Quantum equilibration in finite time

It has recently been shown that small quantum subsystems generically equilibrate, in the sense that they spend most of the time close to a fixed equilibrium state. This relies on just two assumptions: that the state is spread over many different energies, and that the Hamiltonian has non-degenerate energy gaps. Given the same assumptions, it has also been shown that closed systems equilibrate with respect to realistic measurements. We extend these results in two important ways. First, we prove equilibration over a finite (rather than infinite) time-interval, allowing us to bound the equilibration time. Second, we weaken the non degenerate energy gaps condition, showing that equilibration occurs provided that no energy gap is hugely degenerate.


I. INTRODUCTION
It is remarkable that, over one hundred years since its introduction, quantum mechanics still offers new insights into old problems. Recently, major progress has been made towards demonstrating that quantum mechanics alone can justify the methods of statistical physics [1][2][3][4][5][6][7][8][9][10][11][12], something that had previously required additional postulates. Here, we will examine what quantum mechanics tells us about the tendency of systems to equilibrate.
We will focus on extending recent results concerning equilibration over infinite time [1][2][3][4][5]. These results apply to general systems that meet just two very weak criteria: non-degenerate energy gaps (which rules out non-interacting subsystems), and high effective dimension (which means that the state is spread over many different energies) [18]. Using these criteria, it was shown in [2], that small subsystems of a large quantum system equilibrate, in the sense that they evolve towards some static state and stay close to it for the vast majority of time. In [1], it was shown that expectation values of observables on a closed system also equilibrate.
We will extend these results in two important ways. The first is to prove equilibration over a finite timeinterval. This prevents the system from deviating from equilibrium for arbitrarily long periods of time, and sets an upper bound on the equilibration time. The second is to remove the assumption of non-degenerate energy gaps. Instead, we will prove that equilibration occurs as long as no energy gap is hugely degenerate. This still rules out non-interacting subsystems, but is much more likely to be satisfied by real physical Hamiltonians. Finally, we will discuss systems with a series of temporary equilibrium states, and the importance of considering realistic measurements when deriving the Gibbs state.

II. SETUP
We consider a d-dimensional quantum system in the state ρ(t), evolving unitarily via the time independent Hamiltonian H = n E n P n , where E n are distinct energies, and P n are the projectors onto the eigenspaces of H with energy E n . We denote the number of distinct energies by d E (note that d E ≤ d). [19] The crucial feature of the Hamiltonian will be the energy gaps, rather than the energies themselves. The energy gaps are given by E i − E j with i = j, and so each gap is labelled by an ordered pair of integers representing different energy levels. We will call this set of labels G, For clarity, we will use Greek indices solely to denote energy gaps.
We will be particularly interested in the density of energy gaps. We define N (ε) to be the maximum number of energy gaps G α in any interval of size ε > 0, meaning The maximum degeneracy of any energy gap is given by D G ≡ lim ε→0 + N (ε), and the non-degenerate energy gaps condition corresponds to D G = 1. However, note that we will not assume this. In order to prove equilibration, it is important that the system is large in a relevant sense. The crucial quantity turns out to be the effective dimension of the state, given by This measures the number of distinct energies that contribute significantly to the state, which indicates the amount of state-space explored over time. Note that 1 ≤ d eff ≤ d E , and that d eff = N when the state is spread equally over N distinct energy levels. When the effective dimension is large, the probability of finding the system in any one energy level is small. As argued in [1], it is practically impossible to prepare a system with many particles in a state with a low effective dimension.
Finally, we denote the time average of a quantity over the interval [0, T ] by · T . We also define the infinite time-averaged state ω ≡ ρ(t) ∞ .

III. FINITE TIME EQUILIBRATION OF EXPECTATION VALUES
With a time-independent Hamiltonian, there is no way to evolve to an unchanging state if the system did not start in one. Furthermore, for systems whose Hamiltonians have discrete spectra there are recurrences that come arbitrarily close to the initial state [13,14]. Thus, thinking of equilibration as relaxation to to an unchanging state is not appropriate. Instead, we should say that for a system to be in equilibrium it must spend the vast majority of its time in a state close to some fixed state, where 'closeness' is determined by how well we can distinguish the two states.
There are two senses of equilibration in which we will be particularly interested. The first is the equilibration of small subsystems described in [2]. The second is the equilibration of closed systems with respect to some reasonable constraints on our measurement capabilities, described in [1,3]. However, it was shown in [3] that both these results can be derived from the equilibration of expectation values. In particular, they can be proved from the following result (Theorem 1 in [3], based on a slightly weaker version in [1]), which holds for any operator A given the assumption of non-degenerate energy gaps.
where A is the operator norm of A [20]. In real physical situations, we would typically expect d eff ≫ 1. This means that, for most of the time in [0, ∞), tr[ρ(t)A] will be close to its average tr[ωA] relative to the overall scale set by A .
We will now extend this result to show that equilibration happens in a finite time, and also drop the assumption of non-degenerate gaps. Note that we take = 1 throughout.
Theorem 1. Given a quantum system in the state ρ(t) evolving via a time-independent Hamiltonian with d E distinct energies, then for any operator A on the state space, any energy ε > 0 and time (4) where · T denotes the average over the interval [0, T ] and ω ≡ ρ(t) ∞ . The quantities N (ε) and d eff are defined in (1) and (2) respectively.
Proof. We first consider a pure state |ψ(t) and later extend the result to mixed states by purification.
If the Hamiltonian has degenerate energies, we choose an eigenbasis of H such that |ψ(t) has non-zero overlap with only one eigenstate |n for each distinct energy. The state at time t is then given by with c n ≡ n|ψ(0) . It is clear that |ψ(t) evolves in the subspace spanned by {|n } as if it were acted on by the non-degenerate Hamiltonian H ′ = n E n |n n|. In this case the equilibrium state is ω = n |c n | 2 |n n|, and the effective dimension is It is helpful to rewrite this expression in terms of energy gaps, by taking α = (k, l) and β = (i, j). We also define the vector and the Hermitian matrix Adopting a similar approach to [3], equation (6) becomes The last two inequalities follow from the Cauchy-Schwartz inequality for operators with the scalar product tr(A † B), and the fact that for positive operators P, Q, tr(P Q) ≤ P tr(Q). Note that if the Hamiltonian has no degenerate energy gaps, and we consider the infinitetime limit T → ∞, then M becomes the identity matrix and hence M = 1. In this limit we recover the previous result given by (3).
We now consider the general case in which T is finite, and we make no assumptions about the energy gaps in the Hamiltonian. As M is Hermitian, standard matrix norm bounds give [21] M ≤ max The matrix elements of M are Since M αβ is an average of quantities with absolute value 1, note that it must satisfy |M αβ | ≤ 1.
We now break the sum in (10) into intervals of width ε, centered on some given energy gap G β . There are at most 2 )ε and hence from (11), There are d E (d E − 1) terms in the sum α |M αβ |, and it is maximised by having as many terms with small values of |k| as possible. It follows that Where the first term comes from the k = 0 interval, and the second term comes from the intervals with positive and negative k.
Using (10), and the fact that (for d E > 1) which is proven in Appendix A, we find and hence This proves the theorem for pure states. We extend the result to mixed states via purification [3]. Given any initial state ρ(0) on H, we can define a pure state |φ(0) on H ⊗ H such that the reduced state of the first system is ρ(0). By evolving |φ(t) under the joint Hamiltonian H ⊗ I, we will recover the correct evolution ρ(t) of the first system. The expectation value of any operator A for ρ(t) will be the same as the expectation value of A ⊗ I on the total system, and A = A ⊗ I . Note that N (ε) is the same for H ⊗ I as for H and that the effective dimension of the purified system is the same as the effective dimension of the original system because tr[P E ρ(0)] = tr[P E ⊗ I|φ(0) φ(0)|].

IV. DISTINGUISHABILITY
Even if Theorem 1 shows that the expectation value of a particular observable equilibrates, this does not necessarily mean that it is impossible distinguish ρ(t) from ω by measuring that observable [3]. Furthermore it does not tell us that the system as a whole has equilibrated. For example, when ρ(t) is pure, for any time t there exists some measurement which can distinguish ρ(t) and ω with very high probability (p success ≥ 1 − 1 2d eff ) [22]. To discuss an appropriate measure of distinguishability, it is helpful to describe a measurement not by its expectation value, but by a Positive Operator Valued Measure (POVM). Each measurement outcome a is associated with a positive operator M a , such that the probability of obtaining that outcome given the state ρ is tr[ρM a ]. For normalisation, we require a M a = I (assuming a finite number of measurement outcomes).
Suppose we are trying to decide whether a system is in state ρ or state σ by performing a measurement (we will assume that both are equally likely). The best strategy is to guess whichever state has the highest probability of giving the measurement outcome. For example, if tr[ρM a ] > tr[σM a ], we should guess that we have state ρ when we obtain outcome a. With this strategy, our probability of success is This motivates us to define the distinguishability of ρ and σ, given the measurement M , to be (see [3]) We can go a step further, and define the distinguishability of ρ and σ, given the set of measurements M, to be Naturally, if ρ = σ, the distinguishability is zero. Furthermore, it is symmetric and obeys the triangle inequality. But it may not be perfect: there may be states that are different but for which the distinguishability is zero. This is not surprising -the set of measurements may not be good enough to distinguish any two states. The special case where M includes all measurements defines the trace distance, denoted by D(ρ, σ). It follows that 0 ≤ D M (ρ, σ) ≤ D (ρ, σ) ≤ 1.
The trace distance is a good measure of how similar two states are; in fact, it is a metric on the set of density matrices because it also satisfies the property that D(ρ, σ) = 0 only if ρ = σ. But this is a good indicator of why the trace distance may not be suitable from the point of view of statistical physics: we cannot do any measurement we like on 10 23 particles, so we will usually miss out on the fine details of the microstate of the system.

V. EQUILIBRATION OF SYSTEMS AND SUBSYSTEMS
We can use the notion of distinguishability above to give a more precise definition of equilibration. In particular, we note that as the distinguishability is positive, if its average over some time-interval is small, then it must be small for most times. This leads us to the following definition of equilibration. If D M ρ(t), σ T is small, it means that we cannot distinguish ρ(t) from σ with the measurements in M for most times in [0, T ].
In [3], the equilibration of expectation values is used to prove equilibration of a closed system given reasonable constraints on the set of possible measurements M. In particular it is shown that with no degenerate energy gaps, where S(M) is the total number of outcomes of all the measurements that we can do. This is a huge number, but typically it is insignificant compared to d 1/2 eff , [3]. Thus, given the non-degenerate gaps condition, we would expect equilibration to occur for realistic measurements on large systems over the infinite time interval [0, ∞).
Here we use Theorem 1 to extend this result to finite time-intervals, and to Hamiltonians that do not satisfy the non-degenerate energy gaps condition.
Theorem 2. Consider the quantum state ρ(t) evolving via a Hamiltonian with d E distinct energies. For any energy ε > 0 and time T > 0, the average distinguishability of ρ(t) from ω = ρ(t) ∞ over the interval [0, T ] using measurements in the set M is bounded by where S(M) is the total number of measurement outcomes in M, and N (ε) and d eff are defined in (1) and (2) respectively.
This result and Theorem 3 below follow from a straightforward substitution of (4) for (3) in the derivations given in [3]. A detailed proof is given in the Appendix B for completeness.
The main theorem in [2] concerning the equilibration of subsystems is the result that, again with no degenerate energy gaps, where d S is the dimension of the subsystem, ρ S (t) is the state of the subsystem at time t and ω S = ρ S (t) ∞ . This tells us that small enough subsystems equilibrate with respect to all measurements over [0, ∞). This is an extremely strong result. It says that even if we can do any measurement we want on a subsystem, which can be arbitrarily large (as long as d 2 S ≪ d eff ), its state is indistinguishable from ω S for most times in [0, ∞).
Here we can also use Theorem 1 to extend this result to finite time-intervals, and to Hamiltonians that do not satisfy the non-degenerate energy gaps condition.
Theorem 3. Consider a system evolving via a Hamiltonian with d E distinct energies. For any energy ε > 0 and time T > 0, the trace distance between the subsystem state ρ S (t) and ω S = ρ S (t) ∞ averaged over the interval [0, T ] is bounded by where d S is the dimension of the subsystem, N (ε) and d eff are defined in (1) and (2) respectively.

See Appendix C for proof.
To understand how these results compare to the previous ones given by (19) and (21) it is helpful to choose ε equal to the minimum spacing between (non-degenerate) energy gaps. This means setting For this choice of ε, it is easy to see that N (ε min ) is equal to the maximum degeneracy of a single energy gap D G . Considering this in the context of (22), we obtain This choice of ε gives the strongest bound in the limit as T → ∞, yielding a version of (21) which applies to any Hamiltonian.
If the non-degenerate gaps condition is satisfied, then D G = 1 and we recover the previous result. However, this bound shows that subsystems can equilibrate even with large numbers of degenerate energy gaps. Note that D G < d E for all Hamiltonians, with the maximal value of D G = d E − 1 being obtained when all energy levels are equally spaced (e.g. a harmonic oscillator with an energy cut-off). However, almost any non-trivial Hamiltonian will have a much smaller value of D G . A good motivation for the non-degenerate gaps condition was to rule out systems composed of non-interacting subsystems. Here, such systems are allowed, but we see that appending a non-interacting ancilla to the original system does not improve the bound given by (25). In particular, if the ancilla has k distinct energies d eff can increase by a factor of at most k, but D G will also increase by a factor of at least k.
Given that we can prove equilibration in the infinitetime limit for a particular choice of ε, then Theorem 2 tells us that this equilibration will also occur over the finite interval [0, T ] as long as We can obtain the best bound on the timescale for equilibration by choosing ε > ε min . If the energy levels in the Hamiltonian are reasonably evenly distributed over a range ∆E, and d eff ∼ d E , then a choice of ε = η∆E/d E (with η ≪ 1) should be sufficient to ensure equilibration for a small subsystem. This gives a timescale of T = dE log 2 dE η∆E , which is much shorter than the recurrence time (which typically grows exponentially with the dimension [15]). Furthermore, we can find examples which we would expect to take at least time linear in d E to equilibrate, so the bound probably cannot be improved significantly without further assumptions. One such example would be a timer counting down using basis states of a quantum system of dimension d, then exploding a bomb when it reached 0.

VI. DISCUSSION
We have extended previous results concerning the equilibration of systems and subsystems, to Hamiltonians with degenerate energy gaps (so long as no gap is hugely degenerate) and to finite equilibration times.
The equilibration times obtained are still very large, but this is inevitable given the generality of the approach. We hope that this work can provide a starting point for proving much shorter equilibration times given further assumptions on the Hamiltonian (such as locality and translation invariance), or on the accessible measurements M (such as only permitting macroscopic measurements).
Throughout, we have focussed on equilibration to the final (infinite-time) equilibrium state ω. However, in reality, systems often evolve through a series of temporary equilibrium states over time. For example, consider a hot cup of tea. Eventually it will cool to room temperature, so it seems to be in equilibrium for the next hour or so. However, if we wait a lot longer, it will evaporate, and then there will be a new equilibrium state over a longer period. If we wait long enough, even the cup will disintegrate. Defining the time-averaged state over the interval [0, T ] by ω T = ρ(t) T , the characteristic timescales for these temporary equilibrations will correspond to those values of T for which D M (ρ(t), ω T ) T ≪ 1. The analogue of equation (20) in this case would be Note that when the system has a series of different timeintervals [0, T i ] over which it equilibrates, and the corresponding equilibrium states ω Ti are distinguishable, then the timescale for each equilibration must be much longer than the previous one. This is because a state cannot be close to ω Ti for most of [0, T i ] and ω Ti+1 for most of [0, T i+1 ] unless T i+1 ≫ T i . Furthermore, this means that if a system equilibrates over infinite-time, then the timescale for any temporary equilibration must be much less than that given by (26). It would be interesting to study these phenomena more closely in future work, and obtain a strong bound on M .
Another interesting question we have not addressed here is the form of the equilibrium state ω. In [12], given additional assumptions, it is shown that the state ω S of a small subsystem is close to a Gibbs state ρ Gibbs S = e −βHS / tr(e −βHS ), where H S is the Hamiltonian of the subsystem and β is the inverse temperature. However, one key assumption is that the interaction Hamiltonian between the subsystem and its environment is very weak (satisfying H int ≪ 1/β). As H int is extensive and 1/β is intensive, this seems unlikely to hold for subsystems composed of many particles. We now argue that when the system is a lattice of spins, one should not expect it to equilibrate to a Gibbs state in the sense that D(ω S , ρ Gibbs S ) ≪ 1. This is because we would expect each spin on the boundary of S to be slightly different from its state in ρ Gibbs S due to edge effects caused by H int . By measuring all of these boundary particles we would generally expect to be able detect these subtle edge effects, and hence distinguish ω S from ρ Gibbs S , implying D(ω S , ρ Gibbs S ) ≃ 1. In order for standard statistical mechanics to be applied in such cases, it is therefore highly relevant that we cannot make any measurement we like on the system, further motivating the study of restricted measurement sets.