A Framework for Sequential Measurements and General Jarzynski Equations

2.1 Simple Case

We consider two sequential measurements at the same physical system at times t0<t1 with respective outcome sets ℐ and 𝒥. These sets are assumed to be finite or countably infinite. Hence, the joint outcome of the two measurements can be represented by the pair (i,j)∈ℐ×𝒥. We define

as the set of “elementary events” and describe the probability of elementary events by a function

subject to the natural condition

As usual, one defines the first and second marginal probability functions

and

For the sake of simplicity, we will assume

This could be achieved by deleting all outcomes i∈ℐ with p(i) = 0, thereby reducing the set ℐ. Due to (8), the “conditional probability”

can be defined for all (i,j)∈𝖤. It satisfies

and hence can be considered as a stochastic matrix. Note further that

If additionally π is a “doubly stochastic matrix,” i.e.

the triple (ℐ,𝒥,P) will be called a “statistical model of two sequential measurements” (SM2).

In accordance with the usual nomenclature of probability theory, functions X:𝖤→ℝ are also called “random variables.” Their expectation value is defined as

if the series converges. Using a sloppy notation, the expectation value will be sometimes also written as ⟨X(i,j)⟩ if no misunderstanding is likely to occur. We have the following result:

The q(j) will also be called “hypothetical probabilities” in contrast to the “real probabilities” p^(j). The choice q(j)=p^(j) will be referred to as the “case R.” The above proposition essentially says that the J-equation is equivalent to π(j|i) being doubly stochastic. The proof can be found in Appendix A.

We will call (15) and its modified form (27) the “J-equation” as we think that it contains the probabilistic core of the Jarzynski equation but should be distinguished from the latter for the sake of clarity. To illustrate this claim, we note that any sequence p:ℐ→[0,1] of probabilities satisfying (8) may be written in the form

and, analogously,

where the β,Ei,E′,F,F′ are certain real parameters, not uniquely determined by (16) and (17). Then, (15) can be written as

where w:𝖤→ℝ is a random variable defined by w(i,j)≡E′−Ei and ΔF≡F′−F. Indeed, (18) has the form of the standard Jarzynski equation, but in general, the parameters occurring in (18) will not have the physical meaning of inverse temperature β, work w, and difference of free energies ΔF, as required for the Jarzynski equation. Even in the special case where the usual physical interpretation of the parameters β,Ei,E′,F,F′ holds, we have not yet proven the standard Jarzynski equation, because we still would have to confirm the conditions of Proposition 1 for this special case.

Let (ℐ,𝒥,P) be an SM2 and choose q(j)=p^(j) for all j∈𝒥, that is, replace the hypothetical probabilities by the real ones. Then, by Proposition 1,

As the logarithm (with arbitrary basis) is a concave function, Jensen’s inequality yields

for any random variable X:𝖤→ℝ+. If we define the Shannon entropy [19] as usual by

it follows immediately from (20) that S(p) does not decrease between two sequential measurements:

The proof can be found in Appendix A.

It is an obvious question under which circumstances the inequality in (22) will be a strict one. We will answer this question only for the case of finite ℐ=𝒥:

One may ask which assumption is responsible for the asymmetry between the two sequential measurements that appears in (22). Obviously, this is the property (12) of the conditional probability matrix being doubly stochastic that is postulated only for the first conditional probability π and not for the second one π^(i|j)≡P(i,j)p^(j). It will be instructive to consider the situation in which both matrices, π and π^, are doubly stochastic. For the sake of simplicity, we will assume that the outcome sets ℐ and 𝒥 are finite, both containing exactly n elements, and that P(i,j)>0 for all i∈ℐ and j∈𝒥. It follows that, in matrix notation, πp=p^ and π^p^=p, cf. (11); moreover, the double stochasticity may be written as π1=π^1=1, where 1 denotes the constant vector 1=(1,1,…,1). Hence, the matrix Π≡π^π has the two positive invariant distributions 1n1 and p. As P(i,j)>0, the matrix Π is irreducible, and hence, its positive invariant distribution is unique (Theorem 54 of [20]). Consequently, p(i)=p^(j)=1n for all i∈ℐ and j∈𝒥. This characterises the constant distribution with maximal Shannon entropy and hence a completely symmetric situation.

An equation similar to the J-equation (15) considered above has been proven in [21]. In our notation, it can be formulated as

However, the closer comparison of (23) and (15) shows that these equations are not equivalent, which is also clear from the fact that (23) does not presuppose additional assumptions like the double stochasticity of the conditional probability.

2.2 Modified Case

Now we will formulate a slightly more general framework for SM2 that is motivated by applications using quantum theory in Section 3 and partially follows the account of Wolfgang Pauli in [18], Ch. I §2.

When defining the modified framework for sequential measurements, we will again consider the triple (ℐ,𝒥,P) assumed for the simple case and additionally postulate two nonvanishing functions

In Section 3, the d(i) and D(j) will be interpreted as the degeneracies of certain eigenspaces of measured observables. In Appendix B, we will derive the following assumption (25) characterising the modified case by coarse graining of the outcome sets of the simple case. Here, the d(i) and D(j) play the role of cell sizes of the coarse graining.

The 5-quintuple (ℐ,𝒥,P,d,D) will be called a “modified statistical model of two sequential measurements” (mSM2), if the condition of π being doubly stochastic is replaced by the unprimed version of (B9):

We will generally denote a conditional probability function π:𝖤→[0,1] satisfying (25) as being of “modified doubly stochastic” type.

Consequently, we obtain the following variant of Proposition 1:

The proof is completely analogous to that of Proposition 1. Analogously, it follows that the modified Shannon entropy

does not decrease in the modified statistical model of two sequential measurements, i.e.

This equation is analogous to the statement dSdt≥0 after (22) in [18] that has been proven by Pauli using the stronger symmetry condition (in our notation)

see (21) in [18], justified by first-order perturbation theory (“Fermi’s Golden Rule”). We note that in general (31) need not hold, see Section 4.2 for a counter-example, but there are also positive examples beyond the Golden Rule, see Subsection 4.3.

2.3 Symmetric Formulation

In this subsection, we will give a more symmetric and slightly more formal account of the framework theory for the (modified) statistical model of sequential measurements that will be used later in Subsection 4.3.

The basic concepts will be ℐ,𝒥,Π,p,q. As in Section 2.1, ℐ and 𝒥 will be finite or countably infinite sets and

We first postulate

Next, Π will be a function Π:𝖤→ℝ+ called the “conditional matrix” that has no direct physical meaning. Its values will be written as Π(j|i). It is subject to the following.

As an immediate consequence, the “first conditional probability”

satisfies

and

Hence, π is a doubly stochastic matrix in the modified sense. Define

then

and m=(ℐ,𝒥,P,d,D) will be an mSM2 in the sense of Subsection 2.2.

As the axioms Axiom 1 and Axiom 2 are completely symmetric with respect to the transpositions ℐ↔𝒥 and p↔q, we may analogously define a second model m~=(𝒥,ℐ,P~,D,d) by

and

such that

and

Hence, m~=(𝒥,ℐ,P~,D,d) will also be an mSM2 in the sense of Subsection 2.2 called the “reciprocal model” with respect to m and will satisfy a reciprocal J-equation of the form

Let us reconsider the original J-equation (27) and define the corresponding random variable Y (in a less sloppy way than above) as

We then rewrite the expectation value of Y in the following way. For any real number y∈ℝ, we define

Let 𝖸≡{y∈ℝ|𝖤y≠∅}, then

The sum in the brackets can be interpreted as the probability that Y assumes the value y, or, in symbols:

Next, we repeat the above definitions for the reciprocal model m~=(𝒥,ℐ,P~,D,d) setting

for z∈ℝ and

We note that

and formulate the following “C-equation”:

The proof can be found in Appendix A.

From the C-equation, we may again derive the J-equation (27) in the following way:

3.1 General Case

We will investigate how the (modified) statistical model of sequential measurements outlined in the preceding subsections can be realised within the framework of quantum theory. The identification of the respective concepts will be facilitated by denoting them with the same letters. Additionally to a number of usual assumptions, we will use Assumption 1 and Assumption 2 that are highlighted below.

We consider a quantum system with a Hilbert space ℋ and a finite number of mutually commuting self-adjoint operators E∼ 1,…,E∼ L defined on (suitable domains of) ℋ. They are assumed to have a pure point spectrum and hence a family of common eigenprojections (P∼ i)i∈ℐ such that

Here ℐ is a finite or countable infinite index set to be identified with the outcome set of the first measurement according to Section 2. The P∼ i are assumed to be of finite degeneracy,

and are chosen as maximal projections in the sense that i≠j implies Ei(λ)≠Ej(λ) for at least one λ=1,…,L. Note the completeness relation

Physically, the E∼ 1,…,E∼ L correspond to observables that can be jointly measured. We assume a (mixed) state of the system before the time t=t0 described by a density operator ρ and perform a joint Lüders measurement, cf. [14] (10.22), of E∼ 1,…,E∼ L at the time t=t0. The probability of the outcome i∈ℐ will be

satisfying

In accordance with (8), we will make the following assumption:

The validity of this assumption could be achieved by restricting the Hilbert space ℋ to the subspace spanned by the eigenspaces of those P∼ i with p(i)=Tr(ρP∼ i)>0.

After the first measurement of the E∼ 1,…,E∼ L, the system is subject to a further time evolution and a second measurement of (possibly) other observables. Thus, the primary preparation together with the first measurement may be considered as another preparation of a certain state, in general different from the initial state ρ. If a selection according to a particular outcome i∈ℐ is involved, this state will be, according to the assumption of a Lüders measurement, cf. [14] (10.22),

If no selection according to a particular outcome is involved, the state resulting after the first measurement will rather be the mixed state

In order to apply the results of the preceding section we will make the following crucial assumption:

If P∼ i is a one-dimensional projection, i.e. if d(i)=1, the assumption (63) will be automatically satisfied. In the case of d(i)>1, this assumption means that ρ is diagonal with respect to any common eigenbasis of the E∼ 1,…,E∼ L. An important case where (63) holds is given if ρ is a function of the operators E∼ 1,…,E∼ L, say,

This has to be interpreted in the sense of functional calculus as

where Ei𝝀 will be the short-hand notation for (Ei(1),…,Ei(L)). It follows that, in accordance with (63),

as

In what follows, we will refer to the case (64) as the “standard case” (case S). However, we stress that this is not the most general case compatible with Assumption 2 as the counter-example of all d(i)=1 and ρ not commuting with the P∼ i shows.

Next, we consider a second set of observables described by the mutually commuting self-adjoint operators F∼ 1,…,F∼ L subject to analogous assumptions. Hence, the following holds:

and

We have chosen another index set 𝒥 for the second set of observables in order to stress that no natural identification between both index sets is required in what follows. Obviously, 𝒥 has to be identified with the second outcome set introduced in Section 2. In general, the E∼ λ will not commute with the F∼ μ. We assume that a second measurement of the F∼ 1,…,F∼ L will be performed at the time t=t1>t0, not necessarily of Lüders type. Between the two measurements in the time interval (t0,t1), the evolution of the system can be quite arbitrary and will be described by a unitary evolution operator U=U(t1,t0).

Next, we will show that the suitably defined “physical conditional probability” p(j|i) is of modified doubly stochastic type, and hence, the J-equation (27) also holds in cases of physical relevance. Recall that the state of the system immediately after the first measurement at time t=t0 with outcome i∈ℐ is assumed to be of the form (63) and that the time evolution between t=t0 and t=t1 is given by the unitary evolution operator U. Hence, according to the rules of quantum theory

Moreover,

The proof of this Lemma can be found in Appendix A.

Recall that certain “hypothetical probabilities” q(j) occur in Proposition 3 of Section 2.2. For the quantum case, we will always assume that these probabilities are of the following form:

for all j∈𝒥, where the function 𝒢 is chosen to be the same as in (64). We understand the “standard case S” as including the condition (72).

Lemma 1 and Proposition 3 immediately entail the following theorem, referred to as claiming the general Jarzynski equation, which will be formulated only for the standard case S:

We note in passing that the physical probabilities p(i, j) can be written as

where the positive operators

satisfy

and hence constitute a “positive operator valued measure” (POVM) F:𝖤→ℒ(ℋ). Here, ℒ(ℋ) denotes the space of bounded, linear operators defined on ℋ. (For a more general definition, see [14]; note that we use a simplified form of POVM adapted to countably infinite outcome spaces E.) Hence, the various random variables defined on E, including “work,” can be viewed as generalised observables in the sense of [14], albeit generally not “sharp observables” (i.e. observables described by projection valued measures). This observation puts the statement of [13] “work is not an observable” into perspective, see also [15], [16], [22].

For the examples in the following subsections, it suffices to identify the families E∼ 1,…,E∼ L and F∼ 1,…,F∼ L and the function 𝒢. The conditions of Theorem 1 can be easily verified by the reader; it remains to evaluate the general Jarzynski equation (74) for the various examples. Moreover, as in all examples the convex exponential function is involved, we may invoke Jensen’s inequality and obtain special relations that may be viewed as manifestations of the Second Law for nonequilibrium case S scenarios, but have to be distinguished from case R statement of nondecreasing modified Shannon entropy in Subsection 4.1.

3.2 Systems in Local Canonical Equilibrium

We assume that the quantum system consists of N subsystems and consequently ℋ=⊗μ=1Nℋμ. For each subsystem, we assume a, possibly time-dependent, Hamiltonian Hμ(t), where the lift to the total Hilbert space by means of suitable tensor products with identity operators will be tacitly understood. Its spectral composition will be written as

Then, we set L = N, ℐ=𝒥=ℐ1×…×ℐN, and

These Hamiltonians are not necessarily connected with the unitary time evolution operator U=U(t1,t0). Further, we choose

with the usual interpretation of the parameters β_μ > 0 as the inverse temperatures of the subsystems. The generalised Jarzynski equation (74) then assumes the form

where

for all μ=1,…,N. As exp is convex, Jensen’s inequality yields e⟨x⟩≤⟨ex⟩, and hence (81) implies

or, equivalently,

Note that the left-hand side of (86) has the form of a sum of entropy changes in the quasi-static limit and hence can be viewed as a manifestation of the Second Law for the present nonequilibrium scenario. For similar results, see [3] and [23].

3.3 Systems in Microcanonical Equilibrium

We choose L = 1 and a one-parameter family of Hamiltonians H(t) with spectral decomposition

The microcanonical ensemble will not be represented by a characteristic function concentrated on a small energy interval but in the physically equivalent form

where

and E, w > 0 are parameters. The generalised Jarzynski equation (74) then assumes the form

where Δf≡f(t1)−f(t0). Application of Jensen’s inequality analogous to that in Section 3.2 yields

The generalisation to systems in local microcanonical equilibrium analogous to the case treated in Section 3.2 is straightforward and need not be given here in detail.

3.4 Systems in Grand Canonical Equilibrium

The Hilbert space of the system is chosen as the bosonic or fermionic Fock space over the one-particle Hilbert space ℋ:

where ℋ⊗n denotes the n-fold tensor product and 𝒮_± the projector onto the totally symmetric (+) part or the totally antisymmetric (−) part of ℋ⊗n. We choose L = 2 and E∼ 1=H(t0), where H(t) is the canonical lift of a time-dependent one-particle Hamiltonian H1(t) to ℱ±(ℋ). Further, we choose E∼ 2=∼N, the particle number operator in ℱ±(ℋ). By definition, E∼ 1 and E∼ 2 commute. Let the respective spectral decompositions with a common system of eigenprojections be written as

and

Moreover, we set

where

and β,μ,Ω have the usual physical interpretation as inverse temperature, chemical potential, and grand potential, respectively.

The generalised Jarzynski equation (74) then assumes the form

where ΔΩ≡Ω(t1)−Ω(t0). Application of Jensen’s inequality analogous to that in Section 3.2 yields

The generalisation to systems in local grand canonical equilibrium analogous to the case treated in Section 3.2 is straightforward and need not be given here in detail. For similar results, see also [6], [7], [8], [9], [10], [11].

3.5 Application to PeriodicThermodynamics

Analogously to Section 3.2, we consider two systems (i.e. N = 2) and assume that the first system is periodically driven with a Hamiltonian K1(t) satisfying K1(t+T)=K1(t). We have chosen the letter “K” as we will have to distinguish between the Hamiltonian and the (quasi) energy operator H1(t) and want to conform, as far as possible, with the notation introduced in the preceding sections. According to Floquet theory, the general solution of the corresponding Schrödinger equation will be of the form

with time-independent coefficients a_i. Here, the ε_i denote the quasi-energies, unique up to integer multiples of ω≡2πT, and the ui(t) are T-periodic functions of t. We assume a pure point spectrum of the quasi-energies, and accordingly, ℐ₁ will be a countably infinite or possibly finite index set.

Upon choosing a selection of quasi-energies from their equivalence classes, we may define a quasi-energy operator

Hence, TrP∼i(1)(t)=1 for all t and

The first system is coupled to a heat bath with Hamiltonian

where the P∼n(2) are assumed to be finite-dimensional projectors with dimension (degeneracy) d(n)=TrP∼n(2). Without loss of generality, we also assume a pure point spectrum of the heat bath corresponding to a countably infinite index set 𝒩. The outcome sets introduced in Section 2 can be chosen as ℐ=𝒥=ℐ1×𝒩. Note the completeness relation

The total Hamilton operator of the system plus bath will be written as

with some self-adjoint operator H₁₂ defined on the total Hilbert space ℋ=ℋ1⊗ℋ2 describing the system–bath interaction. It is assumed to be valid for t<t0. Strictly speaking, the form of K(t) is irrelevant for the Jarzynski equation to be formulated below. Its only purpose is to motivate the following assumptions about the state of the total system at the time t=t0.

We assume that for times t<t0, the heat bath will be in a thermal equilibrium state

where, as usual, β is the inverse temperature and the heat bath partition function is

The crucial assumption of this subsection will be that also the system assumes, for times t<t0, a quasi-stationary distribution ρ1(t) of Floquet states that will be of Boltzmann type with an inverse quasi-temperature ϑ, namely

and the corresponding time-independent quasi-partition function reads

With respect to the conditions of Theorem 1, we thus may write the initial state as

Whereas the general existence of a quasi-stationary distribution has been made plausible in the literature [24], the more restrictive assumption of a quasi-Boltzmann distribution has been demonstrated for only four kinds of systems:

1. For the particular case of a linearly forced harmonic oscillator, the authors of [24] have shown that the Floquet-state distribution remains a Boltzmann distribution with the temperature of the heat bath, i.e. ϑ = β, see also [25].

2. Similarly, the parametrically driven harmonic oscillator assumes a quasi-stationary state with a quasi-temperature that is, however, generally different from the bath temperature, see [26].

3. A spin s exposed to both a static magnetic field and an oscillating, circularly polarised magnetic field applied perpendicular to the static one, as in the classic Rabi set-up, and coupled to a thermal bath of harmonic oscillators has been shown to approach a quasi-Boltzmann distribution, see [27],

4. And finally, every quasi-stationary distribution of Floquet states of a two-level system, see [25], can be trivially viewed as a quasi-Boltzmann distribution.

As in Section 3, we will assume that at times t=t0 and t=t1 there will be performed measurements of the observables corresponding to the commuting (quasi) energy operators H1(t) and H₂. The interaction between the system and the heat bath in the time interval (t0,t1) can be quite arbitrary and will be described by a Hamiltonian H~(t). It follows that all mathematical assumptions necessary to prove the general Jarzynski equation (74) are satisfied. But note the following difference: Typically, the general Jarzynski equation holds in a situation of local thermal equilibrium at the initial time t=t0. In this section, we will rather apply Theorem 1 to a situation of a quasi-stationary distribution of Floquet states of a periodically driven system in contact with a heat bath. This situation may be far from local thermal equilibrium.

Analogously to Section 3.2, we will set β1=ϑ, the inverse quasi-temperature, whereas β2=β is the ordinary temperature of the heat bath. Consequently, we will rewrite w₁ as the “change of quasi-energy e” and w₂ as the heat q absorbed by the heat bath. Strictly speaking, this would exclude a time-dependent Hamiltonian for the heat bath as otherwise w₂ could also be composed of both heat and work. Nevertheless, we will stick to this more intuitive notation.

As noted above, both partition functions Z₁ and Z₂ are time-independent. Hence, (74) simplifies to

The inequality derived by means of Jensen’s inequality analogous to (86) hence will read

and can again be viewed as a manifestation of the Second Law for periodic thermodynamics.

4.1 A Second Law–like Statement for the Nonstandard Case

In the preceding sections, we have formulated a number of Second Law–like statements, namely (86), (91), (98), and (111), which follow from the respective Jarzynski equations in the standard case S. However, these statements are not special cases of the “Pauli-type” inequalities (22) and (29) as these are based on the assumption q(j)=p^(j) for all j∈𝒥 (case R) and hence do not belong to the standard case that is characterised by (72). In view of the fundamental significance of the Second Law, it will be in order to add a few remarks on the realisation of (22) and (29) in quantum mechanics.

First, we will reformulate (29) in the context of quantum theory:

This statement is certainly not new but has a couple of forerunners albeit formulated in different frameworks [18], [28], [29]. We note that the modified Shannon entropy S′(p) can be identified with the von Neumann entropy −Tr(ρ1logρ1) of the mixed state ρ₁ after the first measurement according to (62). Indeed,

implies

For the modified Shannon entropy S′(p^), this identification is not possible in general. Even if we additionally assume that the second measurement will be of Lüders type, it is not clear whether an assumption analogous to (63) would hold. Below we will consider a simplified scenario where this identification is nevertheless possible.

Next, we note that the Second Law–like statement (114) holds for closed systems irrespective of their size and is in this respect more general than the usual formulations of the Second Law for large systems including small systems coupled to a heat bath. Moreover, (114) is not restricted to sequential energy measurements and e.g. would also hold for (discretised) position measurements, thereby describing the spreading of wave packets, see the example of the following Subsection 4.2. In this context, it might be instructive to discuss the well-known Umkehreinwand (reversibility paradox) of Loschmidt. There exist solutions ψ(t) of, say, the 1-particle Schrödinger equations that are time reflections of spreading wave packets and hence concentrate on smaller and smaller regions. These solutions do not lead to a violation of (114) as after the first measurement this special solution ψ(0) is transformed into a mixed state ρ₁ that again will spread with increasing time. The delicate phase relations of ψ(0) needed for the inverse spreading are destroyed by the first measurement.