ψ = ψ ψ, ψ ψ = 1 (pure state). (4) The states that are not pure states are called mixed states. 2 Measurement axiom Speaking generally, the state of the system cannot be directly observed. Rather, ˆρ describes a two-fold probabilistic outcome of observation (aka measurement), as a result of which an observer (i) finds a certain distinct value 1 Without loss of generality, the system in question can be either an independent isolated system, or a subsystem of a larger system. 2 For a reason that will become clear later. 1
of the quantity being observed/measured and (ii) finds the new state of the system. It is a fundamental feature of quantum mechanics that, speaking generally, an observation changes the state of the system. 3 The measurement axiom of the quantum mechanics states that each measurement is described by corresponding Hermitian operator. If the measurement deals with the operator A (we say: A is being measured ), then possible outcomes of the measurement correspond to a set {λ} of (real) eigenvalues of A, while the probability p λ of finding a particular value λ and the new state ˆρ (new) are defined by λ, ˆρ and the projector ˆP λ of the λ-eigenspace of the operator A: p λ = Tr ˆρ ˆP λ, (5) ˆρ (new) = ˆP λ ˆρ ˆP λ Tr ˆP λ ˆρ ˆP λ = ˆP λ ˆρ ˆP λ p λ. (6) The denominator in (6) is merely a normalization constant necessary to obey (3). At this point we need to explain/remind what we mean by projector ˆP λ. If the eigenvalue λ is not degenerate then ˆP λ = e λ e λ, where e λ is corresponding unit eigenvector of the operator A: A e λ = λ e λ, e λ e λ = 1. In a general case, ˆP λ = e λ,ν e λ,ν, (7) ν where { e λ,ν } is any does not matter which particular one ONB in the λ-eigenspace of the operator A. The following three properties of the projectors associated with certain Hermitian operator A are especially important: A = λ λ ˆP λ, (8) where ˆ1 is the identity operator. λ ˆP λ ˆPλ = δ λ,λ ˆPλ, (9) ˆP λ = ˆ1 (completeness relation), (10) 3 Note a dramatic difference between the quantum-mechanical and classical-mechanical notions of state and observation/measurement. 2
A particularly simple and important case of measurement corresponds to observation of a nondegenerate eigenvalue λ. Here ˆρ (new) is necessarily a pure state exhaustively defined by observed λ: ˆρ (new) = ˆP λ = e λ e λ. (11) Additionally, if the state ˆρ before measurement is pure, ˆρ = ψ ψ, then the expression for the probability (5) to observe λ simplifies to p λ = e λ ψ 2. (12) Equation (11) is very important. It shows how a pure state can be prepared by observing a non degenerate eigenvalue in the process of measurement, no matter what is the state being measured. Finally, given probabilistic character of the measurement, it is often useful to address its statistical properties: the expectation value Ā A λ λp λ and variance ( A) 2 λ(λ Ā)2 p λ. With (5) and (8), we find Ā = Tr Aˆρ, ( A) 2 = Tr (A Ā)2 ˆρ. (13) For a pure state (4), equations (13) become Ā = ψ A ψ, ( A) 2 = ψ (A Ā)2 ψ. (14) Problem 1. Derive equations (8) through (14). Consistently with the measurement axiom, any mixed state ˆρ can be interpreted as a statistical mixture (appropriate random choice) and that is why the term mixed of special set of pure states, namely a random choice with probability w j of one of the pure states { ψ j }, corresponding to the eigenvectors of the density matrix: The interpretation is based on the representation ˆρ ψ j = w j ψ j. (15) ˆρ = j w j ψ j ψ j, ψ j1 ψ j2 = δ j1 j 2, (16) 3
along with w j 0, j w j = 1, allowing to interpret w j as probabilities. Problem 2. Using representation (16) and the measuring postulate, derive the above-mentioned interpretation of generic density matrix as a statistical mixture of corresponding pure states. Problem 3. The Hilbert space of the system X is three-dimensional. The current state of the system is described by the density matrix 3 ˆρ = w j ψ j ψ j, ψ j1 ψ j2 = δ j1 j 2. (17) j=1 The matrix of the observable A in the ONB of eigenvectors of A has the form 2015 0 0 A = 0 2015 0. (18) 0 0 2003 In the same ONB (of eigenvectors of A), the vectors ψ j have the following representation ψ 1 = 1/ 2 1/ 2 0, ψ 2 = 1/ 2 1/ 2 0, ψ 3 = 0 0 1. (19) (a) What is the probability to find the eigenvalue 2015 when measuring A in the state ˆρ? (b) What will be the new ( collapsed ) density matrix after the eigenvalue 2015 is observed? The interplay between destructive and probabilistic aspects of measurement can be quite nontrivial. The following example (Problem 4) shows how a sequence of measurements with an essentially deterministic outcome can be used to perform a dramatic, but absolutely predictable change of a pure 4
state of a system. Problem 4. Let spin-1/2 system be prepared in spin-up state with respect to some axis z. Suppose one performs successive measurements of the system, with each new measurement rotating the axis z with respect to which the spin projection is being measured in a certain fixed plane by a small angle θ/n, where θ 1 is fixed and N is supposed to be very large compared to unity. Prove that in each of the successive N measurements, the probability to find the spin-up state approaches unity as N. That is, given sequence of measurements rotates deterministically the spin-up state by the angle θ. Evolution The dynamic postulate of quantum mechanics states that the time evolution of the density matrix of an isolated system is prescribed by a certain Hermitian operator H (called system s Hamiltonian and acting in the system s Hilbert space) in accordance with the following equation of motion: i h ˆρ = H ˆρ ˆρH [H, ˆρ]. (20) t Equation (20) is consistent with the requirement (3). Moreover, it preserves all the eigenvalues (but not the eigenvectors!) of ˆρ. Both statements are immediately implied by the following result [cf. Eq. (16)] : ˆρ(t) = j w j ψ j (t) ψ j (t), ψ j1 (t) ψ j2 (t) = δ j1 j 2, (21) where ψ j (t) evolves in accordance with the Schrödingier equation for a pure state, i h t ψ j(t) = H ψ j (t). (22) Problem 5. Derive the result (21) (22) from (20). With equations (21) (22) we see that the interpretation of ˆρ as a statistical mixture of pure states applies not only to an instant state, but to the whole 5
process of evolution of ˆρ. Each of the eigenstates ψ j (t) evolves individually in accordance with Schrödingier s equation (22), the probabilities w j remaining intact. The relationship with the Schrödingier equation allows us to invoke the evolution operator ψ j (t) = U(t) ψ j (0), (23) in terms of which Eq. (21) implies ˆρ(t) = U(t)ˆρ(0)[U(t)]. (24) Along with the representation (21) for ˆρ in terms of its eigenvectors, yet another very important representation in the case of time-independent Hamiltonian is in terms of the Hamiltonian (i.e., energy) eigenstates { n }, H n = E n n. (25) Starting from (23) and taking into account that for a time-independent Hamiltonian, we have U(t) = e iht/ h, (26) for the matrix ρ nm (t) = n ˆρ(t) m we readily get ρ nm (t) = ρ nm (0)e it(em En)/ h. (27) This formula is especially important for discussion of the relaxation (an apparent relaxation, to be more accurate) in an isolated system. If, after a long enough time of system s evolution, we completely loose control 4 on the value of the phases t(e m E n )/ h modulo 2π, then, for all practical purposes we should replace our actual density matrix with the one averaged over the values of the phases t(e m E n )/ h. This amounts to vanishing the off-diagonal terms: ρ nm (t) δ nm ρ nn (0), (28) meaning that our state is indistinguishable from a statistical mixture of energy eigenstates, the state n entering the ensemble with the probability ρ nn (0). Composition postulate 4 Due to natural limitations on experimental resolution. 6
The composition postulate deals with the structure of the Hilbert space of a composite system consisting of two or more subsystem. The postulate is naturally formulated for the case of two subsystem, and then straightforwardly generalized (by induction) to the case of many subsystems. The composition postulate states that the Hilbert space of a system consisting of two subsystems is a direct product of corresponding two Hilbert spaces. This, in particular, means that, if { e (I) n } and { e (II) n } are ONBs in the Hilbert spaces of the system I and II, respectively, then the set of all the vectors e nm = e (I) n e (II) m, e n1 m 1 e n2 m 2 = δ n1,n 2 δ m1,m 2 (29) forms the basis in the Hilbert space of the composite system, so that a vector state in this space can be represented as the following linear combination ψ = nm c nm e nm nm c nm e (I) n e (II) m. (30) Depending on the structure of the Hamiltonian, the two subsystems either interact with each other or not. The two subsystems do not interact with each other if and only if the Hamiltonian of the composed system reduces to a direct sum of two independent Hamiltonians, each dealing with corresponding subsystem: H = H (I) + H (II) (noninteracting subsystems). (31) A distinctively different from the issue of interaction is the notion entanglement between the two subsystems. The notion of entanglement applies to the density matrix 5 of the composed system. The two subsystems are called disentangled (with respect to each other) if the density matrix ˆρ of their composed state reduces to a direct product of individual density matrices: ˆρ = ˆρ (I) ˆρ (II) (disentangled subsystems). (32) If two subsystems are noninteracting and disentangled in the initial state, then the equation of motion (20) readily implies that the two subsystems remain disentangled, with individual density matrices evolving in accordance 5 Including the wave function the pure state as a particular case of the density matrix. 7
with individual Hamiltonians. Problem 6. Make sure that this is the case. Interaction leads to entanglement, by which, qualitatively speaking, one means such a structure of ˆρ that cannot be reduced to (32). As opposed to interaction, which can vanish at a certain point of evolution, the entanglement, if created, will persist. Physically, the entanglement implies certain correlations between results of measurements dealing with the observables of both systems. If we perform a measurement dealing with the observables of only one of the subsystems (subsystem I, for definiteness, then from the measurement postulate, it follows that the outcome of this measurement is exhaustively described by the density matrix of the subsystem, related to the total density matrix by tracing out the variables of the other subsystem (note that the result of doing the trace is still an operator, not number): ˆρ (I) = Tr (II) ˆρ m e (II) m ˆρ e (II) m. (33) In the matrix form, ρ (I) n 1 n 2 = m ρ n1 m; n 2 m, ρ n1 m 1 ; n 2 m 2 = e n1 m 1 ˆρ e n2 m 2. (34) Problem 7. Derive the above-mentioned fact from the measurement axiom. In particular, make sure that the new ˆρ (I) following by applying the measuring axiom to ˆρ and only then using (33) is identical to the result of formally applying the measurement axiom directly to ˆρ (I) (despite of the fact of entanglement between the systems, a priori preventing one from applying the measurement axion to the subsystem I). Measuring an observable of the system I often (but not necessarily always!) leads to disentanglement of originally entangled systems. A simple and very important sufficient condition for disentanglement upon measurement is that the observed eigenvalue is nondegenerate. In this case, the state of the systems disentangles into a product of corresponding pure state ψ (I) of the system I and a certain new state (not necessarily pure!) of the system II: ˆρ new = ψ (I) ψ (I) ˆρ (II) new. (35) 8
Problem 8. Check the above statement. Thermal equilibrium, Gibbs distribution In classical mechanics, the statistical description and the full dynamical description naturally deal with distinctively different (though related to each other) mathematical objects. The dynamical description the canonical formalism operates with canonical dynamical variables (i.e., generalized coordinates and momenta), while the statistical description is all about the distribution functions for the canonical variables. In quantum mechanics thanks to its probabilistic (i.e., quasi-statistical) nature, on one hand, and linearity, on the other hand the statistical description operates with the very same object the density matrix 6 as the dynamic theory. The only (minor) difference is that, in the dynamic theory, one normally addresses the density matrix (or the wave function, in the case of a pure state) of the full system, while quantum statistics is naturally constructed in terms of the reduced density matrix (33). In the quantum-statistical context, the reduced density matrix ˆρ (I) (statistical operator of subsystem I) describes either a smaller subsystem of a macroscopic system, or a system (no matter microscopic or macroscopic) weakly coupled to a heat bath. In the case of equilibrium statistics considered below, both cases are essentially equivalent, since a smaller subsystem of a macroscopic system can be viewed as a part of a bigger subsystem, the latter being weakly coupled to a heat bath effectively replacing the rest of the macroscopic system. For example, in the case of ideal Bose or Fermi gases, these subsystems are nothing but single-particle eigenmodes. 7 Our goal is to establish the form of the density matrix of a system in equilibrium with a heat bath. First, let us elaborate on what do we mean by the heat bath. Think of a macroscopic number of subsystems of arbitrary nature weakly coupled to each other, our subsystem of interest being one of them. The condition of infinitesimal weakness of coupling is crucial for the subsystems to preserve their individuality. In quantum-mechanical language, this means that the entanglement between any two subsystems is negligibly 6 Often referred to, in this context, as statistical operator. 7 Recall that, in the case of bosons, a single-particle eigenmode is equivalent to a quantum harmonic oscillator; and, in the case of fermions, a single-particle eigenmode is equivalent to a two-level system. 9
small, so that the density matrix ˆρ AB of any two subsystems A and B (viewed as a untied system AB) splits into a direct product of individual density matrices [cf. (32)]: ˆρ AB = ˆρ A ˆρ B. (36) Apart from the rather simple but quite fundamental relation (36), the only two related circumstances one needs to take into account to fix the form of equilibrium density matrix, say, ˆρ A is that, by definition, the equilibrium matrix does not evolve in time and that, because of the infinitesimally weak interaction with the rest of the system, this condition implies that ˆρ A is a conserved quantity ( constant of motion ) of the system A. From the equation of motion (20) we thus have where H A is the Hamiltonian of the subsystem A. [H A, ˆρ A ] = 0, (37) Problem 9. Based on the equation of motion (20), verify that, for an operator A to be a conserved quantity (in the sense that the expectation value of any function of A stays constant in time), it is necessary and sufficient that A and H (the system s Hamiltonian) commute, provided H is time independent. Now introducing, for convenience, logarithm of the density matrix, and rewriting (36) and (37) as ln ˆρ AB = ln ˆρ A + ln ˆρ B, [H A + H B, ln ˆρ AB ] = 0, (38) we conclude that the logarithm of the equilibrium density matrix is an operator of an additive conserved quantity. In the most general case, there is only one universal additive conserved quantity (up to a global scaling factor and a global constant): the energy, with corresponding operator the system s Hamiltonian being the sum of individual Hamiltonians for all the subsystems. In this case, for any subsystem A of our weakly-coupled macroscopic system, ˆρ A = e βh A Z A, Z A = Tr e βh A, (39) where the value of the parameter β is the same for each subsystem. Here we took into account the requirement Tr ˆρ A allowing us to fix one of the 10
two free constants, namely, the additive constant in the expression ˆρ A = βh A + const. We see that, in the case when the energy is the only additive constant, the equilibrium statistics is exhaustively characterized by the system s Hamiltonian and a global (for all the subsystems, or, equivalently, the heat bath) parameter β. Expression (39) is known as Gibbs distribution for a quantum system. The parameter β, playing the role of a parameter of the Gibbs distribution, can be also replaced with an equivalent parameter called temperature, T, and related to β as T = 1/β, (40) provided T is measured in the same units as energy. In practice but not in theory (!), it is somewhat convenient to measure T in units different from energy ones, in which case one has to introduce the Boltzmann constant: β = 1/(k B T ). Problem 10. There is a three-level system described, in a certain fixed representation, by the Hamiltonian H = ε 2 1 0 1 2 0 0 0 3. (41) The system is in equilibrium with a heat bath at the temperature T = 0.5ε (in energy units). (a) What is the expectation value of the system s energy? (b) What is the probability that a measurement of the system s energy will produce the result 3ε? (c) What is the probability to find the system in the state ψ 1 = 1 0 0? (42) Yet another important and quite general case is when we are dealing with a 11
system of particles, the total number of which is a conserved quantity. 8 Now the most general expression for the logarithm of the density matrix should contain an extra term: the one proportional to yet another additive conserved quantity, the total number of particles. The standard convention is to achieve this goal by using Gibbs distribution (39) with modified Hamiltonian, H A H A: H A = H A µn A, (43) with N A the operator of the total number of particles of the subsystem A, and µ some global parameter (having the dimensions of energy) referred to as chemical potential. Corresponding Gibbs distribution is called grand canonical distribution, and the Gibbs distribution (39) is called canonical distribution. The Hamiltonian H A is often referred to as grand canonical Hamiltonian. Especially simple and transparent is the form of the Gibbs distribution in terms of the eigenstates of energy { n } (25) [for clarity, below we omit the subscript A and do not distinguish between canonical and grand canonical cases]: ˆρ = n w n n n, w n = e βen /Z, Z = n e βen. (44) In accordance with the previously-discussed statistical interpretation of the density matrix (see Problem 2 and the text preceding it), Eq. (44) allows one to interpret quantum equilibrium statistics as that of an ensemble of pure states { n }. Such an ensemble is called the (grand) canonical ensemble. Based on Eq. (44), one can introduce a number of generic thermodynamic quantities and establish fundamental relations between them. The partition function Z is one of such quantities. Another one is the entropy defined as S = Tr ˆρ ln ˆρ = n w n ln w n. (45) Corresponding derivations belong to the course of statistical mechanics, where it is shown that, starting from an expression for Z as a function of T, µ, and 8 In this respect, very instructive is the difference between the quantum statistics of photons in a cavity (or phonons in a solid or liquid) and the statics of ideal gas of bosonic atoms. In the former case, the interaction with the other degrees of freedom of the system (or the heat bath) does not conserve the total number of photons (phonons), and, for each eigenmode, we should use Eq. (39), while in the latter case, the conservation of the total number of particles has a profound effect on statistics, leading, in particular, to the phenomenon of Bose-Einstein condensation. 12
the systems s volume V, one can restore the rest of thermodynamic quantities by doing partial derivatives with respect to T, µ,and V. Einstein Podolsky Rosen paradox Consider the singlet state of two spins 1/2: ψ 0 = 1 2 1 2 (singlet state of two spins 1/2). (46) Problem 11. Show that ψ 0 has the same form, Eq. (46), for any choice of the direction of the quantization axis. To this end, find the two eigenstates â ± of the operator σ â [the operator of the projection of the spin onto the axis of the unit vector â; the eigenvalues are ±1: ( σ â) â ± = ± â ± ] and re-expand ψ 0 in terms of them. Problem 12. Show that the reduced density matrix of each of the two spins obtained by tracing the full density matrix ˆρ = ψ 0 ψ 0 over the variables of the other spin is nothing but the unity operator. The result of Problem 12 means that, a priori, we can predict absolutely nothing about the outcome of an individual measurement of any of the two spins, provided its counterpart is excluded from the measurement. In a shocking contrast to that, we can exactly predict the outcome of measuring the projection of any of the two spins onto any given axis by simply measuring the projection of its counterpart on the very same axis: The two projections will necessarily be found opposite to each other. Problem 13. Verify the above statement by applying the measuring axiom to the state (46). This situation is known as Einstein Podolsky Rosen (EPR) paradox. Before John Bell has proven his famous theorem (see next section), the EPR paradox 9 was interpreted by many as the indication of existence of hidden local 9 Note, however, that the EPR setup does not involve any formal logical contradiction. 13
variables deterministically prescribing the outcome of any individual measurement of each of the two spins. The locality means that measuring one spin has no effect on the state of its counterpart. The assumption of locality is most natural, given that the spatial distance between the two spins can be arbitrarily large. Furthermore, in the absence of locality, hidden variables can hardly be of any non-academic importance. Bell s theorem Let ξ be the set of all relevant local hidden variables, defining the projection A (j) ξ (â) of the j-th spin (j = 1, 2) on the axis of the unit vector â. To comply with quantum mechanics, for two spins 1/2 in the singlet state, we have A (1) ξ (â) = A (2) ξ (â) A ξ (â), A ξ (â) = ±1. (47) Hence, for any fixed value of ξ, the function A ξ (â) prescribes the deterministic outcome of measuring the projection of any of the two entangled spins onto any axis â. Consider the observable Q(â, ˆb) = ( σ 1 â)( σ 2 ˆb). (48) Its quantum-mechanical expectation value in the state (46) is Q(â, ˆb) ψ 0 ( σ 1 â)( σ 2 ˆb) ψ 0 = â ˆb. (49) Problem 14. Prove the last equality in (49). Within the hidden-variable picture, the result of measuring Q(â, ˆb) in the state ξ equals A ξ (â)a ξ (ˆb), and thus Q(â, ˆb) = A ξ (â)a ξ (ˆb) ξ, (50) where (...) ξ stands for the averaging over a representative ensemble of hidden variables. 14
It can be shown (John Bell, 1964) that Eq. (50) is inconsistent with Eq. (49), meaning that quantum mechanics does not allow local hidden variables. The proof of Bell s theorem is very simple. 10 To construct a contradiction, consider Q(â, ˆb) Q(â, ĉ) = Aξ (â)a ξ (ĉ) A ξ (â)a ξ (ˆb) ξ Aξ (â)a ξ (ĉ) A ξ (â)a ξ (ˆb) ξ. (51) Then, taking into account that A ξ 1, observe that A ξ (â)a ξ (ˆb) A ξ (â)a ξ (ĉ) = A ξ (â)a ξ (ˆb) [ 1 A ξ (ˆb)A ξ (ĉ) ], A ξ (â)a ξ (ĉ) A ξ (â)a ξ (ˆb) = 1 A ξ (ˆb)A ξ (ĉ). (52) Plugging (52) into (51) and using (50), we arrive at Bell s inequality Q(â, ˆb) Q(â, ĉ) 1 + Q(ˆb, ĉ). (53) If local hidden variables exist, inequality (53) has to be satisfied for any three unit vectors â, ˆb, and ĉ. Meanwhile, with the explicit form (49), we have â ˆb â ĉ 1 ˆb ĉ, (54) and, choosing â = ( 1 2, 1 2, 0), ˆb = (1, 0, 0), and ĉ = (0, 1, 0), get the desired contradiction: 2 1. Quantum mechanics of measurement: Schrödinger cats The process of measurement is supposed to be consistent with quantum mechanics. 11 It is important thus to understand what type of evolution stands behind the measurement. Let us take a generic pure state of spin- 1/2: ψ = c 1 + c 2, (55) 10 But not at all immediately obvious! 11 As long as we believe which we do in this course that the measuring device and the observer obey quantum mechanics. 15
and some measuring device 12 that is supposed to measure the projection of the spin onto the quantization axis. For simplicity, assume that the initial state of the device, 0, is pure. The initial state of the system spin+device is init. = ψ 0. The final state is supposed have the form: final = c 1 + 1 + c 2 1, (56) where + 1 and 1 are two distinct new states of the measuring device. Distinct and new means that the states are orthogonal to each other and to the initial state of the device. Due to the linearity of quantum mechanics, the coefficients c 1 and c 2 are fixed by the condition that the spin-up and spindown states are the eigenstates of the measurement. Any external observer i.e., the one different from the device who can measure the new state of the spin but does not have an access to the state of the device, has to deal with the reduced density matrix of the spin, ˆρ spin, obtained by tracing the full density matrix ˆρ = final final (57) over the variables of the device: ˆρ spin = c 1 2 + c 2 2. (58) Problem 15. Trace ˆρ over the variables of the device to get (58). In accordance with the statistical interpretation of the density matrix (see Problem 2), Eq. (58) is consistent with the statement that the spin-up (spindown) outcome of the measurement comes with the probability c 1 2 ( c 2 2 ). This is already something, but not the end of the story, because, in reality, we do want to see the reading of the measuring device (same as to talk to the observer who just measure the projection of the spin). We thus need to make sure that for all practical purposes, 13 the pure state (57) is equivalent to the effective mixed state ˆρ eff = c 1 2 ˆρ (dev) +1 + c 2 2 ˆρ (dev) 1, (59) where ˆρ (dev) +1 and ˆρ (dev) 1 are distinctively different (in the sense of orthogonality: ˆρ (dev) +1 ˆρ (dev) 1 = 0) states of the measuring device/observer. 12 Without loss of generality, the word device here means both the device and the observer. 13 The term coined by John Bell. 16
Arriving at (59) from (57) is not possible without an extra requirement distinguishing the measuring device from just a generic (say, three-level) quantum-mechanical system. The requirement (condition) we need is the macroscopicity of the measuring device, in pretty much the same sense that we introduced the notion in the context of quantum statistics. Specifically, we want our measuring device to consist of a macroscopically large number of subsystems (enumerated below with natural numbers), so that 0 1 0 2 0 3 0 4 0 5 0, ±1 1 ±1 2 ±1 3 ±1 4 ±1 5 ±1, (60) j 0 j ±1 = 0, j +1 j 1 = 0 (j = 1, 2, 3, 4, 5...). (61) Now tracing ˆρ with respect to at least one subsystem j which makes perfect sense since, in practice, we never measure all the subsystems of a macroscopic system leads (same as in Problem 15) to (59). The extremely fragile (with respect to tracing out a tiny portion of variables) superposition of macroscopic states (60) is often referred to as Schrödingier cat state. The name comes from the famous gedanken experiment with a cat and a spin-1/2 particle proposed by Schrödingier. The above discussion was based on a two-level system. Generalization to the case of arbitrary system (and mixed initial state) is straightforward. It will automatically follow from dynamical consideration of the next section. Dynamics of quantum measurement The requirement that the measurement leaves intact the eigenstates of the observable A implies a rather simple and essentially universal for all possible measuring devices form of the Hamiltonian describing the evolution of the system and device during the measurement process: H mes = λ ˆP λ H (dev) λ (system + device). (62) Here the projector ˆP λ has precisely the same meaning as in the formulation of the measuring axiom, and H (dev) λ is a certain (rather general) Hermitian operator acting in the Hilbert space of the measuring device. 14 14 In particular, H (dev) λ can be time dependent. 17
Looking at (62), one can raise a question of why the Hamiltonian H mes contains only the terms describing the interaction between the system and device but not the terms describing the self-evolution of the system, H sys, and the terms describing the self-evolution of the device, H dev. As for the Hamiltonian H dev, it is naturally absorbed in (62) by the property (10) of the projectors: H dev ˆP λ λ H dev. The Hamiltonian H sys can be absorbed into (62) only in a rather special case when the observable A is compatible with it (i.e., [A, H sys ] = 0), so that one can represent H sys = λ E λ ˆPλ. In a general case of [A, H sys ] 0, the reason for not including H sys into (62) is rather prosaic. The measurement has to be performed during the time interval short enough to guarantee that self-evolution of the system does not distort the result of measurement. Quantitatively, this means that H sys should be much smaller than H mes to the extent that it can be simply neglected during the time of performing the measurement. The simple generic form (62) of the measuring Hamiltonian implies equally simple and generic form of corresponding evolution operator: U mes = λ ˆP λ U (dev) λ. (63) Here U (dev) λ is the evolution operator (in the Hilbert space of the device) corresponding to the Hamiltonian H (dev) λ. Problem 16. Show that (63) follows from (62). Note that H (dev) λ necessarily time-independent. is not In a standard measurement setup, the system and the devise are initially disentangled, so that initial total density matrix, ˆρ 0, is simply a direct product of corresponding two density matrices: ˆρ 0 = ˆρ (sys) 0 ˆρ (dev) 0 (before measurement). (64) After the measurement, the total density matrix becomes ˆρ = U mes ˆρ 0 U mes = λλ ( ˆPλ ˆρ (sys) 0 ( (dev) ˆP λ ) U λ ˆρ (dev) 0 [U (dev) λ ] ). (65) The requirement of the macroscopicity of the device along with the requirement of macroscopic distinguishability of the states of the device corresponding to different λ s cf. the discussion in the previous section implies that 18
tracing out a tiny amount of the variables of the device results in vanishing the off-diagonal (λ λ) terms. Indeed, splitting measuring device into systems, analogously to Eqs. (60) (60)], we see that tracing out the variables of the j-th subsystem yields: Tr (j) ˆρ = Tr (j) λλ (...) j λ j λ = λλ (...) j λ j λ = λλ (...)δ λλ. (66) This leaves us with the effective density matrix ˆρ eff = λ ( ˆPλ ˆρ (sys) 0 ) (dev) ˆP λ ˆρ λ, ˆρ (dev) λ = U (dev) λ ˆρ (dev) 0 [U (dev) λ ]. (67) Note that ˆρ (dev) λ has a very transparent meaning of the final state of measuring device corresponding to the evolution operator U (dev) λ. Finally, observe that [cf. Eq. (59)] ˆρ eff = λ p λ ˆρ (sys) λ ˆρ (dev) λ. (68) That is, consistently with both aspects of measuring axiom, ˆρ eff has the form of statistical mixture, with the weights p λ, of (corresponding to each other) the new states of the system and measuring device. Needless to say that the new states of the measuring device are distinctively different (orthogonal): ˆρ (dev) λ ˆρ (dev) λ = 0, if λ λ. (69) Schmidt decomposition Suppose we split a system into two subsystems, I and II. With a general choice of ONBs in the Hilbert spaces of the subsystems I and II, an expansion of any pure state ψ of the total system has the generic form (30). There is, however, a very special choice of orthonormal sets of vectors in two subsystems, { φ (I) n } and { φ (II) n } (same subscript is not a typo!) dictated by the particular form of the pure state ψ. With this special choice, one has ψ = n a n φ (I) n φ (II) n. (70) 19
In principle, the coefficients a n can be rendered real and nonnegative, since the phase factor can be always absorbed into the basis vector. In mathematics, the representation (70) is known as Schmidt decomposition. It has quite important physical implications. Namely, for the two reduced density matrices it yields ˆρ (I) = n a n 2 φ (I) n φ (I) n, ˆρ (II) = n a n 2 φ (II) n φ (II) n, (71) revealing an instructive fact that, despite all possible qualitative and quantitative differences between the two subsystems including different dimensions of Hilbert spaces the two reduced density matrices always feature remarkable correspondence between their eigenvectors, with the same eigenvalues given by the squares of absolute values of coefficients a n. The proof of Schmidt decomposition is also very physical in the sense that it utilizes the mathematical structure known in physics as reduced density matrix. Let the subsystem I have the dimension of its Hilbert space lower or equal to that of subsystem II. Given the pure state ψ, construct the reduced density matrix ˆρ (I) = Tr (II) ˆρ with ˆρ = ψ ψ. Let { φ (I) n } be an ONB of the eigenvectors of ˆρ (I). Then, in accordance with (30), we have ψ = n φ (I) n ϕ (II) n, (72) where { ϕ (II) n } are certain states of the system II. Correspondingly, ˆρ = n 1 n 2 φ (I) n 1 ϕ (II) n 1 ϕ (II) n 2 φ (I) n 2, (73) ˆρ (I) = Tr (II) ˆρ = n 1 n 2 φ (I) n 1 ϕ (II) n 2 ϕ (II) n 1 φ (I) n 2. (74) So far, we have used only the fact that { φ (I) n } is a certain ONB in the Hilbert space of the subsystem I. Now we recall that the vectors { φ (I) n } are the eigenvectors of ˆρ (I), so that all the n 1 n 2 term in (74) have to be identically equal to zero: ˆρ (I) = n φ (I) n ϕ (II) n ϕ (II) n φ (I) n. (75) This brings us to ϕ (II) n 2 ϕ (II) n 1 = 0 if n 1 n 2, (76) 20
and thus to (71) with φ (II) n = ϕ (II) n ϕ (II) n ϕ (II) n. (77) For a density matrix ˆρ, the quantity Tr ˆρ ln ˆρ = j w j ln w j is called entropy [recall (45)]. The entanglement between two subsystems in a pure global states is characterized by entanglement entropy S entang = Tr ˆρ (I) ln ˆρ (I) = Tr ˆρ (II) ln ˆρ (II) = n a n 2 ln a n 2. (78) Problem 17. Argue that the minimum possible value of S entang equals zero. What case does it correspond to? Show that the maximal possible value of S entang equals ln N, where N is the smallest dimension of the Hilbert space out of the two subsystems. For a pure state of a macroscopic system and a macroscopically large interface between the two macroscopic subsystems, the entanglement entropy between the two subsystems is proportional to the size of the interface: the length of the interface in 2D and the surface of the interface in 3D. (The so-called area law.) Permanent observation For a small enough time interval t, we have ( h = 1) ˆρ(t + t) = ˆρ(t) i t[h, ˆρ(t)] + O [ ( t) 2]. (79) Now suppose that before the evolution (79) has started, we have measured some quantity A observing certain eigenvalue λ. Then we let the system evolve for a short time t and measure A once again. What will be the probability p λ ( t) of observing the same eigenvalue λ? From (79) and the measuring axiom, one readily concludes that p λ ( t) = 1 O [ ( t) 2]. (80) Problem 18. Derive Eq. (80). Hint. The fact that, at the moment t, the eigenvalue λ has been observed implies (explain why!) ˆρ(t) ˆP λ ˆρ(t) ˆP λ, 21
with ˆP λ the corresponding projector. Now let us fix some finite time interval τ and perform N 1 measurements of the same quantity A, with the time interval τ = τ/n between successive measurements. From (80) we then see 15 that in the limit N, each successive measurement will deterministically yield one and the same eigenvalue λ found in the very first measurement. This fact is known as quantum Zeno effect. 16 The limit N corresponds to permanent observation. In the light of the Zeno effect, a question arises of whether constant observation implies complete suppression of system s evolution. If the eigenvalue λ (found in the first measurement) is nondegenerate, then the obvious answer is yes. The system simply stays in the corresponding pure state 17 ψ λ ψ λ, such that A ψ λ = λ ψ λ. If the eigenvalue λ is degenerate, 18 the system keeps evolving. The law of this permanent-measurement evolution is very intuitive (we restore h): where i h t ˆρ = [H(λ), ˆρ], (81) H (λ) = ˆP λ H ˆP λ (82) is the projected onto the λ-subspace Hamiltonian. Equations (81) (82) apply to the case of nondegenerate λ as well, with H (λ) being just a number, resulting in [H (λ), ˆρ] 0 and thus the suppression of any evolution. Problem 19. Derive the law (81) (82). Hint. Similar to Problem 18, the identity ˆρ(t) ˆP λ ˆρ(t) ˆP λ plays the key role. A rathe instructive projected Hamiltonian arises in the case when the observable A deals only with the variables of a certain subsystem call it subsystem II of a composite system (consisting of subsystems I and II) and 15 Observe close similarity with Problem 4 in terms of taking the limit of N. 16 The name comes by the striking analogy with the arrow paradox by Zeno of Elea (ca. 490 430 BC). 17 Note, however, that the phase of the state even with respect to the phase of the initial state is fundamentally indefinite. 18 Note that the degeneracy is unavoidable in the fundamentally important case of composite system, provided the observable A deals only with the variables of a certain subsystem. 22
the permanently observed eigenvalue λ is nondegenerate within the Hilbert space of subsystem II. The Hamiltonian of interaction between subsystems I and II can be represented as H int = j H (I) j H (II) j, (83) where H (I) j and H (II) j are certain Hermitian operators acting in the Hilbert spaces I and II, respectively. Permanent observation of the above-mentioned eigenvalue λ results in the following projected interaction Hamiltonian H (λ) int = j h j H (I) j, h j = ψ (II) λ H(II) j ψ (II) λ, A ψ(ii) λ = λ ψ(ii). (84) λ Problem 20. Derive (84). The physical meaning of the interaction Hamiltonian (84) is as follows. The subsystem II gets frozen in the state ψ (II) λ. The evolution of the system I is driven by the effective Hamiltonian equal to the sum of the systems own Hamiltonian plus H (λ) int. The numbers h j have the meaning of generalized external classical fields. In the spirit of Problem 4, the fields h j in the effective interaction Hamiltonian (84) can be rendered time-dependent by appropriately changing the observable A in time. Problem 21. For two spins-1/2 interacting via the Hamiltonian σ 1 σ 2, show that permanently measuring the projection of the second spin onto the time-dependent axis ˆn(t) results in the effective time-dependent magnetic field acting on the first spin, provided the dependence of the unit vector ˆn on time is smooth. 23