Phenomenological measure of quantum non-Markovianity
PPhenomenological measure of quantum non-Markovianity
Adam Winick, Joel J. Wallman, and Joseph Emerson
1, 2 Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo, Waterloo, Canada Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
Non-Markovian effects are ubiquitous in physical quantum systems and remain a significant chal-lenge to achieving high-quality control and reliable quantum computation, but due to their inherentcomplexity, are rarely characterized. Past approaches to quantifying non-Markovianity have con-centrated on small, simple systems, and we show that such measures can capture properties thatare irrelevant to applications. With this insight, we propose a method for constructing relevantquantifiers of non-Markovian dynamics and illustrate the scheme’s computability by characterizinga model quantum dot array.
Dynamical semigroups are often applied to describethe interaction of a quantum system with its environ-ment. The corresponding Lindblad equation [1, 2] de-picts memoryless dynamics and an irreversible loss of in-formation through decoherence mechanisms. However, inmany settings, the assumptions that justify a semigroupfail, and systems exhibit memory effects that give riseto a resurgence of coherence and information flow backinto the system. These memory effects are the principalcharacteristics of non-Markovian dynamics in quantumsystems.Identifying non-Markovian effects and their respectivemagnitudes and timescales is critical to the success ofquantum computation [3], where such effects can havea significant impact on the performance of fault toler-ant quantum error correction [4, 5] and the robustnessof error characterization methods such as randomizedbenchmarking [6–10]. Characterizing these memory ef-fects is an essential first step towards their suppression.Non-Markovian effects also play a role in, for example,quantum biology [11], quantum key distribution [12], andquantum metrology [13]. Moreover, recent results showthat the manipulation of reservoir spectral properties canimprove quantum control [14, 15], and hence character-izing systems beyond Lindblad master equations is thepathway to unlock these capabilities.Quantifying non-Markovian processes in quantum sys-tems has been a standing challenge for several decades.Recent work has resolved the theoretical problem of de-scribing non-Markovian processes that reduces to its clas-sical counterpart in an appropriate limit [16, 17]. The re-quired experimental resources become practically insur-mountable for complex multipartite systems, and there isa need for so-called measures of non-Markovianity. Sev-eral approaches have been proposed to construct mea-sures of non-Markovianity based on the geometry ofstates [18, 19], the violation of CP-divisibility [20, 21],monotonicity under CP maps [3, 12, 22–26], and otherprinciples [27, 28]. To date, all proposed measures re-quire full process tomography which is unrealistic foreven moderately sized systems of a few qubits. All butone of the measures above further require an optimization that becomes intractable for systems beyond one or twoqubits. The remaining measure [20] assumes knowledgeof the dynamical map or an explicit form of the mas-ter equation and is therefore not measurable for complexsystem-environment couplings and internal environmentdynamics.In this Letter, we introduce a new approach to quantifynon-Markovianity that is directly related to how it mani-fests under common metrics of interest. We begin with areview of the interpretation and measure of the degree ofnon-Markovianity proposed by Breuer et al. [22], whichwe denote BLP below. By considering the behavior of themeasure for a toy model, we highlight some of its limi-tations. We then use this insight to define our method.A numerical experiment demonstrates that our techniquecan be used to gauge the effect of non-Markovian noise onsystems that are orders of magnitude larger than thoseanalyzed in previous studies.A key concept for our approach is that memory effectsthat appear in non-Markovian dynamics arise from anexchange of information between a system and its envi-ronment. The exchange is recognizable in several suit-able quantities, e.g., quantum relative entropy, but forconcreteness, we consider the trace distance D ( ρ , ρ )between quantum states ρ and ρ . Among the manyproperties of the trace distance, it is an operationallymeaningful quantifier of the distinguishability betweentwo states. Suppose Alice prepares a system in either thestate ρ or ρ , each with probability and sends it toBob who performs a measurement to identify if the trans-mitted state was ρ or ρ . It can be shown that with anoptimal measurement, Bob can successfully identify thestate with probability [1 + D ( ρ , ρ )].It is well-known that there is no quantum operation,defined on the system Hilbert space, that can increasethe trace distance between states. More precisely, a com-pletely positive non-trace-increasing map Φ is a contrac-tion on the trace distance metric, D (Φ ρ , Φ ρ ) ≤ D ( ρ , ρ ) . (1)Consider a Markovian master equation for the reducedstate ρ of an open system characterized by the Lind-blad generator L . The solution is a dynamical semigroup a r X i v : . [ qu a n t - ph ] J un Φ( t ) = exp( L t ). As a consequence of the semigroup prop-erty Φ( τ + t ) = Φ( τ )Φ( t ) for all τ, t ≥ D (Φ( t + τ ) ρ , Φ( t + τ ) ρ ) ≤ D (Φ( t ) ρ , Φ( t ) ρ ) . (2)The inequality holds for the larger class of positivity pre-serving maps. For example, a time-inhomogeneous Lind-blad generator L ( t ) describes time-dependent Markovianprocesses where the map Φ( t ) need not be a dynamicalsemigroup. If instead we define a two-parameter family[22] of dynamical maps Φ( t , t ) where Φ( t,
0) = Φ( t ), weobtain the comparable semigroup property Φ( t + τ,
0) =Φ( t + τ, τ )Φ( τ, ρ , is σ ( t, ρ , ) = ddt D (Φ( t ) ρ , Φ( t ) ρ ) . (3)At a time t >
0, if σ ( t, ρ , ) <
0, the distinguishabilitybetween the pair of states is decreasing and informationflows from the system. The interpretation motivates theBLP measure of non-Markovianity, N BLP (Φ) = sup ρ , (cid:90) σ> dtσ ( t, ρ , ) , (4)where the supremum is taken over all mixed states.Because of the maximization inherent to the measure,for this model it is independent of N ! Next, suppose thatwe then apply a strongly decohering Markovian processto the N uncoupled qubits over some time. Again, theBLP measure remains constant. An immediate conse-quence is that if one only looks at the BLP measure inthe limit N → ∞ , then one concludes that a process ishighly non-Markovian when actually, except with van-ishing probability, every state evolves under an effectiveMarkovian channel!In a generic quantum computing problem, we selectsome initial state ρ and apply a noisy operation Φ. Ifthe process has ‘concentrated’ non-Markovianity, like inthe preceding example, the BLP measure can be inac-curate when trying to understand the degree to whichnon-Markovianity discernably affects that overall system.The imprecision is unsurprising for an optimal state pair ρ , need not relate to the set S of valid initial states forthe computing problem.Building upon the information flow description of non-Markovianity, what aspect of that flow is discernable inpractice? Rather than probing the distinguishability be-tween an optimal state pair, we examine the expecteddistinguishability between states in S . Without an a pri-ori set, we focus on the expected distinguishability of all pure states and introduce the average pure state distin-guishability, D avg (Φ) = (cid:90) (cid:90) dψ , D (Φ ψ , Φ ψ ) , (5)where dψ , denotes the natural invariant Haar measure.At a time t , if ∂ t D avg <
0, then the expected distin-guishability between any pair of states is decreasing andinformation flows only out from the system. There is anopposite interpretation of ∂ t D avg > net flux, σ avg ( t ) = ∂D avg ∂t = (cid:90) (cid:90) dψ , σ ( t, ψ , ) , (6)which we can decompose into its positive and negativecontributions, σ avg ( t ) = (cid:90) (cid:90) σ> dψ , σ ( t, ψ , ) + (cid:90) (cid:90) σ< dψ , σ ( t, ψ , )= σ + ( t ) + σ − ( t ) . (7)In the above, we implicitly defined σ + and σ − to denotethe overall strength of non-Markovian and Markovian likeprocesses respectively. According to these measures, aprocess is non-Markovian at a time t if σ + > purely non-Markovian at a time t if σ + > σ − = 0. We call a process strongly non-Markovianat a time t if σ + > σ − .By integrating the information flux, we arrive at twopossible measures of the strength of non-Markovianity: ameasure of average non-Markovianity, N avg (Φ) = (cid:90) σ avg > dtσ avg ( t ) , (8)and of pure non-Markovianity, N p (Φ) = (cid:90) dtσ + ( t ) . (9)To illustrate the significance of the new measures, werevisit the N + 1 qubit toy model. While the BLP mea-sure was constant, with only the ZZ coupling, both mea-sures tend to zero like 1 /N as N → ∞ . Instead of de-scribing extremal behavior these measures relate to typ-ical characteristics. Now suppose we gradually turn on astrongly decohering Markovian channel that acts on the N uncoupled qubits. Both the BLP and pure measure re-main constant, and the average measure decreases. Thusby examining N avg ,p , we can identify the common non-Markovianity qualities of a process. In general, the BLPmeasure is a lower bound on our measures with N avg (Φ) ≤ N p (Φ) ≤ N BLP (Φ) , (10)where the leftmost inequality follows from Jensen’s in-equality. -4 -3 -2 -1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 (cid:15) rel h/τ c FIG. 1. Relative error in approximating σ as a function of thefinite difference step size h . The figure numerically demon-strates that τ c describes an upper bound on the order of thestep size required for the accurate estimation of σ . We now demonstrate the practicality of our approachand examine 2 N + 1 spin-1 / H = N +1 (cid:88) k =1 ω k Z k + N (cid:88) k =1 J k ( X k Y k +1 + Y k X k +1 ) , (11)where X k , Y k , Z k denote Paulis on the k -th particle, and ω k , J k denote the frequency of the k -th particle and theinteraction strength between the k -th and ( k +1)-th parti-cle respectively. This type of Hamiltonian models a quan-tum dot array where Heisenberg interactions decay expo-nentially over inter-particle distances. Suppose that theparticles form a chain of N system qubits coupled to N +1environmental qubits arranged E − S − E − · · · − E where E and S denote the environmental and system qubitsrespectively. The arrangement corresponds to environ-mental ‘defects’ that mediate the interactions betweensystem qubits. In Fig. 2 we sample ω k ∼ N (0 . , . J k ∼ N ( µ, σ ) and consider a total time T = 5. InFig. 2.a) we set σ = 0 .
05 and vary the mean couplingstrength µ . With 2000 samples we determine a relativelysmall 90% confidence interval for the non-Markovianityof a system. The size of the system’s Hilbert space is 4 times that of the largest space probed numerically by aprevious study [29]. The plot shows that non-Markovianbackflow begins at about µ = 0 .
5. In Fig. 2.b) we varythe standard deviation while fixing µ = 0 .
8. As the vari-ance of the noise grows, the dynamics of each qubit be-come distinctive. The process deviates from strict non-Markovianity, and the measures diverge.The expression for σ is experimentally intractable andfurther requires knowledge of the dynamical map, makingit numerically cumbersome. Rather than pursuing an ex-act solution, we apply the central difference approxima-tion with a step size h . We need to bound the timescaleof the features in σ so that we can determine an upperbound on suitable values for h . Without a loss of gener- sw N µ a) sw σ b)FIG. 2. (a) Non-Markovianity vs. mean coupling strength µ for a 10 qubit system coupled to an 11 qubit environment.The plot demonstrates that with 2000 random samples ourmeasures of non-Markovianity can be computed. (b) Non-Markovianity vs. standard deviation σ . The plot illustratesthat a discrepancy between our measures signifies a varyingconcentration of non-Markovianity over the space of states.The error bars denote 90% confidence intervals. ality, we consider an interaction Hamiltonian of the form J = (cid:80) i J i ⊗ B i where the internal evolution of the envi-ronment has been removed by requiring that tr( J i ) = 0for all i . In quantum error-correction theory, the quan-tity λ = (cid:107) J (cid:107) is a measure of the overall noise-strength[30]. Thus τ c = 1 /λ is the order of the shortest correla-tion time scale present in the interaction and provides anupper bound for h , and one should usually pick h (cid:28) τ c .It is unclear how to bound the second time derivativeof the trace distance, and we cannot rigorously boundthe local truncation error. 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