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arXiv:1201.0477 · Journal of Quantum Information Science

Interplay of Quantum Stochastic and Dynamical Maps to Discern Markovian and Non-Markovian Transitions

A. R. Usha Devi, A. K. Rajagopal, S. Shenoy, R. W. Rendell

Published Jan 1, 2012• 17 citations
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Abstract

It is known that the dynamical evolution of a system, from an initial tensor product state of system and environment, to any two later times, t1, t2 (t2 > t1), are both completely positive (CP) but in the intermediate times between t1 and t2 it need not be CP. This reveals the key to the Markov (if CP) and non Markov (if it is not CP) avataras of the intermediate dynamics. This is brought out here in terms of the quantum stochastic map A and the associated dynamical map B—without resorting to master equation approaches. We investigate these features with four examples which have entirely different physical origins: 1) A two qubit Werner state map with time dependent noise parameter; 2) Phenomenological model of a recent optical experiment (Nature Physics, 7, 931 (2011)) on the open system evolution of photon polarization; 3) Hamiltonian dynamics of a qubit coupled to a bath of N qubits; 4) Two qubit unitary dynamics of Jordan et al. (Phys. Rev. A 70, 052110 (2004) with initial product states of qubits. In all these models, it is shown that the positivity/negativity of the eigenvalues of intermediate time dynamical B map determines the Markov/non-Markov nature of the dynamics.

DOI
10.4236/jqis.2012.23009

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It is known that the dynamical evolution of a system, from an initial tensor product state of system and environment, to any two later times, t1, t2 (t2 > t1), are both completely positive (CP) but in the intermediate times between t1 and t2 it need not be CP. This reveals the key to the Markov (if CP) and non Markov (if it is not CP) avataras of the intermediate dynamics. This is brought out here in terms of the quantum stochastic map A and the associated dynamical map B—without resorting to master equation approaches. We investigate these features with four examples which have entirely different physical origins: 1) A two qubit Werner state map with time dependent noise parameter; 2) Phenomenological model of a recent optical experiment (Nature Physics, 7, 931 (2011)) on the open system evolution of photon polarization; 3) Hamiltonian dynamics of a qubit coupled to a bath of N qubits; 4) Two qubit unitary dynamics of Jordan et al. (Phys. Rev. A 70, 052110 (2004) with initial product states of qubits. In all these models, it is shown that the positivity/negativity of the eigenvalues of intermediate time dynamical B map determines the Markov/non-Markov nature of the dynamics.
It is known that the time evolution of a subsystem from an initial state to two later times, t1, t2 (t2 > t1), are both completely positive (CP) but it is shown here that in the intermediate times between t1 and t2, in general, it need not be CP. This reveals the key to the Markov (if CP) and nonMarkov (if NCP) avataras of the intermediate dynamics. This is brought out based on A and B dynamical maps - without resorting to Master equation approach. The choice of tensor product form for the global initial state points towards the system-environment interaction dynamics as the sole cause for Markovianity/non-Markovianity. A succinct summary of the results is given in the form of a table.
Dynamical AAA and BBB maps have been employed extensively by Sudarshan and co-workers to investigate open-system evolution of quantum systems. A canonical structure of the AAA map is introduced here. It is shown that this canonical AAA map enables us to investigate whether the dynamics is completely positive (CP) or not completely positive (NCP) in an elegant way and, hence, it subsumes the basic results on open-system dynamics. Identifying memory effects in open-system evolution is gaining increasing importance recently and, here, a criterion of non-Markovianity, based on the relative entropy of the dynamical state is proposed. The relative entropy difference of the dynamical system serves as a complementary characterization---though not related directly---to the fidelity difference criterion proposed recently. Three typical examples of open-system evolution of a qubit, prepared initially in a correlated state with another qubit (environment), and evolving jointly under a specific unitary dynamics---which corresponds to a NCP dynamical map---are investigated by employing both the relative entropy difference and fidelity difference tests of non-Markovianity. The two-qubit initial states are chosen to be (i) a pure entangled state, (ii) the Werner state, which exemplifies both entangled and separable states of qubits, depending on a real parameter, and (iii) a separable mixed state. Both the relative entropy and fidelity criteria offer a nice display of how non-Markovianity manifests itself in all three examples.
We introduce a method to construct non-Markovian variants of completely positive (CP) dynamical maps, particularly, qubit Pauli channels. We identify non-Markovianity with the breakdown in CP divisibility of the map, i.e., appearance of a not-completely positive intermediate map. In particular, we consider the case of non-Markovian dephasing in detail. The eigenvalues of the Choi matrix of the intermediate map crossover at a point which corresponds to a singularity in the canonical decoherence rate of the corresponding master equation and thus to a momentary noninvertibility of the map. Thereafter, the rate becomes negative, indicating non-Markovianity. We quantify the non-Markovianity by two methods, one based on CP divisibility [Hall et al., Phys. Rev. A 89, 042120 (2014)], which does not require optimization but requires normalization to handle the singularity, and another method, based on distinguishability [Breuer et al. Phys. Rev. Lett. 103, 210401 (2009)], which requires optimization but is insensitive to the singularity.
A universal definition of non-Markovianity for open systems dynamics is proposed. It is extended from the classical definition to the quantum realm by showing that a `transition' from the Markov to the non-Markov regime occurs when the correlations between the system and the environment, generated by their joint evolution, can no longer be neglected. The suggested definition is based on the comparison between measured correlation functions and those built by assuming that the system is in a Markov regime thus giving a figure of merit of the error coming from this assumption. It is shown that the knowledge of the dynamical map and initial condition of the system is not enough to fully characterise the non-Markovian dynamics of the reduced system. The example of three exactly solvable models, i.e. decoherence and spontaneous emission of the qubit in a bosonic bath and decoherence of the photon's polarization induced by interaction with its spacial degrees of freedom, reveals that previously proposed Markovianity criteria and measures which are based on dynamical map analysis fail to recognise non-Markov behaviour.
Quantum Markov models are employed ubiqui-tously in quantum physics and in quantum information theory due to their relative simplicity and analytical tractability. In particular, these models are known to give accurate approxi-mations for a wide range of quantum optical and meso scopic systems. However, in general, the validity of the Markov approximation entails assumptions regarding properties of the system of interest and its environment, which may not be satisfied or accurate in arbitrary physical systems. Therefore, developing useful modelling tools for general non-Markovian quantum systems for which the Markov approximation is inappropriate or deficient is an undertaking of significant importance. This work considers non-Markovian principal quantum systems that can be embedded in a larger Markovian quantum system with one or more compound baths consisting of an auxiliary quantum system and a quantum white noise field, and derives a set of coupled stochastic and quantum master equations for embedded non-Markovian quantum systems. The case of a purely Hamiltonian coupling between the principal and auxiliary systems as a closed system without coupling to white noises is included as a special case. The results are expected to be of interest for (open-loop and feedback) control of continuous-time non-Markovian systems and studying reduced models for numerical simulation of such systems. They may also shed more light on the general structure of continuous-time non-Markovian quantum systems.
The theoretical description of quantum dynamics in an intriguing way does not necessarily imply the underlying dynamics is indeed intriguing. Here we show how a known very interesting master equation with an always negative decay rate [eternal non-Markovianity (ENM)] arises from simple stochastic Schrödinger dynamics (random unitary dynamics). Equivalently, it may be seen as arising from a mixture of Markov (semi-group) open system dynamics. Both these approaches lead to a more general family of CPT maps, characterized by a point within a parameter triangle. Our results show how ENM quantum dynamics can be realised easily in the laboratory. Moreover, we find a quantum time-continuously measured (quantum trajectory) realisation of the dynamics of the ENM master equation based on unitary transformations and projective measurements in an extended Hilbert space, guided by a classical Markov process. Furthermore, a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) representation of the dynamics in an extended Hilbert space can be found, with a remarkable property: there is no dynamics in the ancilla state. Finally, analogous constructions for two qubits extend these results from non-CP-divisible to non-P-divisible dynamics.