It is well known in the realm of quantum mechanics and information theory that the entropy is non-decreasing for the class of unital physical processes. However, in general, the entropy does not exhibit monotonic behavior. This has restricted the use of entropy change in characterizing evolution processes. Recently, a lower bound on the entropy change was provided in the work of Buscemi, Das, and Wilde [Phys. Rev. A 93(6), 062314 (2016)]. We explore the limit that this bound places on the physical evolution of a quantum system and discuss how these limits can be used as witnesses to characterize quantum dynamics. In particular, we derive a lower limit on the rate of entropy change for memoryless quantum dynamics, and we argue that it provides a witness of non-unitality. This limit on the rate of entropy change leads to definitions of several witnesses for testing memory effects in quantum dynamics. Furthermore, from the aforementioned lower bound on entropy change, we obtain a measure of non-unitarity for unital evolutions.

Entropy is a fundamental quantity that is of wide interest in physics and information theory.1–4 Many natural phenomena are described according to laws based on entropy, like the second law of thermodynamics,5–7 entropic uncertainty relations in quantum mechanics and information theory,8–12 and area laws in black holes and condensed matter physics.13–16 

No quantum system can be perfectly isolated from its environment. The interaction of a system with its environment generates correlations between the system and the environment. In realistic situations, instead of isolated systems, we must deal with open quantum systems, that is, systems whose environment is not under the control of the observer. The interaction between the system and the environment can cause loss of information as a result of decoherence, dissipation, or decay phenomena.17–19 The rate of entropy change quantifies the flow of information between the system and its environment.

In this work, we focus on the von Neumann entropy, which is defined for a system in the state ρ as S(ρ) := −Tr{ρ log ρ}, and from here onwards we refer to it as the entropy. The entropy is non-decreasing under doubly stochastic, also called unital, physical evolutions.20,21 This has restricted the use of entropy change in the characterization of quantum dynamics only to unital dynamics.22–26 Recently, Ref. 27 gave a lower bound on the entropy change for any positive trace-preserving map. Lower bounds on the entropy change have also been discussed in Refs. 22, 26, 28, and 29 for certain classes of time evolution. Natural questions that arise are as follows: What are the limits placed by the bound30 on the entropy change on the dynamics of a system? And can it be used to characterize evolution processes?

We delve into these questions, at first, by inspecting another pertinent question: At what rate does the entropy of a quantum system change? Although the answer is known for Markovian one-parameter semigroup dynamics of a finite-dimensional system with full-rank states,31 the answer in full generality has not yet been given. In Ref. 32, the result of Ref. 31 was extended to infinite-dimensional systems with full-rank states undergoing Markovian one-parameter semigroup dynamics (cf., Ref. 33). We now prove that the formula derived in Ref. 31 holds not only for finite-dimensional quantum systems undergoing Markovian one-parameter semigroup dynamics but also for arbitrary dynamics of both finite- and infinite-dimensional systems with states of arbitrary rank. We then derive a lower bound on the rate of entropy change for any memoryless quantum evolution, also called a quantum Markov process. This lower bound is a witness of non-unitality in quantum Markov processes. Interestingly, this lower bound also helps us to derive witnesses for the presence of memory effects, i.e., non-Markovianity, in quantum dynamics. We compare one of our witnesses to the well-known Breuer-Laine-Piilo (BLP) measure34 of non-Markovianity for two common examples. As it turns out, in one of the examples, our witness detects non-Markovianity even when the BLP measure does not, while for the other example, our measure agrees with the BLP measure. We also provide bounds on the entropy change of a system. These bounds are witnesses of how non-unitary an evolution process is. We use one of these witnesses to propose a measure of non-unitarity for unital evolutions and discuss some of its properties.

The organization of the paper is as follows. In Sec. II, we introduce standard definitions and facts that are used throughout the paper. In Sec. III, we discuss the explicit form (Theorem 1) for the rate of entropy change of a system in any state undergoing arbitrary time evolution. In Sec. IV, we provide a brief overview of quantum Markov processes. We state Theorem 2 that provides a lower limit on the rate of entropy change for quantum Markov processes. We show that this lower limit provides a witness of non-unitality. We also discuss the implications of the lower limit on the rate of entropy change in the context of bosonic Gaussian dynamics (Sec. IV A). In Sec. V, based on the necessary conditions for the Markovianity of quantum processes as stated in Theorem 2, we define some witnesses of non-Markovianity and also a couple of measures of non-Markovianity based on these witnesses. We apply these witnesses to two common examples of non-Markovian dynamics (Secs. V A 1 and V A 2) and illustrate that they can detect non-Markovianity. In Sec. V A 2, we consider an example of a non-unital quantum non-Markov process whose non-Markovianity goes undetected by the BLP measure while it is detected by our witness. In Sec. VI, we derive an upper bound on entropy change for unital evolutions. We also show the monotonic behavior of the entropy for a wider class of operations than previously known. In Sec. VII, we provide a measure of non-unitarity for any unital evolution based on the bounds on the entropy change obtained in Sec. VI. We also discuss the properties of our measure of non-unitarity. We give concluding remarks in Sec. VIII.

We begin by summarizing some of the standard notations, definitions, and lemmas that are used in Secs. III–VII.

Let B(H) denote the algebra of bounded linear operators acting on a Hilbert space H, with 1H denoting the identity operator. Let dim(H) denote the dimension of H, and note that this is equal to + in the case that H is a separable, infinite-dimensional Hilbert space. If the trace Tr{A} of AB(H) is finite, then we call A trace-class. The subset of B(H) containing all trace-class operators is denoted by B1(H). The subset containing all positive semi-definite operators is denoted by B+(H). We write P ≥ 0 to indicate that PB+(H). Let B1+(H):=B+(H)B1(H). Elements of B1+(H) with unit trace are called density operators, and the set of all density operators is denoted by D(H). The Hilbert space associated with a quantum system A is denoted by HA. The state of a quantum system A is represented by a density operator ρAD(HA). We let HABHAHB denote the Hilbert space of a composite system AB. The density operator of a composite system AB is denoted by ρABD(HAB), and the partial trace TrA over the system A gives the local density operator ρB of system B, i.e., ρB = TrA{ρAB}. A pure state ψA:= |ψ⟩ψ|A is a rank-one element in D(HA).

Let NAB:B(HA)B(HB) denote a linear map (also called superoperator) that maps elements in B(HA) to elements in B(HB). It is called positive if it maps elements of B+(HA) to elements of B+(HB) and completely positive if idRNAB is positive for a Hilbert space HR of any dimension, where id is the identity superoperator. A positive map NAB:B1+(HA)B1+(HB) is called trace non-increasing if Tr{NAB(σA)}Tr{σA} for all σAB1+(HA), and it is called trace-preserving if Tr{NAB(σA)}=Tr{σA} for all σAB1+(HA). Where confusion does not arise, we omit identity operators in expressions involving multiple tensor factors so that, for example, NAB(ρRA) is understood to mean idRNAB(ρRA).

A linear map NAB:B(HA)B(HB) is called sub-unital if NAB(1A)1B, unital if NAB(1A)=1B and super-unital if NAB(1A)1B, where for C,DB(H), CD is defined to mean CD ≥ 0. Note that it is possible for a linear map to be neither unital, sub-unital, nor super-unital. A positive trace-preserving map can be sub-unital only if the dimension of the output Hilbert space is greater than or equal to the dimension of the input Hilbert space. A positive trace-preserving map can be super-unital only if the dimension of the output Hilbert space is less than or equal to the dimension of the input Hilbert space. Positive trace-preserving maps between two finite-dimensional Hilbert spaces of the same dimension that are both sub-unital and super-unital are unital.

The evolution of a quantum state is described by a quantum channel, which by definition is a linear, completely positive, and trace-preserving (CPTP) map. A quantum operation is defined to be a linear, completely positive, and trace non-increasing map. An isometry U:HH is a linear map such that UU=1H.

The adjoint N:B(HB)B(HA) of a linear map N:B1(HA)B1(HB) is the unique linear map that satisfies

XAB1(HA),YBB(HB):   YB,N(XA)=N(YB),XA,
(2.1)

where ⟨C, D⟩ = Tr{CD} is the Hilbert-Schmidt inner product. The adjoint of a trace-preserving map is unital, the adjoint of a trace-non-increasing map is sub-unital, and the adjoint of a trace-non-decreasing map is super-unital.

Let A be a self-adjoint operator acting on a Hilbert space H. The support supp(A) of A is the span of the eigenvectors of A corresponding to its non-zero eigenvalues, and the kernel of A is the span of the eigenvectors of A corresponding to its zero eigenvalues. There exists a spectral decomposition of A,

A=kλk|kk|,
(2.2)

where {λk}k are the eigenvalues corresponding to an orthonormal basis of eigenvectors {|k}k of A. The projection Π(A) onto supp(A) is then

Π(A)=k:λk0|kk|.
(2.3)

Let rank(A) denote the rank of A. If A is positive definite, i.e., A > 0, then rank(A)=dim(H), Π(A)=1H, and we say that the rank of A is full. If f is a real-valued function with domain Dom(f), then f(A) is defined as

f(A)=k:λkDom(f)f(λk)|kk|.
(2.4)

The von Neumann entropy of a state ρA of a quantum system A is defined as

S(A)ρS(ρA)=Tr{ρAlogρA},
(2.5)

where log denotes the natural logarithm. In general, the state of an infinite-dimensional quantum system need not have finite entropy.35 For any finite-dimensional system A, the entropy is upper-bounded by logdim(HA). The quantum relative entropy of any two operators ρ,σB1+(H) is defined as36–38 

D(ρσ)=i,jϕi|ψj2p(i)logp(i)q(j),
(2.6)

where ρ=ip(i)|ϕiϕi| and σ=jq(j)|ψjψj| are spectral decompositions of ρ and σ, respectively, with {|ϕi}i and {|ψj}j orthonormal bases for H. From the above definition, it is clear that D(ρ||σ) = + if supp(ρ) ⊈ supp(σ). Another common way to write the relative entropy is as follows:

D(ρ||σ)=Tr{ρ(logρlogσ)}ifsupp(ρ)supp(σ),+otherwise,
(2.7)

when ρ(logρlogσ)B1(H), where the trace is understood to be with respect to the orthonormal basis {|ϕi}i. In general, the formula (2.7) has to be evaluated using (2.6). For any two positive semi-definite operators ρ and σ, D(ρσ) ≥ 0 if Tr{ρ} ≥ Tr{σ}, D(ρσ) = 0 if and only if ρ = σ, and D(ρσ) < 0 if ρ < σ. The quantum relative entropy is non-increasing under the action of positive trace-preserving maps,39 that is, D(ρσ)D(N(ρ)N(σ)) for any two density operators ρ and σ and positive trace-preserving map N.

The Schatten p-norm of an operator AB(H) is defined as

ApTr|A|p1p,
(2.8)

where |A|AA and p ∈ [1, ). If {σi(A)}i are the singular values of A, then

Ap=iσi(A)p1p.
(2.9)

A:=limpAp is the largest singular value of A. Let Bp(H) be the subset of B(H) consisting of operators with finite Schatten p-norm.

Lemma 1
(Hölder’s inequality40–42). For allABp(H), BBq(H), and p, q ∈ [1, ) such that1p+1q=1, it holds that
|A,B|=TrABApBq.
(2.10)

The following important lemma can be found in Ref. 43, Corollary 5.2.

Lemma 2.
LetN:B+(HA)B+(HB)be a linear, positive, and sub-unital map. Then, for allσAB+(HA), it holds that
NAB(log(σA))log(NAB(σA)).
(2.11)

We now define entropy change, which is the main focus of our work.

Definition 3
(Entropy change). LetN:B1+(H)B1+(H)be a positive trace-non-increasing map. The entropy changeΔS(ρ,N)of a system in the stateρD(H)under the action ofNis defined as
ΔS(ρ,N)SN(ρ)S(ρ)
(2.12)
whenever S(ρ) andS(N(ρ))are finite.

It should be noted that N(ρ) is a sub-normalized state, i.e., Tr{N(ρ)}1, if N is a positive trace-non-increasing map.

It is well known that the entropy change ΔS(ρ,N) of ρ is non-negative, i.e., the entropy is non-decreasing, under the action of a positive, sub-unital, and trace-preserving map N20,21 (see also Ref. 27, Sec. III, and Ref. 44, Theorem 4.2.2). Recently, a refined statement of this result was made in Ref. 27, which is the following:

Lemma 4
(Lower bound on entropy change). LetN:B1+(H)B1+(H)be a positive, trace-preserving map. Then, for allρD(H),
ΔS(ρ,N)DρNNρ.
(2.13)

Lemma 4 gives a tight lower bound on the entropy change. As an example of a map saturating the inequality (2.13), take the partial trace NABB=TrA, which is a quantum channel that corresponds to discarding system A from the composite system AB. Its adjoint is N(ρB)=1AρB. Then, we have SNρABSρAB=S(ρB)S(ρAB)=DρAB1AρB=DρABNNρAB.

In general, physical systems are dynamical and undergo evolution processes with time. An evolution process for an isolated and closed system is unitary. However, no quantum system can remain isolated from its environment. There is always an interaction between a system and its environment. The joint evolution of the system and environment is considered to be a unitary operation whereas the local evolution of the system alone can be non-unitary. This non-unitarity causes a flow of information between the system and the environment, which can change the entropy of the system.

For any dynamical system with associated Hilbert space H, the state of the system depends on time. The time evolution of the state ρt of the system at an instant t is determined by dρtdt when it is well defined.46 The state ρT at some later time t = T is determined by the initial state ρ0, the evolution process, and the time duration of the evolution. Since the time evolution is a physical process, the following condition holds for all t:

Trρ̇t=0,
(3.1)

where ρ̇t:=dρtdt.

It is known from Refs. 31 and 45 that for any finite-dimensional system, the following formula for the rate of entropy change holds for any state ρt whose kernel remains the same at all times and whose support Πt is differentiable:

ddtS(ρt)=Trρ̇tlogρt.
(3.2)

The above formula has also been applied to infinite-dimensional systems for Gaussian states evolving under a quantum diffusion semigroup32,33 whose kernels do not change in time.

Here, we derive the formula (3.2) for states ρt having fewer restrictions, which generalizes the statements from Refs. 31 and 45. In particular, we show that the formula (3.2) can be applied to quantum dynamical processes in which the kernel of the state changes with time, which can happen because the state has time-dependent support.

Theorem 1.
For any quantum dynamical process withdim(H)<+, the rate of entropy change is given by
ddtS(ρt)=Trρ̇tlogρt,
(3.3)
wheneverρ̇tis well defined. The above formula also holds whendim(H)=+given thatρ̇tlogρtis trace-class and the sum of the time derivative of the eigenvalues of ρtis uniformly convergent49on some neighborhood of t, however small.

Proof.
Let Spec(ρt) be the set of all eigenvalues of ρtD(H), including those in its kernel. Let
ρt=λ(t)Spec(ρt)λ(t)Pλ(t)
(3.4)
be a spectral decomposition of ρt, where the sum of the projections Pλ(t) corresponding to λ(t) is
λ(t)Spec(ρt)Pλ(t)=1H.
(3.5)
The following assumptions suffice to arrive at the statement of the theorem when dim(H)=+. We assume that ρ̇t is well defined. We further assume that λ(t)Spec(ρt)λ̇(t) is uniformly convergent on some neighborhood of t, and ρ̇tlogρt is trace-class. We note that when dim(H)<+, λ(t)Spec(ρt)λ̇(t) and ρ̇tlogρt are always uniformly convergent and trace-class, respectively.
Now, we define the function s: [0, ) × (−1, ) → (0, ) by
s(t,h):=Tr{ρt1+h}=λ(t)Spec(ρt)λ(t)1+h.
(3.6)
Noting that ddxax=axloga for all a > 0 and xR, we have that
ddhρth+1=ρth+1logρt.
(3.7)
Applying (A3) in  Appendix A, we find that
ddts(t,h)=ddtTr{ρth+1}=h+1Tr{ρthρ̇t},
(3.8)
ddhs(t,h)=ddhTr{ρth+1}=Tr{ρth+1logρt}.
(3.9)
Then the entropy is
S(ρt)=ddhs(t,h)h=0=Tr{ρtlogρt}=λ(t)Spec(ρt)λ(t)logλ(t),
(3.10)
where by definition 0 log 0 = 0.
We note that ρth is an infinitely differentiable, i.e., a smooth function of h, and a differentiable function of t for all t, h. Note that the trace is also a continuous function. Since ddhddts(t,h) exists and is continuous for all (t, h) ∈ [0, ) × (−1, ), the following exchange of derivatives holds for all (t, h) ∈ (0, ) × (−1, ):
ddhddts(t,h)=ddtddhs(t,h).
(3.11)
This implies that
ddhddts(t,h)h=0=ddtddhs(t,h)h=0.
(3.12)
From (3.8), we see that ddts(t,h) is a smooth function of h. Therefore, the Taylor series expansion of this function in the neighborhood of h = 0 is
ddts(t,h)=ddts(t,h)h=0+ddhddts(t,h)h=0h+O(h2).
(3.13)
From (3.6), we find
ddts(t,h)h=0=ddtλ(t)Spec(ρt)λ(t)1+hh=0=λ(t)Spec(ρt)ddtλ(t)1+hh=0
(3.14)
=λ(t)Spec(ρt)(1+h)λ(t)hλ̇(t)h=0
(3.15)
=λ(t)0λ̇(t).
(3.16)
The second equality follows from Ref. 47, Theorem 7.17, due to the uniform convergence of λ(t)Spec(ρt)λ̇(t) on some neighborhood of t. To obtain the last equality, we use the following fact: since λ(t) ≥ 0 for all t and λ(t) is differentiable, if λ(t*) = 0 for some time t = t*∈ (0, ), then λ̇(t*)=0. From (3.8) and (3.16), we obtain
Tr{Πtρ̇t}=λ(t)0λ̇(t)=ddtλ(t)0λ(t)=ddtTr{ρt}=0,
(3.17)
where Πt is the projection onto the support of ρt. The second equality holds because λ̇(t*)=0 whenever λ(t*) = 0 for all λ(t*)Spec(ρt*) and all t*∈ (0, ).
Employing (A4), we find
ddhddts(t,h)=ddhh+1Tr{ρthρ̇t}
(3.18)
=Tr{ρthρ̇t}+h+1Trρthlogρtρ̇t.
(3.19)
Therefore,
ddtS(ρt)=ddtddhs(t,h)h=0
(3.20)
=ddhddts(t,h)h=0
(3.21)
=Tr{Πtρ̇t}+Trρ̇tΠtlogρt
(3.22)
=Tr{ρ̇tlogρt},
(3.23)
where to obtain the last equality we used (3.17) and the fact that logρt is defined on supp(ρt). This concludes the proof.

As an immediate application of Theorem 1, consider a closed system consisting of a system of interest A and a bath (environment) system E in a pure state ψAE, for which the time evolution is given by a unitary UAE. Under unitary evolution, the entropy of the composite system AE does not change. Also, for a pure state, the entropy of the composite system is zero, and S(ρA) = S(ρE), where ρA and ρE are the reduced states of the systems A and E, respectively. Now, it is often of interest to determine the amount of entanglement in the reduced state ρA of the system A. Several measures of entanglement have been proposed,48 among which the entanglement of formation,50,51 the distillable entanglement,50,52 and the relative entropy of entanglement53,54 all reduce to the entropy S(ρA) of the system A in the case of a closed bipartite system.55 Thus, in this case, the rate of entropy change of the system A is equal to the rate of entanglement change (with respect to the aforementioned entanglement measures) caused by unitary time evolution of the pure state of the composite system, and Theorem 1 provides a general expression for this rate of entanglement change.

In  Appendix B, we discuss how (3.3) generalizes the development in Refs. 31 and 45. We consider examples of dynamical processes in which the support and/or the rank of the state change with time, but the formula (3.3) is still applicable according to the above theorem.

The dynamics of an open quantum system can be categorized into two broad classes, quantum Markov processes and quantum non-Markov processes, based on whether the evolution process exhibits memoryless behavior or has memory effects.

Here, we consider the dynamics of an open quantum system in the time interval I=[t1,t2)R for t1 < t2. We assume that the state ρtD(H) of the system at time tI satisfies the following differential master equation:

ρ̇t=Lt(ρt)tI,
(4.1)

where Lt is called the generator,56 or Liouvillian, of the dynamics and can in general be time-dependent.57 A state ρeq is called a fixed point, or an invariant state of the dynamics, if ρ̇eq=0, or

Lt(ρeq)=0,tI.
(4.2)

In general, the evolution of systems governed by the master equation (4.1) is given by the two-parameter family {Mt,s}t,sI of maps Mt,s:B(H)B(H) defined by18 

Mt,s=TexpstLτ dτ,  s,tI,st,Mt,t=id,  tI,
(4.3)

where T is the time-ordering operator so that the state ρt of the system at time t is obtained from the state of the system at time st as ρt=Mt,s(ρs). The maps {Mt,s}ts satisfy the following composition law:

srt:  Mt,s=Mt,rMr,s,
(4.4)
tI:  Mt,t=id,
(4.5)

and in terms of these maps, the generator Lt is given by

Lt=limε0+Mt+ε,tidε.
(4.6)

For the maps {Mt,s}ts to represent physical evolution, they must be trace-preserving. This implies that for all ρD(H), the generator Lt has to satisfy

TrLt(ρ)=0,tI.
(4.7)

When the intermediate maps Mt,r and Mr,s are positive and trace-preserving for all srt, the condition (4.4) is called P-divisibility. If the intermediate maps Mt,r and Mr,s are CPTP (i.e., quantum channels) for all srt, the condition (4.4) is called CP-divisibility.58,59 Based on the notion of CP-divisibility, we have the following definition of a quantum Markov process.

Definition 5

(Quantum Markov process60). The dynamics of a system in a time interval I are called a quantum Markov process when they are governed by (4.1) and they are CP-divisible (i.e., the intermediate maps in (4.4) are CPTP).

An important fact is that the dynamics governed by the master equation (4.1) are CP-divisible (hence Markovian) if and only if the generator Lt of the dynamics has the Lindblad form
Lt(ρ)=ι[H(t),ρ]+iγi(t)Ai(t)ρAi(t)12Ai(t)Ai(t),ρ,
(4.8)
with H(t) being a self-adjoint operator and γi(t) ≥ 0 for all i and for all tI. The operators Ai(t) are called Lindblad operators. In the time-independent case, this result was independently obtained by Gorini et al.61 for finite-dimensional systems and by Lindblad62 for infinite-dimensional systems. For a proof of this result in the time-dependent scenario, see Refs. 18 and 24. In finite dimensions, necessary and sufficient conditions for Lt to be written in the Lindblad form have been given in Ref. 63. It should be noted that in general, for some physical processes, γi(t) can be temporarily negative for some i and the overall evolution still CPTP.64,65
Given the generator Lt of the dynamics (4.1) and the corresponding positive trace-preserving maps {Ms,t}s,tI, it holds that the adjoint maps {Ms,t}s,tI are positive and unital. Furthermore, the adjoint maps {Ms,t}s,tI are generated by Lt, where Lt is the adjoint of Lt. The Lindblad form (4.8) of the generator Lt is
Lt(X)=ι[H(t),X]+iγi(t)Ai(t)XAi(t)12X,Ai(t)Ai(t).
(4.9)
Now, let us consider the rate of entropy change ddtS(ρt) of a system in state ρt at time t evolving under dynamics with Liouvillian Lt. Theorem 1 implies the following equality:
ddtS(ρt)=TrLt(ρt)logρt,tI.
(4.10)

We now derive a limitation on the rate of entropy change of quantum Markov processes using the lower bound in Lemma 4 on entropy change.

Theorem 2
(Lower limit on the rate of entropy change). The rate of entropy change of any quantum Markov process (Definition 5) is lower bounded as
ddtS(ρt)limε0+ddεTrΠt(Mt+ε,t)Mt+ε,t(ρt)=TrΠtLt(ρt),
(4.11)
where Πtis the projection onto the support of the state ρtof a system. In general, (4.11) also holds for dynamics that obey (4.1) and are P-divisible.

Proof.
First, since the system is governed by (4.1), we have that ρt+ε=Mt+ε,t(ρt) for any ε > 0. Also, since Mt+ε,t is CPTP (hence positive and trace-preserving), we can use Lemma 4 to get that
S(Mt+ε,t(ρt))S(ρt)Dρt(Mt+ε,t)Mt+ε,t(ρt).
(4.12)
Therefore, by definition of the derivative, we obtain
ddtS(ρt)=limε0+S(ρt+ε)S(ρt)ε
(4.13)
limε0+1εDρt(Mt+ε,t)Mt+ε,t(ρt)
(4.14)
=limε0+S(ρt)Trρtlog(Mt+ε,t)Mt+ε,t(ρt)ε
(4.15)
=limε0+ddεTrρtlog(Mt+ε,t)Mt+ε,t(ρt)
(4.16)
=limε0+Trρtdlog(Mt+ε,t)Mt+ε,t(ρt)dε
(4.17)
=limε0+ddεTrΠt(Mt+ε,t)Mt+ε,t(ρt),
(4.18)
where we used the definition of the derivative to get (4.16) from (4.15). From  Appendix A and noting that limε0(Mt+ε,t)Mt+ε,t(ρt)=ρt, we arrive at (4.18). Then, using the definition of the adjoint and the master equation (4.1), we get
limε0+ddεTrΠt(Mt+ε,t)Mt+ε,t(ρt)       =limε0+ddεTrMt+ε,t(Πt)Mt+ε,t(ρt)
(4.19)
       =limε0+TrddεMt+ε,t(Πt)Mt+ε,t(ρt)
(4.20)
       =limε0+TrddεMt+ε,t(Πt)Mt+ε,t(ρt)+Mt+ε,t(Πt)ddεMt+ε,t(ρt).
(4.21)
Employing (4.6) and the fact that Mt,t=id for all tI, we get
Lt=limε0+Mt+ε,tidε=limε0+ddεMt+ε,t.
(4.22)
Therefore,
limε0+ddεTrΠt(Mt+ε,t)Mt+ε,t(ρt)=TrLt(Πt)ρt+ΠtLt(ρt)
(4.23)
=TrΠtLt(ρt),
(4.24)
where we used the fact (3.17) that Tr{ΠtLt(ρt)}=TrΠtρ̇t=0.

Quantum dynamics obeying (4.1) are unital in a time interval I if Lt(1)=0 for all tI, which implies that Tr{ΠtLt(ρt)}=0 for any initial state ρ0 and for all tI. The deviation of Tr{ΠtLt(ρt)} from zero is therefore a witness of non-unitality at time t. One can find the maximum deviation of Tr{ΠtLt(ρt)} away from zero by maximizing over all possible initial states and over states at any time tI to obtain a measure of non-unitality.

Remark 3.
When ρt > 0, the rate of entropy change of any quantum Markov process is lower bounded as
ddtS(ρt)limε0ddεTr(Mt+ε,t)Mt+ε,t(ρt)=TrLt(ρt).
(4.25)

Given a quantum Markov process and a state described by a density operator ρt > 0 that is not a fixed (invariant) state of the dynamics, we can make the following statements for tI and for all ε > 0 such that [t, t + ε) ⊂ I:

  • If Mt+ε,t is strictly sub-unital, i.e., Mt+ε,t(1)<1, then its adjoint is trace non-increasing, which means that Tr{Lt(ρt)}<0. This implies that the rate of entropy change is strictly positive for strictly sub-unital Markovian dynamics.

  • If Mt+ε,t is unital, i.e., Mt+ε,t(1)=1, then its adjoint is trace-preserving, which means that Tr{Lt(ρt)}=0. This implies that the rate of entropy change is non-negative for unital Markovian dynamics.

  • If Mt+ε,t is strictly super-unital, i.e., Mt+ε,ε(1)>1, then its adjoint is trace-increasing, which means that Tr{Lt(ρt)}>0. This implies that it is possible for the rate of entropy change to be negative for strictly super-unital Markovian dynamics.

Using the Lindblad form of Lt in (4.9), we find that
Tr{Lt(ρt)}=iγi(t)Ai(t),Ai(t)ρt,
(4.26)
where ⟨Aρ = Tr{}. Using this expression, the lower bound on the rate of entropy change for quantum Markov processes when the state ρt > 0 is
ddtS(ρt)iγi(t)Ai(t),Ai(t)ρt.
(4.27)
The inequality (4.27) was first derived in Ref. 66 and recently discussed in Ref. 67.

When the generator LtL is time-independent and I = [0, ), it holds that the time evolution from time sI to time tI is determined merely by the time difference ts, that is, Mt,s=Mts,0 for all ts. The evolution of the system is then determined by a one-parameter semi-group. We let Mt:=Mt,0 for all t ≥ 0.

Remark 4.
If the dynamics of a system are unital and can be represented by a one-parameter semi-group{Mt}t0of quantum channels such that the generatorLis self-adjoint, then for ρ0 > 0,
Tr{ρ0logρ2t}S(ρt)Tr{ρ2tlogρ0}.
(4.28)
This follows from Lemma 2, (2.1), and the fact thatMt=Mt. In particular,
S(ρt)=S(Mt(ρ0))=Tr{Mt(ρ0)logMt(ρ0)}Tr{Mt(ρ0)Mt(logρ0)}
(4.29)
=Tr{MtMt(ρ0)logρ0}
(4.30)
=Tr{ρ2tlogρ0}.
(4.31)
Similarly,
S(ρt)=S(Mt(ρ0))=Tr{Mt(ρ0)logMt(ρ0)}=Tr{ρ0Mt(logMt(ρ0))}
(4.32)
Tr{ρ0log(MtMt(ρ0))}
(4.33)
=Tr{ρ0logρ2t}.
(4.34)

Remark 5.
If the dynamics of a system are unital and can be represented by a one-parameter semi-group{Mt}t0of quantum channels such that the generatorLis self-adjoint, then the entropy change is lower bounded as
S(ρt)S(ρ0)D(ρ0ρ2t).
(4.35)
This follows using Lemma 4. Under certain assumptions, when the dynamics of a system are described by Davies maps,68the same lower bound (4.35) holds for the entropy change.28 

From the above remark, we see that the entropy change in a time interval [0, t] is lower bounded by the relative entropy between the initial state ρ0 and the evolved state ρ2t after time 2t. In the context of information theory, the relative entropy has an operational meaning as the optimal type-II error exponent (in the asymptotic limit) in asymmetric quantum hypothesis testing.69,70 The entropy change in the time interval [0, t] is thus an upper bound on the optimal type-II error exponent, where ρ0 is the null hypothesis and ρ2t is the alternate hypothesis.

Let us consider Gaussian dynamics that can be represented by the one-parameter family {Gt}t0 of phase-insensitive bosonic Gaussian channels Gt (cf. Ref. 71). It is known that all phase-insensitive gauge-covariant single-mode bosonic Gaussian channels form a one-parameter semi-group.72 The Liouvillian for such Gaussian dynamics is time-independent and has the following form:

L=γ+L++γL,
(4.36)

where

L+(ρ)=âρâ12ââ,ρ,
(4.37)
L(ρ)=âρâ12ââ,ρ,
(4.38)

â is the field-mode annihilation operator of the system, and the following commutation relation holds for bosonic systems:

â,â=1.
(4.39)

The state ρt of the system at time t is

ρt=Gt(ρ0)=etL(ρ0).
(4.40)

The thermal state ρth(N) with mean photon number N is defined as

ρth(N):=1N+1n=0NN+1n|nn|,
(4.41)

where N ≥ 0 and {|n}n0 is the orthonormal, photonic number-state basis. Using (4.26), we have

Tr{L(ρt)}=γ+â,âρtγâ,âρt
(4.42)
=γ+γ.
(4.43)

Therefore, by Remark 3, we find that if ρt > 0, then

dS(Gt(ρ0))dtγ+γ.
(4.44)

The lower bound γ+γ is a witness of non-unitality. It is positive for strictly sub-unital, zero for unital, and negative for strictly super-unital dynamics. For example, when the dynamics are represented by a family {At}t0 of noisy amplifier channels At with thermal noise ρth(N), we have γ+ = N + 1 and γ = N, which implies that the dynamics are strictly sub-unital. When the dynamics are represented by a family {Bt}t0 of lossy channels Bt (i.e., beam splitters) with thermal noise ρth(N), we have γ+ = N, γ = N + 1, which implies that the dynamics are strictly super-unital. When the dynamics are represented by a family {Ct}t0 of additive Gaussian noise channels Ct, we have γ+ = γ, which implies that the dynamics are unital.

Dynamics of a quantum system that are not a quantum Markov process as stated in Definition 5 are called a quantum non-Markov process. Among these two classes of quantum dynamics, non-Markov processes are not well understood and have attracted increased focus over the past decade. Some examples of applications of quantum Markov processes are in the fields of quantum optics, semiconductors in condensed matter physics, the quantum mechanical description of Brownian motion, whereas some examples where quantum non-Markov processes have been applied are in the study of a damped harmonic oscillator, or a damped driven two-level atom.17–19 

There can be several tests derived from the properties of quantum Markov processes, the satisfaction of which gives witnesses of non-Markovianity. Based on Theorem 2, we mention here a few tests that will always fail for a quantum Markov process. Passing of these tests guarantees that the dynamics are non-Markovian.

An immediate consequence of Theorem 2 is that only a quantum non-Markov process can pass any of the following tests:

  • ddtS(ρt)+limε0+ddεTrΠt(Mt+ε,t)Mt+ε,t(ρt)<0,
    (5.1)
  • ddtS(ρt)+TrΠtLt(ρt)<0,
    (5.2)
  • limε0+ddεTrΠt(Mt+ε,t)Mt+ε,t(ρt)TrΠtLt(ρt).
    (5.3)

If the dynamics of the system satisfy any of the above tests, then the process is non-Markovian. Based on the description of the dynamics and the state of the system, one can choose which test to apply. In the case of unital dynamics, (5.1) and (5.2) reduce to ddtS(ρt)<0. The observation that the negativity of the rate of entropy change is a witness of non-Markovianity for random unitary processes, which are a particular kind of unital processes, was made in Ref. 73.

Based on the above witnesses of non-Markovianity, we can introduce different measures of non-Markovianity for physical processes. Here, we introduce two measures of non-Markovianity that are based on the channel and generator representation of the dynamics of the system:

  1. M(L)maxρ0t:dS(ρt)dt+TrΠtLt(ρt)<0dS(ρt)dt+TrΠtLt(ρt),
    (5.4)
  2. M(M)maxρ0t:f(t)<0f(t),
    (5.5)
    where
    f(t):=ddtS(ρt)+limε0+ddεTrΠt(Mt+ε,t)Mt+ε,t(ρt).
    (5.6)

In the case of unital dynamics, the above measures are equal. It should be noted that the above measures of non-Markovianity are not faithful. This is due to the fact that the statements in Theorem 2 do not provide sufficient conditions for the evolution to be a quantum Markov process. In other words, if the measure ∁M (5.4) is non-zero, then the dynamics are non-Markovian, but if it is equal to zero, then that does not in general imply that the dynamics are Markovian.

In this section, we consider two common examples of quantum non-Markov processes: pure decoherence of a qubit system (Sec. V A 1) and a generalized amplitude damping channel (Sec. V A 2). In order to characterize quantum dynamics, several witnesses of non-Markovianity and measures of non-Markovianity based on these witnesses have been proposed.34,60,63,65,73–81 Many of these measures are based on the fact that certain quantities are monotone under Markovian dynamics, such as the trace distance between states,34 entanglement measures,60,75,76 Fisher information and Bures distance,74,77,78 and the volume of states.79 Among these measures, the one proposed in Ref. 60 based on the Choi representation of dynamics is both necessary and sufficient. The measure proposed in Ref. 65 is also necessary and sufficient and is based on the values of the decay rates γi(t) appearing in the Lindblad form (4.8) of the Liouvillian of the dynamics.

Here, we compare our measures of non-Markovianity with the widely considered Breuer-Laine-Piilo (BLP) measure of non-Markovianity.34 This is a measure of non-Markovianity defined using the trace distance and is based on the fact that the trace distance is monotonically non-increasing under quantum channels. Our measure agrees with the BLP measure in the case of pure decoherence of a qubit. In the case of the generalized amplitude damping channel, our witness is able to detect non-Markovianity even when the BLP measure does not.

1. Pure decoherence of a qubit system

Consider a two-level system with ground state and excited state +. We allow this qubit system to interact with a bosonic environment that is a reservoir of field modes. The time evolution of the qubit system is given by

dρtdt=ι[H(t),ρt]+γ(t)σρtσ+12{σ+σ,ρt},
(5.7)

where σ+=|+|, σ=|+| and t ≥ 0. If H(t) = 0, then the system undergoes pure decoherence and the Liouvillian reduces to

Lt(ρt)=γ(t)2(σzρtσzρt),
(5.8)

where σz = [σ+, σ]. The decoherence rate is given by γ(t), and it can be determined by the spectral density of the reservoir.34 

One can verify that Tr{ΠtLt(ρt)}=0 for all t ≥ 0 and any initial state ρ0. This implies that the dynamics are unital for all t ≥ 0. In this case, for t > 0, our witness (5.2) reduces to ddtS(ρt)<0. For qubit systems undergoing the given unital evolution, it holds that ρt > 0 for all t > 0, and thus for t > 0, our measures (5.4) and (5.5) are equal and reduce to the measure in Ref. 80, Eq. (15), which was based on the fact that the rate of entropy change is non-negative for unital quantum channels. As stated therein, these measures of non-Markovianity are positive and agree with those obtained by the BLP measure [Ref. 34, Eq. (11)].

2. Generalized amplitude damping channel

In this example, we consider non-unital dynamics that can be represented as a family of generalized amplitude damping channels {Mt}t0 on a two-level system.78 These channels have Kraus operators82 

Mt1=pt100ηt,Mt2=pt01ηt00,Mt3=1ptηt001,Mt4=1pt001ηt0,
(5.9)

where pt = cos2(ωt), ωR, and ηt = et. Then, for all t ≥ 0, Mt(ρ)=i=14Mtiρ(Mti). Mt is unital if and only if pt=12 or ηt = 1. When ηt = 1, Mt=id for all ω.

It was shown in Ref. 78 that the BLP measure34 does not capture the non-Markovianity of the dynamics given by (5.9).

Let the initial state ρ0 be maximally mixed, that is, ρ0=121. The evolution of this state under Mt is then

ρt:=Mt(ρ0)=121+Wt001Wt,
(5.10)

where Wt = (2pt − 1)(1 − ηt) = cos(2ωt)(1 − et). Note that ρt > 0 for all t ≥ 0. The evolution of these states for an ε > 0 time interval is

ρt+ε=Mε(ρt)=121+Wε+ηεWt001WεηεWt.
(5.11)

To check whether or not the given dynamics are non-Markovian, let us apply the test in (5.1). First, we evaluate

MεMε(ρt)=12at00bt,
(5.12)

where

at:=pt(1+Wε+ηεWt)+(1pt)ηt(1+Wε+ηεWt)+(1pt)(1ηt)(1Wε+ηεWt),
(5.13)
bt:=ptηt(1Wε+ηεWt)+pt(1ηt)(1+Wε+ηεWt)+(1pt)(1Wε+ηεWt).
(5.14)

Then,

limε0+ddεTrMεMε(ρt)=Wt.
(5.15)

It should be noted that the deviation of Wt from zero is a witness of non-unitality. For a unital process, for any initial state ρ0 and for all time t, we should have limε0+ddεTrΠtMεMε(ρt)=0. For a non-unital process, there will exist some initial state such that for some time t, limε0+ddεTrΠtMεMε(ρt)0. Next, we evaluate the entropy of the state ρt to be

S(ρt)=12(1+Wt)log1+Wt2+(1Wt)log1Wt2.
(5.16)

This implies that the rate of entropy change is

dS(ρt)dt=12dWtdtlog1+Wt2+12dWtdtlog1Wt2
(5.17)
=12dWtdtlog1Wt1+Wt,
(5.18)

where

dWtdt=2ωsin(2ωt)(1et)+cos(2ωt)et.
(5.19)

Therefore, the test in (5.1) reduces to

f(t)=12dWtdtlog1+Wt2+12dWtdtlog1Wt2+Wt<0,
(5.20)

where f is defined in (5.6). For values of ω such that the dynamics are non-unital, we find that f can be negative in several time intervals; for example, see Fig. 1 for the case ω = 5.

FIG. 1.

Negative values of f, as given in (5.20), indicate non-Markovianity. We have taken ω = 5.

FIG. 1.

Negative values of f, as given in (5.20), indicate non-Markovianity. We have taken ω = 5.

Close modal

In this section, we give bounds on how much the entropy of a system can change as a function of the initial state of the system and the evolution it undergoes.

Lemma 6.
LetN:B1+(H)B1+(H)be a positive, trace-non-increasing map. Then, for allρD(H)such thatN(ρ)>0,
ΔS(ρ,N)DρNNρ.
(6.1)

Proof.
Using the definition (II.1) of the adjoint, we obtain
ΔS(ρ,N)=S(N(ρ))S(ρ)=Tr{ρlogρ}TrN(ρ)logN(ρ)=Tr{ρlogρ}TrρN(logN(ρ))Tr{ρlogρ}TrρlogNN(ρ)=DρNNρ.
(6.2)
The inequality follows from Lemma 2 applied to N, which is positive and sub-unital since N is positive and trace non-increasing.

Note that for a quantum channel N, ΔS(ρ,N)=0 for all ρ if and only if ρ=NN(ρ), which is true if and only if N is a unitary operation [Ref. 83, Theorem 2.1, Ref. 18, Theorem 3.4.1]. We use this fact to provide a measure of non-unitarity in Sec. VII.

As an application of the lower bound in Lemma 4, let us suppose that a quantum channel EAB can be simulated as follows:
ρAD(HA):EAB(ρA)=FACB(ρAθC),
(6.3)
for a fixed interaction channel FACB and a fixed ancillary state θC. By applying Lemma 4 to F and the state ρAθC, we obtain
ΔS(ρA,E)=S(F(ρAθC))S(ρ)A
(6.4)
S(ρAθC)S(ρA)+DρAθCFF(ρAθC)
(6.5)
=S(θC)+DρAθCFF(ρAθC).
(6.6)
Equality holds, i.e., ΔS(ρ,E)=S(θC), if and only if the interaction channel F is a unitary interaction. If F is a sub-unital channel, then ΔS(ρ,E)S(θC) because the relative entropy term is non-negative. This result is of relevance in the context of quantum channels obeying certain symmetries (cf. Ref. 84).

Lemma 7.
LetN:B+(H)B+(H)be a sub-unital channel. Then, for allρD(H)such that ρ > 0,
ΔS(ρ,N)TrρNN(ρ)logρ.
(6.7)
This also holds for any positive sub-unital map satisfying the above conditions.

Proof.
By applying Lemma 2 to N, we get
ΔS(ρ,N)=Tr{ρlogρ}TrN(ρ)logN(ρ)Tr{ρlogρ}TrN(ρ)N(logρ)=TrρNN(ρ)logρ.
(6.8)
This concludes the proof.

By applying Hölder’s inequality (Lemma 1) to this upper bound, we obtain the following.

Corollary 8.
LetN:B+(H)B+(H)be a sub-unital channel. Then, for allρD(H)such that ρ > 0,
ΔS(ρ,N)ρNN(ρ)1logρ.
(6.9)
Now, if we let N be a sub-unital quantum operation, then as a consequence of Lemma 4 and Corollary 8, we have, for all states ρ > 0 such that N(ρ)>0 and the entropies S(ρ) and S(N(ρ)) are finite,
D(ρNNρ)S(N(ρ))S(ρ)ρNN(ρ)1logρ.
(6.10)
It is interesting to note that (6.10) implies
ρNN(ρ)11logρD(ρNNρ)
(6.11)
for a sub-unital quantum operation N and a state ρ > 0 such that N(ρ)>0. This inequality has the reverse form of Pinsker’s inequality,85 which in this case is
D(ρNNρ)12ρNN(ρ)12.
(6.12)
In general, the relationship between relative entropy and different distance measures, including trace distance, has been studied in Refs. 86–88.

In this section, we introduce a measure of non-unitarity for any unital quantum channel that is inspired by the discussion at the end of Sec. VI. A measure of unitarity for channels N:D(HA)D(HA), where HA is finite-dimensional, was defined in Ref. 89. A related measure for non-isometricity for sub-unital channels was introduced in Ref. 27. A measure of non-unitarity for a unital channel is a quantity that gives the distinguishability between a given unital channel with respect to any unitary operation. It quantifies the deviation of a given unital evolution from a unitary evolution. These measures are relevant in the context of cryptographic applications90,91 and randomized benchmarking.89 

We know that any unitary evolution is reversible. The adjoint of a unitary operator is also a unitary operator, and a unitary operator and its adjoint are the inverse of each other. These are the distinct properties of any unitary operation. Let us denote a unitary operator by UAB, where dim(HA)=dim(HB). Then a necessary condition for the unitarity of UAB is (UAB)UAB=1A. The unitary evolution UAB of a quantum state ρA is given by

UAB(ρA)=UAB(ρA)(UAB).
(7.1)

From the reversibility property of a unitary evolution, it holds that (UAB)UAB=idA. It is clear that (UAB) is also a unitary evolution, and (UAB) and UAB are the inverse of each other.

Contingent upon the above observation, we note that a measure of non-unitarity for a unital channel NAB should quantify the deviation of (NAB)NAB from idA and is desired to be a non-negative quantity. We make use of the trace distance, which gives a distinguishability measure between two positive semi-definite operators and appears in the upper bound94 on entropy change for a unital channel (Sec. VI), to define a measure of non-unitarity for a unital channel called the diamond norm of non-unitarity.

Definition 9
(Diamond norm of non-unitarity). The diamond norm of non-unitarity of a unital channelNABis a measure that quantifies the deviation of a given unital evolution from a unitary evolution and is defined as
N=idNN,
(7.2)
where the diamond norm92of a Hermiticity-preserving mapMis defined as
M=maxρRAD(HRA)(idM)(ρRA)1.
(7.3)
In other words,
N=maxρRAD(RA)(id(idNN))(ρRA)1.
(7.4)

The diamond norm of non-unitarity of any unital channel N has the following properties:

  1. N0.

  2. N=0 if and only if NN=id, i.e., the unital channel N is unitary.

  3. In (7.4), it suffices to take ρRA to be rank one and to let dim(HR)=dim(HA).

  4. N2.

Once we note that NN:D(HA)D(HA) is a quantum channel, properties 1, 3, and 4 are direct consequences of the properties of the diamond norm.93 For property 3, the reference system R has to be comparable with the channel input system A, following from the Schmidt decomposition. So HR should be countably infinite if HA is. Property 2 follows from Ref. 83, Theorem 2.1, and Ref. 18, Theorem 3.4.1.

The diamond norm has an operational interpretation in terms of channel discrimination93,95 (see also Refs. 96 and 97 for state discrimination). Specifically, the optimal success probability psucc(N1,N2) of distinguishing between two channels N1 and N2 is
psucc(N1,N2):=121+12N1N2.
(7.5)
It follows that the optimal success probability of distinguishing between the identity channel and NN is
psucc(id,NN)=121+12idNN
(7.6)
=121+12N.
(7.7)

Proposition 10.
LetN:D(H)D(H)be a unital channel. If there exists a unitary operatorUB(H)such that
NUδ,
(7.8)
whereU:D(H)D(H)is the unitary evolution (7.1) associated with U, thenN2δ+δ.

Proof.
We have that
idNN=idNU+NUNN
(7.9)
idNU+N(UN)
(7.10)
idNU+δ.
(7.11)
To obtain these inequalities, we have used the following properties of the diamond norm:93 
  1. Triangle inequality: N1+N2N1+N2.

  2. Sub-multiplicativity: N1N2N1N2.

  3. For all channels M, M=1.

In particular, to use the third fact, we observe that N is a channel since N is unital. We have also made use of our assumption that UNδ.
Now, from the assumption UNδ, it follows by unitary invariance of the diamond norm that
idUNδ.
(7.12)
By the operational interpretation of the diamond distance, this means that the success probability of distinguishing the channel UN from the identity channel, using any scheme whatsoever, cannot exceed psucc(id,UN) as defined in (7.5). In other words, the success probability cannot exceed 121+12δ. One such scheme is to send in a bipartite state |ψRA on a reference system R and the system A on which the channel acts and perform the measurement defined by the positive operator-valued measure {|ψψ|RA,1RA|ψψ|RA}. If the outcome of the measurement is |ψψ|RA, then we guess that the channel is the identity channel, and if the outcome of the measurement is 1RA|ψψ|RA then we guess that the channel is UN. The success probability of this scheme is
     12Tr{|ψψ|RAidRA(|ψψ|RA)}     +Tr{1RA|ψψ|RAidR(UN)A(|ψψ|RA)}
(7.13)
     =122ψ|RAidR(UN)A(|ψψ|RA)|ψRA.
(7.14)
By employing the above, we find that
122ψ|RAidR(UN)A(|ψψ|RA)|ψRA121+12δ
(7.15)
ψ|RAidR(UN)A(|ψψ|RA)|ψRA112δ.
(7.16)
By employing the definition of the channel adjoint, we find that
ψ|RAidR(UN)A(|ψψ|RA)|ψRA  =ψ|RAidR(NU)A(|ψψ|RA)|ψRA112δ.
(7.17)
This holds for all input states, so we can conclude that the following inequality holds:
minψRAψ|RAidR(NU)A(|ψψ|RA)|ψRA112δ.
(7.18)
Now, by the definition (7.3) of the diamond norm and the fact that it suffices to take the maximization in the definition of the diamond norm over only pure states, we have
idNU=maxψRAidR(idNU)A(|ψψ|RA)1.
(7.19)
By the Fuchs-van de Graaf inequality,100 we obtain
idR(idNU)A(|ψψ|RA)1
(7.20)
=|ψψ|RAidR(NU)A(|ψψ|RA)1
(7.21)
21ψ|RAidR(NU)A(|ψψ|RA)|ψRA.
(7.22)
It follows that
idNU21minψRAψ|RAidR(NU)A(|ψψ|RA)|ψRA.
(7.23)
Using (7.18), we therefore obtain
idNU212δ=2δ.
(7.24)
Finally, from (7.11), we arrive at
idNN2δ+δ,
(7.25)
as required.
Let us now quantify the non-unitarity of the qubit depolarizing channel Dd,q defined as98 
Dd,q(ρ)=(1q)ρ+q1d1,ρD(HA),
(7.26)
where dim(HA)=d and q0,d2d21. The input state ρ remains invariant with probability (11d2)q under the action of Dd,q.

Proposition 11.
For the depolarizing channelDd,q, the diamond norm of non-unitarity is
Dd,q=2q(2q)11d2.
(7.27)

Proof.

The result follows directly from Ref. 99, Sec. V.A, but here we provide an alternative proof argument that holds for more general classes of channels.

The depolarizing channel is self-adjoint, that is, Dd,q=Dd,q for all q, which means that Dd,qDd,q=Dd,q2=Dd,2qq2. Therefore,
Dd,q=idDd,q2=2qq2maxψAAψAAψA1d1,
(7.28)
where ψAA=|ψψ|AA is a pure state and dim(HA)=dim(HA)=d.

The identity channel and the depolarizing channel are jointly teleportation-simulable [Ref. 84, Definition 6] with respect to the resource states, which in this case are the respective Choi states (because these channels are also jointly covariant [Ref. 84, Definitions 7 and 12]). We know the trace distance is monotonically non-increasing under the action of any channel. Therefore, we can conclude from the form [Ref. 84, Eq. (3.2)] of the action of jointly teleportation-simulable channels that the diamond norm between any two jointly teleportation-simulable channels is upper bounded by the trace distance between the associated resource states.

Since dim(HA) is finite, the maximally entangled state |ΦAA:=1di=1d|i|i, where {|i}i=1d is any orthonormal basis in HA, is an optimal state in (7.28). It is known that
1d1d=1d2x=0d21σAxΦAAσAx,
(7.29)
where {σAxΦAAσAx}x=0d21 forms an orthonormal basis for HAHA and {σx}x=0d21 forms the Heisenberg-Weyl group. We denote the identity element in {σx}x=0d21 by σ0. Using this, we get
Dd,q=(2qq2)ΦAA1d1d1
(7.30)
=(2qq2)11d2ΦAA1d2x=1d21σAxΦAAσAx1
(7.31)
=(2qq2)11d2+d21d2
(7.32)
=2q(2q)11d2.
(7.33)
We conclude that Dd,q=2q(2q)11d2. See Fig. 2 for a plot of D2,q as a function of q.

FIG. 2.

The measure D2,q of non-unitarity for the qubit depolarizing channel D2,q as a function of the parameter q0,43.

FIG. 2.

The measure D2,q of non-unitarity for the qubit depolarizing channel D2,q as a function of the parameter q0,43.

Close modal

In this paper, we discussed the rate of entropy change of a system undergoing time evolution for arbitrary states and proved that the formula derived in Ref. 31 holds for both finite- and infinite-dimensional systems undergoing arbitrary dynamics with states of arbitrary rank. We derived a lower limit on the rate of entropy change for any quantum Markov process. We discussed the implications of this lower limit in the context of bosonic Gaussian dynamics. From this lower limit, we also obtained several witnesses of non-Markovianity, which we used in two common examples of non-Markovian dynamics. Interestingly, in one example, our witness turned out to be useful in detecting non-Markovianity. We generalized the class of operations for which the entropy exhibits monotonic behavior. We also provided a measure of non-unitarity based on bounds on the entropy change, discussed its properties, and evaluated it for the depolarizing channel.

We thank Francesco Buscemi, Nilanjana Datta, Jonathan P. Dowling, Omar Fawzi, Eric P. Hanson, Kevin Valson Jacob, Felix Leditzky, Milán Mosonyi, A. R. P. Rau, Cambyse Rouzé, Punya Plaban Satpathy, and Mihir Sheth for insightful discussions. We are also grateful to Stanislaw Szarek for noticing an error in our previous justification for Proposition 10. S.D. acknowledges support from the LSU Graduate School Economic Development Assistantship. S.K. acknowledges support from the LSU Department of Physics and Astronomy. G.S. acknowledges support from the U.S. Office of Naval Research under Award No. N00014-15-1-2646. M.M.W. acknowledges support from the U.S. Office of Naval Research and the National Science Foundation.

In this section, we recall Ref. 42, Theorem V.3.3.

If f is a continuously differentiable function on an open neighborhood of the spectrum of some self-adjoint operator A, then its derivative Df(A) at A is a linear superoperator and its action on an operator H is given by

Df(A)(H)=λ,ηf[1](λ,η)PA(λ)HPA(η),
(A1)

where A = λλPA(λ) is the spectral decomposition of A and f[1] is the first divided difference function.

If tA(t)B+(H) is a continuously differentiable function on an open interval in R, with derivative A:=dAdt, then

f(A(t)):=ddtf(A(t))=Df(A)(A(t))=λ,ηf[1](λ,η)PA(t)(λ)A(t)PA(t)(η).
(A2)

In particular, (A2) implies the following:

ddtTr{f(A(t))}=Tr{f(A(t))A(t)},
(A3)
TrB(t)f(A(t))=Tr{B(t)f(A(t))A(t)},
(A4)

where B(t) commutes with A(t).

Here, we continue the discussion from Sec. III. We discuss the subtleties involved in determining the rate of entropy change using the formula (3.3) (Theorem 1) by considering some examples of dynamical processes.

Let us first consider a system in a pure state ψt undergoing a unitary time evolution. In this case, the entropy is zero for all time, and thus the rate of entropy change is also zero for all time. Note that even though the rank of the state remains the same for all time, the support changes. This implies that the kernel changes with time. However, ψ̇t is well defined. This allows us to invoke Theorem 1, so the formula (3.3) is applicable.

Formula (3.3) is also applicable to states with higher rank whose kernel changes in time and have non-zero entropy. For example, consider the density operator ρtD(H) with the following time-dependence:

t0:ρt=iIλi(t)Ui(t)Πi(0)Ui(t),
(B1)

where I={i:1id,d<dim(H)}, iIλi(t)=1, λi(t) ≥ 0 and the time-derivative λ̇i(t) of λi(t) is well defined for all iI. The operators Ui(t) are time-dependent unitary operators associated with the eigenvalues λi(t) such that the time-derivative U̇i(t) of Ui(t) is well defined and [Ui(0), Πi(0)] = 0 for all iI. The operators Πi(0) are projection operators associated with the eigenvalues λi(0) such that the spectral decomposition of ρt at t = 0 is

ρ0=iIλi(0)Πi(0),
(B2)

where 1<rank(ρ0)<dim(H). The evolution of the system is such that rank(ρt) = rank(ρ0) for all t ≥ 0. It is clear from (B1) and (B2) that the projection Πt onto the support of ρt depends on time,

Πt=iIUi(t)Πi(0)Ui(t),
(B3)

and the time-derivative Π̇t of Πt is well defined. The entropy of the system is zero if and only if the state is pure.

Let us consider a qubit system A undergoing a damping process such that its state ρt at any time t ≥ 0 is as follows:

ρt=(1et)|00|+et|11|,
(B4)

where {|0,|1} is a fixed orthonormal basis of HA. The entropy S(ρt) of the system at time t is

S(ρt)=(1et)log(1et)etlog(et),
(B5)

which is continuously differentiable for all t > 0 and not differentiable at t = 0. At t = 0, Π0=|11| and rank(ρ0) = 1. At t = 0+, there is a jump in the rank from 1 to 2, and the rank and the support remains the same for all t ∈ (0, ). In this case, the formula (3.3) agrees with the derivative of (B5).

Now, suppose that the system A undergoes an oscillatory process such that for any time t ≥ 0, the state ρt of the system is given by

ρt=cos2(πt)|00|+sin2(πt)|11|.
(B6)

In this case, for all t ≥ 0, the entropy S(ρt) is

S(ρt)=cos2(πt)logcos2(πt)sin2(πt)logsin2(πt),
(B7)

and its derivative is

ddtS(ρt)=πsin(2πt)logcos2(πt)logsin2(πt),
(B8)

which exists for all t ≥ 0. At t=n2 for all nZ+{0}, there is a jump in the rank from 1 to 2 and the support changes discontinuously at these instants. One can check that (3.3) and (B8) are in agreement for all t ≥ 0.

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