Dynamical Maps, Quantum Detailed Balance and Petz Recovery Map
Markovian master equations (formally known as quantum dynamical semigroups) can be used to describe the evolution of a quantum state when in contact with a memoryless thermal bath. This approach has had much success in describing the dynamics of real-life open quantum systems in the lab. Such dynamics increase the entropy of the state and the bath until both systems reach thermal equilibrium, at which point entropy production stops. Our main result is to show that the entropy production at time is bounded by the relative entropy between the original state and the state at time . The bound puts strong constraints on how quickly a state can thermalise, and we prove that the factor of is tight. The proof makes use of a key physically relevant property of these dynamical semigroups – detailed balance, showing that this property is intimately connected with the field of recovery maps from quantum information theory. We envisage that the connections made here between the two fields will have further applications. We also use this connection to show that a similar relation can be derived when the fixed point is not thermal.
It is very often observed in nature that physical systems relax to an equilibrium state. This phenomenon, which has very evident consequences at the macroscopic scales of our everyday experience, ultimately relies on the dynamics of the microscopic components. This fact was understood in the early days of statistical mechanics, and since then a large amount of work has been produced with the aim of trying to understand how exactly physical systems reach thermal equilibrium.
Any such evolution will be ultimately generated through some reversible dynamics on a large composite system, that is effectively irreversible as seen by a smaller part of that composite system. This irreversibility means that, in a coarse-grained sense, entropy will be produced throughout the process. The entropy production can be linked to the fact that correlations between a big thermal object (a heat bath) and one smaller subsystem are increasingly harder to access, which forces the coarse-graining of the description Esposito et al. (2010). Intuitively, the more irreversible a process is, the more entropy is produced, and the closer a particular system will be to equilibrium.
In this work we look at a commonly used family of quantum evolutions that model the dynamics of a system weakly coupled to a thermal bath, and show explicitly how the amount of entropy produced along a particular evolution is related to how much a state changes along that evolution. These maps were first studied by Davies Davies (1974) and are a quantum generalization of the classical Glauber dynamics.
In the limit of a large thermal bath, the total entropy produced by such a process is given by how much the free energy of a system decreases with time Spohn (1978). The free energy for a state at time is defined as
where is the Hamiltonian of the subsystem of interest, and is the temperature of the bath. Moreover, for an evolution from time to , the total amount of Von Neumann entropy produced, the so-called Entropy Production is given by with , the changes in mean energy and Von Neumann entropy of the system. Due to the contractivity property of the quantum relative entropy, this quantity is non-negative and non-decreasing with .
The reason for this name is as follows. For a large thermal reservoir, small changes of energy (that is, heat transferred to the system) are proportional to changes of entropy in it, with proportionality constant . Hence, we can identify the change in energy in the system with a change of entropy in the reservoir , so that the difference in free energy of the system for a time interval is equal to the total entropy generated during the interval in system and bath. Therefore, this entropy production constitutes a natural measure of the irreversibility of the process.
Our main result is Theorem 2, which states that under the condition that the interaction between system and bath is time-independent, we can lower-bound the entropy production at time by the state at time .
This sharpens some intuitive notions, namely that if not much entropy is produced during a time interval , the state will not change very much during the time interval , but if it does, then a large amount of entropy must have been produced at an earlier time, namely during the time interval .
Recovery maps have found many applications in quantum information theory, such as coding theorems Beigi et al. (2016); Brandao and Kastoryano (2016), approximate error correction Ng and Mandayam (2010) or asymmetry Marvian and Lloyd (2016). They also appear in the derivation of quantum fluctuation theorems Aberg (2016); Alhambra et al. (2016).
Our results, are inspired by findings in quantum information theory about recovery maps. Specifically, they are a consequence of the observation that if a dynamical map satisfies quantum detailed balance (QDB), a property of thermodynamical processes, then this implies that the map is its own recovery map. The connection between information theory and thermodynamics goes back a long way, to the seminal work of Landauer Landauer (1961) and has furthered our understanding of both significantly. Within the current surge of information-theory approaches to quantum thermodynamics (see Goold et al. (2016) for a review), our result provides another example of how ideas from one may find definite applications in the other.
We shall first introduce Davies maps, outline their properties. This is followed by the statement of the main result and a discussion on the bound itself. We finally conclude with some suggestions for open questions.
Ii Davies maps and entropy production
Davies maps are a particular set of quantum dynamical semigroups that describe the evolution of a system on a dimensional Hilbert space that is weakly interacting with a heat bath. The first rigorous derivation of their form was given in Davies (1974) (see Alicki and Lendi (2007); Temme (2013) for more modern treatments). As they are time-continuous quantum semigroups, their generator takes the form of a Lindbladian operator, which we define as
where is called the Lindbladian and is called the unitary part, with the effective Hamiltonian. The solution is a one-parameter family of CPTP maps , which governs the dynamics, . We will not delve into the full details here, but instead highlight the important properties the canonical form of Davies maps, denoted , possess:
They arise from the weak system-bath coupling limit
They can be written in the form , with and time independent
They have a thermal fixed point: , where is the Gibbs state of the system at temperature .
Their Lindbladians and unitary part satisfy Quantum detailed balance (QDB):
for all , where is the adjoint Lindbladian. can be any quantum state. However, in the case of Davies maps, . The scalar product in Eq. (3) is defined as
This is sometimes referred to as reversibility or KMS condition. It is stronger than 4), since it has as a consequence that is the fixed point, as .
In Appendix A we give a more detailed account of the microscopic origin of these maps, and of the form of the weak coupling limit, property 1). In the literature, there are various different definitions of QDB which are in general not equivalent. We show in Appendix D that for maps satisfying time translation symmetry, such as Davies maps, definition 5) is equivalent to the definition of QDB in Alicki and Lendi (2007); Kossakowski et al. (1977).
In addition to the properties above, it is sometimes assumed that:
The dynamics associated with Davies maps converge to the fixed point, .
Since we wish to bound the distance from the state at time to the fixed point, we need a distance measure. For this we use the relative entropy . This measure is meaningful since it is non-negative, zero iff , and is contractive under CPTP maps. For the special case that is a Gibbs state, it has an interpretation in terms of a free energy,
where is the partition function of the system, which we assume is constant. We can thus write the entropy production in terms of a difference in relative entropy, as
As one intuitively might expect, this entropy production only depends on the dissipative part of the dynamics, as we explain in Section A.3 of the appendix. Therefore, we will assume for simplicity that in the next Section unless stated otherwise.
If one were to change the initial state of the environment for the maximally mixed state, then the system can only exchange entropy, but not heat/energy with it. These correspond to unital maps, in which case the free energy is replaced with the entropy gain of the system alone. In that case, a lower bound on the entropy they produce in terms of the adjoint of the unital map can be found in Buscemi et al. (2016).
Iii Main results
Our main result is a tight lower bound on the change of free energy and total entropy produced, within a finite time. We start with a Lemma for Davies maps which is an initial step in its derivation:
All Davies maps , satisfy the inequality
where is the time-reversed map or Petz recovery map, defined as
with denoting the adjoint of .
See Appendix A.2. ∎
Eq. 9 proves a physically relevant particular case of an open conjecture about general quantum maps first formulated in Li and Winter (2014). The strongest possible version of the conjecture is known to not be true in full generality Brandao et al. (2015), although it has been shown for particular sets such as unital maps Buscemi et al. (2016), classical stochastic matricesLi and Winter (2014), catalytic thermal operations Alhambra et al. (2015) and we here show it for Davies maps. All these results relate the decrease of relative entropy with a measure of how well a given pair of states can be recovered through a particular recovery map, and are generalizations of an early result by Petz Petz (1986). For the best results up to date on general quantum maps, see Wilde (2015); Junge et al. (2015); Sutter et al. (2016a, b).
For Lemma 1 to hold, only properties 1) and 4) are required. In addition, we find that there is a connection between property 4) and the Petz recovery map which we will now explain. A quantum dynamical semi-group which obeys QDB has a Petz recovery map which is equal to the map itself (See Theorem 8 in Appendix). Petz derived his famous recovery map in 1986 Petz (1986) while the first appearance of the detailed balance condition goes back at least to the work of Boltzmann in 1872 Boltzmann (1964) and QDB to Alicki in 1976 Alicki (1976). To the best of the authors knowledge, this connection between results from the communities of quantum information theory and quantum dynamical semi-groups was previously unknown. Perhaps the closest previous work, is Fagnola and Umanità (2007), which define Detailed Balance as the property that the recovery map is equal to the map itself. Our work implies that for the special case of the Petz recovery map, the Detail Balance definition of Fagnola and Umanità (2007), is equal to definition 5) which is satisfied by Davies maps.
The classical definition of detailed balance, in terms of the transition probabilities of a classical Master equation, implies that, at equilibrium, a particular jump between energy levels has the same total probability as the opposite jump , such that . The condition in Eq. (3) is the most natural quantum generalization of that (although as shown in Temme et al. (2010) different ones are also possible). In that sense, QDB can be understood as the fact that a particular thermalization process coincides with its own time-reversed map, which is defined as in Eq. (9) (for more details, see e.g. Crooks (2008); Ticozzi and Pavon (2010)).
This map is such that we have that iff then and . It appears in quantum information theory when one tries to find the best possible way to recover data after it is processed Barnum and Knill (2002); Wilde (2013).
All Davies maps , satisfy the inequality
See Appendix A.3. ∎
In addition to assuming detailed balance, condition 5), we have also used condition 2). If the Lindbladian is time dependent, i.e. 2) is not satisfied, Eq. (11) holds but with replaced with .
While, as mentioned at the end of Section II, entropy production is invariant under a change in the unitary part of the dynamics, it is interesting to find the Petz recovery map when is not set to zero. We show in Lemma 9 in the appendix, that the Petz recovery map of a map satisfying QDB and for which and commute [property 3) of Davies maps], reverses the unitary part of the dynamics, while keeping the same dissipative part, that is
and thus . So not only is the l.h.s. of Eq. 11 invariant under a change in the unitary part of the dynamics; but so is the r.h.s.
In Fig. 1 we show a simple example of the inequality for the case of Davies maps applied on a qutrits. Eq. (11) is tight at and also in the large time limit, as long as condition 6) is satisfied. In this limit, the total entropy that has been produced is equal to , which both sides of the inequality approach as .
On the other hand, for very short times, the lower bound becomes trivial. In particular, in Appendix A.4 we show what both sides of the inequality tend to in the limit of infinitesimal time transformations. The entropy production becomes a rate, and the lower bound to it approaches .
Non-trivial lower bounds on the rate of entropy production, in the form of log-Sobolev inequalities Kastoryano and Temme (2013) can be used to derive bounds on the time it takes to converge to equilibrium for particular instances of Davies maps. Hence, given that Theorem 2 is completely general, and holds also for Davies maps that do not efficiently reach thermal equilibrium, the fact that the lower bound vanishes for infinitesimal times is not surprising.
Recall that the factor of in Eq. (11) is a consequence of the observation that the Petz recovery map is equal to the map itself. A natural question is then, is the factor fundamental? We show that this is indeed the case with the following Theorem.
[Tightness of the entropy production bound] The largest constant such that
holds for all Davies maps, is .
Iv Beyond Davies maps
We now turn our attention to what recent developments from quantum information theory can say about convergence of dynamical semi-groups in general. A recent advancement in quantum information is the development of universal recoverability maps Wilde (2015); Junge et al. (2015); Sutter et al. (2016b). By universal recoverability, it is meant that given a state and a CPTP map , one can use the recovery map to lower bound the relative entropy difference for all quantum states . In general the lower bound takes on a complicated form (see Appendix C). However, for the case of dynamical semi-groups satisfying QDB and the following property, the bound is more explicit.
Let us assume that we have a one-parameter dynamical semi-group equipped with a fixed point that satisfies a condition we call Time-translation symmetry w.r.t. fixed point (TTSFP),
This condition is satisfied for example by dynamical semi-groups which arise naturally in the weak coupling limit or the low-density limit. Davies maps are one such example, but there are others Wolf et al. (2008).
The properties lead to the following result:
Let the Quantum Dynamical Semigroup satisfy QDB and TTSFP. Then the following holds
where is the quantum fidelity. Moreover, if the generators are time-independent we may write .
One of the main features in the study of dynamical thermalisation processes, such as Davies maps, is QDB. By using tools from quantum information theory, we show that the entropy produced after a time , is lower bounded by how well one can recover the initial state from the state via a recovery map. We then show that, due to QDB, the best way to perform the recovery is to time evolve forward in time an amount to the state . What’s more, if one time evolves for time , a worse bound is generated, while if one evolves for , the bound is not true for all Davies maps; thus showing that the connection between reversibility and recoverability suggested by QBD leads to tight dynamical bounds.
One of the important questions regarding Davies maps is how fast they converge to equilibrium. There have been several approaches to this question, mostly inspired by their classical analogues, which include the computation of the spectral gaps Alicki et al. (2009); Temme et al. (2010); Temme (2013) or the logarithmic-Sobolev inequalities Kastoryano and Temme (2013); Temme et al. (2014). In particular we note that the latter take the form of upper bounds on distance measures between and the thermal state. Likewise Eq. (11) can be re-arranged to give an upper bound in terms of the relative-entropy to the Gibbs state, . It would be interesting to know if the bound of Eq. (11), for primitive Davies Maps, i.e. the dynamics converge to a unique fixed point, contains information about their asymptotic convergence. For instance, one could look at how fast is the inequality saturated in particular cases. We however leave this for future work.
Another potential application of our work in open quantum systems, is to use a tightened monotonicity inequality to find when information backflow occurs in non-Markovian dynamics Rivas et al. (2014, 2010); Laine et al. (2010).
The condition of detailed balance is ubiquitous in thermalization processes, and in particular, current algorithms for simulating thermal states on a quantum computer, such as the quantum Metropolis algorithm Temme et al. (2011), obey it, which makes it all the more interesting. As such, the useful connection we establish here between the Petz recovery map and QDB, is likely to have further implications for both thermodynamics and information theory.
Acknowledgements.The authors would like to thank David Sutter, Michael Wolf, Nilanjana Datta, Mark Wilde and Toby Cubitt for helpful discussions. AA and MW acknowledge support from FQXi and EPSRC. This work was partially supported by the COST Action MP1209.
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Appendix A Technical results
a.1 Davies maps and conditions for Lemma 1
Davies maps are derived from considering the dynamics of a state where is of finite dimension , in contact with a thermal bath on an infinite dimensional Hilbert space . We will here specify the minimal assumptions about the bath and its interaction with the system necessary for the derivation of Lemma 5 and Lemma 1. In order to guarantee other properties, such as the existence of a fixed point or detailed balance, more subtle constraints are also necessary.
Let be a self-adjoint Hamiltonian on . Since we want states on to be thermodynamically stable, we assume that for all . must therefore have a purely discrete spectrum, which is bounded below and has no finite limit points; that is, there are only a finite number of energy levels in any finite interval . The quantum state with its free self-adjoint Hamiltonian of finite dimension interacts with the system via a bounded interaction term , with a parameter determining the interaction strength as follows
The initial state on is assumed to be product, , with the Gibbs state at inverse temperature . The dynamics of the system at time is given by the unitary operator
after tracing out the environment. More precisely, by
where denotes the adjoint of .
The Davies map is defined by taking the limit that the interaction strength goes to zero, while the time goes to infinity while maintaining fixed. More concisely,
It is assumed that in this limit and its inverse are still unitary operators mapping states on to states on . To gain more physical insight into this construction, we refer to [15, 47, 2]. We remind the reader that the conditions described in Section A.1 are not sufficient for the map to satisfy other properties, such as the convergence to a fixed point or detailed balance, more subtle constraints are also necessary. We will not go into the details of these additional conditions, since only sufficient (but perhaps not necessary) conditions are known, e.g. . In other Sections, we will additionally take advantage of the known fact that Davies maps satisfy quantum detailed balance.
a.2 Proof and statement of Lemma 1
In order to prove the main theorem we need a lemma about Davies maps first.
We show that in the weak coupling limit, correlations between the system and the environment (the bath) are not created if both start as initially uncorrelated thermal states. In order to do this, we will need to introduce a finite dimensional cut-off on and prove the results for the truncated space, and finally proving uniform convergence in the bath system size by removing the cut-off by taking the infinite dimensional limit.
Let denote the projection onto a finite dimensional Hilbert Space . Furthermore, assume that and that . For concreteness (although not strictly necessary), one could let where are the eigenvectors of ordered in increasing eigenvalue order.
We define the truncated self-adjoint Hamiltonians on as with a corresponding Gibbs state denoted by . Similarly, we construct unitaries on by
and define . We recall the definition of the thermal state of the system , which is given by
for some inverse temperature
The lemma is the following:
Lemma 5 (Correlations at the fixed point).
Let , and the constant . Then, for all , we have the bound
where , are thermal states at inverse temperatures , respectively, and , is the one-norm and operator norm respectively.
The result is a consequence of mean energy conservation under the unitary transformation and Pinsker’s inequality.
Define the shorthand notation and , , . By direct evaluation of the relative entropy,
where we have used unitary in-variance of the von Neumann entropy . Thus since
Energy conservation implies
Pinsker inequality states that for any two density matrices , ,
It follows from it, and from Eq. (28),
This lemma may be of independent interest, as it makes explicit the idea mentioned in previous work such as  of how Davies maps, in the weak coupling limit, can be taken as free operations in the resource theory of athermality .
With it at hand, we can prove the central lemma.
Lemma 6 (Lemma 1 of main text).
Assume conditions in Section A.1 hold. Then all maps satisfy the inequality
where is the Petz recovery map corresponding to ,
with denoting the adjoint of .
Had there been no interaction term (i.e. ) and the bath been finite dimensional, the proof of this Lemma would have been straightforward, using the techniques developed in  involving simple manipulations of the relative entropy, and the data processing inequality for finite dimensional baths. The added difficulty here will be in proving monotone convergence as the bath Hilbert space tends to infinity. To achieve this, we will use Lemma 5 and continuity arguments. We will perform the calculations for the map rather than itself. We will finally take the limit described in Eq. (19) to conclude the proof.
Noting that the relative entropy between two copies is zero, followed by using its additivity and unitarity invariance properties, we find for ,
With the identity for bipartite states , , we have that
where and in the last line we have used the unitarity invariance of the relative entropy followed by the data processing inequality. Plugging Eq. (36) into Eq. (39) followed by taking the limit, we obtain
for all with the r.h.s. independent. By definition of , it follows that it is the partition function of a tensor product of thermal states on at inverse temperatures . Since the Hamiltonians by definition have well defined thermal states (finite partition functions) for all positive temperatures, it follows that for all . Thus noting that by definition, and that , is a bounded operator, it follows that
Thus since is finite dimensional and Hermitian, and the eigenvalues of finite dimensional Hermitian matrices are continuous in their entries [50, 51], it follows, since has full support, that there exists such that for all , has full support. Thus for all , the r.h.s. of Eq. (41) is upper bounded by a finite quantity uniformly in and thus since relative entropies are non-negative by definition, Eq. (41) is well defined for all .
We now set appearing in to followed by taking the limit while keeping fixed in Eq. (41) thus achieving
where we have used that by definition,
We now proceed to calculate the Petz’s recovery map for the map . The adjoint map is . Hence from the definition in Eq.(65) it follows that the Petz recovery map for is
which are well defined since they are comprised of products of bounded operators. Similarly to before, in Eq. (48) we now set appearing in to followed by taking the limit while keeping fixed achieving
In the above proof, we have taken two independent limits, namely 1st the infinite bath volume limit () followed by the Van Hove limit ( while keeping fixed). This is the order in which Davies performed the limits [2, 47] when defining the Davies map. From physical reasoning, one would expect the Davies map to be equally valid if the order of the limits is reversed. We note that the proof of Theorem 2 follows also if the order of the these two limits is reversed but now with the new definitions