1. IntroductionThe dynamical behavior of open quantum systems plays a core role in quantum mechanics and quantum information.[1–11] In the open system, the prototype of a Markovian process is given by the solution of a master equation for the reduced density matrix with Lindblad structure.[12,13] However, in most realistic physical systems, Markovian time evolution cannot predict many dynamical processes.[1–4,12,14–17] This is general because the relevant environmental correlation times are not smaller than the systems’ relaxation or decoherence time, making the Markov process impossible;[1] for example, in the cases of strong system–environment couplings, structured or finite reservoirs, low temperatures, or large initial system environment correlations. For these results, a lot of research has focused on the development of analysis and quantification of the non-Markovian process. In fact, both Markovian and non-Markovian processes have been well studied in the classical stochastic process. While the quantum non-Markovian process cannot be directly analogous to classical stochastic processes, quantum extensions of non-Markovianity remain elusive and subtle.
Based on the different motivations, several witnesses and measures of the non-Markovian process have been proposed. Riva et al. proposed two non-Markovian measures based on the divisibility of the dynamical map and entanglement of environment.[18] Based on the trace distance norm, Breuer et al. studied a non-Markovian measure in terms of the increasing of distinguishability between different evolving states.[2] Luo et al. introduced a non-Markovian measure in terms of mutual information.[19] Lu et al. studied non-Markovianity by the quantum Fisher information flow.[20] He et al. introduced an alternative method based on local quantum uncertainty to measure non-Markovianity.[21] All these measures above do not coincide exactly in quantifying non-Markovian processes, although there are some coincided instances. At present, exploring the hierarchy of these measures and finding a universal definition of non-Markovianity for open quantum processes is still an important context of research.
In this paper, we investigate the quantification of non-Markovian processes via max-relative entropy.[22] Max-relative entropy has been investigated in Refs. [22–25]. The well-known (smooth) conditional and unconditional max-entropy[26,27] can be obtained from this quantity. It has been shown that max-relative entropy is of operational significance in applications ranging from data compression[28,29] to state merging[30] and security of keys.[26,31] In addition, max-relative entropy has been used to define entanglement and coherence monotones and their operational significance in the manipulation of entanglement and coherence.[22,32] Here, we find that max-relative entropy can be a witness of non-Markovian processes. The max-relative entropy quantifies the maximum probability with which a state ρ can appear in a convex decomposition of a state σ,[33] which can be seen as a “distinguishability”. The non-Markovian process implies that the increasing of distinguishability—a certain flow of information from environment to system as time increases. We also combine the results with other kinds of quantum non-Markovianity, i.e., based on mutual information, Fisher information, trace norm, and entanglement).
This paper is organized as follows. In Section 2, the witness and measure of non-Markovian processes based on max-relative entropy are introduced. In Section 3, we illustrate some examples of the new witness. We summarize our results in Section 4.
2. Quantifying the non-Markovian process via max-relative entropyWe first introduce the necessary notations. We consider a Hilbert space
of finite dimension d. Throughout this paper, we take the logarithm to base 2 and all Hilbert spaces considered are finite dimensional. Let ρ and σ be the density matrixes in the Hilbert space
. The support set supp ρ of a density operator ρ is the vector space spanned by eigenvectors of the operator with non-zero eigenvalues.
The max-relative entropy is given by[22]
Note that
is well defined if
. An equivalent definition of max-relative entropy is
where
is the maximum eigenvalue of the operator
X.
It is well known that the max-relative entropy is an upper bound of relative entropy
:[34]
The max-relative entropy is of operational significance in applications ranging from data compression[28,29] to state merging[30] and security of keys.[26,31]
A remarkable feature of max-relative entropy is that for a completely positive and trace preserving (CPTP) map
, we have
Now, we will introduce the witness and measure of the non-Markovian process. We recall here that the CPTP map
is Markovian in an open system in the sense that
for some operations
, then we have
Therefore, the max-relative entropy is a decreasing function for
, which implies
for any Markovian processes. Any violation of this monotonicity,
i.e.,
is a witness of the non-Markovian process.
We also note an interesting operational meaning for max-relative entropy, which quantifies the maximum probability with which a state ρ can appear in a convex decomposition of a state σ.[33] A non-Markovian process means the maximum probability with which ρ can appear in a convex decomposition of σ could be increased under such dynamics.
Now, we can introduce a measure for the non-Markovian process via the max-relative entropy as
where
is taken over all the states. In particular, if
,
i.e., the max-relative entropy is not well defined), then we could replace the two density operators such that the max-relative entropy is well defined.
3. Some applicationsIn this section, we explore some examples as applications for the quantification of non-Markovian processes based on the max-relative entropy.
Example 1 [Phase-damping channel]
Consider the dynamical evolution
in the case that a qubit undergoes phase-damping noises, which can be represented as the master equation
where
for completely positive dynamics. For the initial state
then the dynamics can be expressed as
with
and note that
. Let the initial states be superposition states
and
in this dynamical evolution, respectively. We have
and
The max-relative entropy of
and
is
Therefore, the witness for the non-Markovian process based on the max-relative entropy is
From the equation given above, we note that
is equivalent to
. Thus, the condition
can be a witness for the non-Markovian process.
First, from Ref. [19], we know that the witness for the non-Markovian process based on mutual information is
Second, from Ref. [35], the witness for the non-Markovian process based on Fisher information is
Third, from Ref. [2], the witness for the non-Markovian process based on the trace norm is
Finally, from Ref. [18], the witness for the non-Markovian process based on entanglement is
To compare the witness based on the max-relative entropy with other witnesses, as suggested in Ref. [36], we can set the time-dependent rate
where
γ is the free scaling parameter with
The corresponding evolution is completely positive and non-Markovian. As shown in Fig.
1, we have plotted the five kinds of witness in the quantum non-Markovian process. All of them coincide when they are witnesses for this evolution.
Example 2 [Amplitude-damping channel]
Consider the dynamical evolution
in the case that a qubit undergoes the amplitude-damping channel, which can be represented as the master equation
where
and
p(
t) is described by the following intergro–differential equation
with condition
p(0)=1. The kernel is related to the spectral density
of the reservoir via
.
For the initial state
then the dynamics can be expressed as
Let the initial states be superposition states
and
in this dynamical evolution, respectively. We have
and
Therefore
and
From the above equation, we note that
is equivalent to
with
Thus, the condition
can be a witness for the non-Markovian process.
Example 3 [Random unitary channel]
Consider the dynamical evolution
in the case that a qubit undergoes a stochastic unitary channel, which can be represented as the master equation
where
are time-dependent decaying rates and the Pauli matrix.
For a single-qubit state
with
and
. After the stochastic unitary channel, we have
with
and
for
i=1,2,3.
Now, consider that the initial states are superposition states
and
in this dynamical evolution, respectively. We have
and
The max-relative entropy of
and
is
Therefore
where
.
From the equation given above, we note that
is equivalent to
. Thus, the condition
can be a witness for the non-Markovian process.
4. Discussion and conclusionWe have investigated the quantum non-Markovian process based on max-relative entropy in open quantum systems. The max-relative entropy quantifies the maximum probability with which a state ρ can appear in a convex decomposition of a state σ. This operational interpretation provides a new view for non-Markovian processes. We also find that the max-relative entropy can be a witness of non-Markovian processes. As applications, some examples are also given and the results are combined with other kinds of quantum non-Markovianity, i.e., the witnesses based on trace norm, mutual information, entanglement, and Fisher information). For the phase-damping channel and stochastic unitary channel, the proposed non-Markovian witness is consistent with the other non-Markovian witnesses. But for the random unitary channel, not all the non-Markovian witnesses are consistent. The relation between those non-Markovian witnesses above is a worthwhile subject for study in the future.
Meanwhile, we note that the min-relative entropy of two operators ρ and σ is given[22]
where
is the projector onto supp
ρ. For a CPTP map Λ, we also have the monotonicity for the min-relative entropy
As in the analysis of max-relative entropy, the min-relative entropy is a decreasing function for
, which implies
for any Markovian process. Any violation of this monotonicity,
i.e.,
is a witness of the non-Markovian process. However, as in the examples shown in Section
3, the witness of non-Markovian processes based on the min-relative entropy will expire. This is because the excited state and the ground state we considered satisfy
. From Eq. (
25), we know that if
, then
Thus, the min-relative entropy cannot be a suitable witness of the non-Markovian process in some cases.
It was shown that the superposition states
and
are the optimal states for measuring non-Markovian processes in some cases, such as the trace norm.[2,37] Due to the hardness for optimizing the function Eq. (7), in future research we will consider studying whether the excited state
and ground state
are also the optimal states as a measure of non-Markovian processes based on the max-relative entropy of the examples.
Finally, we hope that non-Markovianity based on the max-relative entropy can be useful for future research in open quantum systems. Despite the quantity of new results in the quantification of non-Markovian processes in recent years, the relation between those non-Markovian quantifications is still a worthwhile subject for study in the future.
