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arXiv:1810.12540 · Physical Review A

Dynamics of initially correlated open quantum systems: Theory and applications

Gerardo A. Paz-Silva, Michael J. W. Hall, Howard M. Wiseman

Published Oct 25, 2019• 33 citations
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Abstract

We show that the dynamics of any open quantum system that is initially correlated with its environment can be described by a set of ${d}^{2}$ (or less) completely positive maps, where $d$ is the dimension of the system. Only one such map is required for the special case of no initial correlations. The same maps describe the dynamics of any system-environment state obtained from the initial state by a local operation on the system. The reduction of the system dynamics to a set of completely positive maps allows known numerical and analytic tools for uncorrelated initial states to be applied to the general case of initially correlated states, which we exemplify by solving the qubit dephasing model for such states, and provides a natural approach to quantum Markovianity for this case. We show that this set of completely positive maps can be experimentally characterized using only local operations on the system, via a generalization of noise spectroscopy protocols. As further applications, we first consider the problem of retrodicting the dynamics of an open quantum system which is in an arbitrary state when it becomes accessible to the experimenter and explore the conditions under which retrodiction is possible. We also introduce a related one-sided or limited-access tomography protocol for determining an arbitrary bipartite state, evolving under a sufficiently rich Hamiltonian, via local operations and measurements on just one component. We simulate this protocol for a physical model of particular relevance to nitrogen-vacancy centers and in particular show how to reconstruct the density matrix of a set of three qubits, interacting via dipolar coupling and in the presence of local magnetic fields, by measuring and controlling only one of them.

DOI
10.1103/physreva.100.042120

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We show that the dynamics of any open quantum system that is initially correlated with its environment can be described by a set of d2{d}^{2}d2 (or less) completely positive maps, where ddd is the dimension of the system. Only one such map is required for the special case of no initial correlations. The same maps describe the dynamics of any system-environment state obtained from the initial state by a local operation on the system. The reduction of the system dynamics to a set of completely positive maps allows known numerical and analytic tools for uncorrelated initial states to be applied to the general case of initially correlated states, which we exemplify by solving the qubit dephasing model for such states, and provides a natural approach to quantum Markovianity for this case. We show that this set of completely positive maps can be experimentally characterized using only local operations on the system, via a generalization of noise spectroscopy protocols. As further applications, we first consider the problem of retrodicting the dynamics of an open quantum system which is in an arbitrary state when it becomes accessible to the experimenter and explore the conditions under which retrodiction is possible. We also introduce a related one-sided or limited-access tomography protocol for determining an arbitrary bipartite state, evolving under a sufficiently rich Hamiltonian, via local operations and measurements on just one component. We simulate this protocol for a physical model of particular relevance to nitrogen-vacancy centers and in particular show how to reconstruct the density matrix of a set of three qubits, interacting via dipolar coupling and in the presence of local magnetic fields, by measuring and controlling only one of them.
Abstract We investigate the dynamics of open quantum systems which are initially correlated with their environment. The strategy of our approach is to analyze how given, fixed initial correlations modify the evolution of the open system with respect to the corresponding uncorrelated dynamical behavior with the same fixed initial environmental state, described by a completely positive dynamical map. We show that, for any predetermined initial correlations, one can introduce a linear dynamical map on the space of operators of the open system which acts like the proper dynamical map on the set of physical states and represents its unique linear extension. Furthermore, we demonstrate that this construction leads to a linear, time-local quantum master equation with generalized Lindblad structure involving time-dependent, possibly negative transition rates. Thus, the general non-Markovian dynamics of an open quantum system can be described by means of a time-local master equation even in the case of arbitrary, fixed initial system–environment correlations. We present some illustrative examples and explain the relation of our approach to several other approaches proposed in the literature.
In the present thesis, we examine two subjects in the theory of open quantum systems. The first subject deals with the description of the dynamics of open systems initially correlated with the environment. In the theory of open quantum systems, maps characterizing the dynamics of a quantum system in contact with the environment are usually thought to be completely positive. However, this is not necessarily true if the system and its environment are initially correlated, unless we restrict the domain on which the map acts, in other words, only a subset of the set of states of the system gets mapped to other states by the dynamical map. We present a framework for the construction of reduced dynamical maps for subsystems of indistinguishable fermionic particles. We show that in this scenario, a reduced map in the Kraus representation is possible for some sets of states where the only non-classical correlation present is exchange. The second subject studied is related to the characterization of non-Markovian dynamics with the divisibility and entanglement criteria. We obtain the analytical expression for the Kraus decomposition of the quantum map of an environment modeled by an arbitrary quadratic fermionic Hamiltonian acting on one or two qubits, we derive simple functions to examine the non positivity of the intermediate map. In the particular case of an environment represented by the Ising Hamiltonian, we discuss the two sources of non-Markovianity in the model, one due to the finite size of the lattice, and another due to the kind of interactions.
In this thesis we address three topics of open quantum systems theory. \nThe first concerns the operational characterization of an open quantum system. We develop a graphical calculus for open quantum systems and use it to review the operational representation and characterization of noisy evolution using completely-positive maps, or quantum channels. These graphical techniques facilitate an intuitive unification of the various representations of CP-maps, and the transformations between these representations. We generalize the channel formalism by introducing quantum superchannels --- effective quantum channels which take another channel as input. Superchannels may be constructed using graphical techniques to rearrange the tensor network for a sequence of channels, and these constructions allow for the description of a strictly greater set of dynamics than is possible with the standard formalism. We demonstrate this by providing a method for the complete characterization of an open system initially correlated with its environment. In doing so we introduce and develop several novel quantitative measures for characterizing the strength and presence of initial correlations in a quantum system. These techniques may be implemented using measurement of the system alone, and do not require any access or prior knowledge about the environment. \n \nThe second topic we discuss is the parallel initialization of an ensemble quantum system into a high purity state. We describe a dissipative state engineering technique where an ensemble of identical spins may be prepared in its ground state by coupling the spins to the resolved sidebands of a high-Q resonator. This allows for parallel removal of entropy from a large number of quantum systems, enabling an ensemble to achieve a polarization greater than thermal equilibrium, and in principle on a time scale much shorter than thermal relaxation processes. This is achieved by the collective enhancement in coupling strength of the coupled angular momentum subspaces which behave as larger effective spins. Cavity cooling is shown to cool each of these subspaces to their respective ground state, however by including a local T2T_2T2​ dephasing dissipator on each spin we show how one may use this to break the identical spin degeneracy and couple the subspaces to enable cooling to the full ground state of the joint system. \n \nThe third topic concerns quantum correlations between the momentum and spin subsystems of a neutron in a neutron interferometer. We explore the robustness of these correlations in the presence of dephasing noise due to randomized phase differences between interferometer paths. We demonstrate that even in the presence of strong noise quantum correlations persist and can be detected by introducing post-selected spin measurements at the output of the interferometer. We relate these post-selected experiments to which-way measurements and a quantum eraser.
In standard treatments of open quantum systems, the reduced dynamics is described starting from the assumption that the system and the environment are initially uncorrelated. This assumption, however, is not always guaranteed in realistic scenarios and several theoretical approaches to characterize initially correlated dynamics have been introduced. For the uncorrelated scenario, stochastic unravelings are a powerful tool to simulate the dynamics. So far they have not been used in the most general case in which correlations are initially present since they cannot be applied to nonpositive operators or noncompletely positive maps. In our work, we employ the bath positive (B+)(\mathrm{B}+)(B+) or one-sided positive decomposition (OPD) formalism as a starting point to generalize stochastic unraveling in the presence of initial correlations. Noticeably, our approach does not depend on the particular unraveling technique, but holds for both piecewise deterministic and diffusive unravelings. This generalization allows not only for more powerful simulations for the reduced dynamics, but also for a deeper theoretical understanding of open system dynamics.
Quantum mechanics has both deepened our understanding of the physical reality and allowed development of quantum technologies beating their classical counterparts in efficiency, security, and sensitivity. As a major downside, the quantum properties behind these technologies are very sensitive to the effects the system’s interaction with its environment. The theory of open quantum systems describes how such interaction influences the system, and as a consequence, how the quantum features are lost, controlled, and possibly returned by manipulating the environment. This thesis begins with a short description of the most relevant features and results of the open quantum system theory. We continue by presenting some commonly used measures to quantify the information encoded in quantum systems and the correlations between spatially or temporally separated parties. The literature review ends with discussion of non-Markovianity and memory effects which combine the fields of open quantum systems and quantum information in a natural way. The rest of the thesis presents the main results of Publications I – VII. We study the memory effects and physicality conditions of open system dynamics described by the vast family of phase-covariant master equations. We show how the open quantum system picture can be exploited to design probing protocols to extract information without knowing how the probe and system interact. Such protocols are considered also from the point-of view of efficiently obtaining the information required in the probing, both in theory and by experimental implementation. The experiment serves also as evidence for applicability of recently developed geometric tools to design informationally incomplete measurements for solving so-called membership problems. We present and experimentally realize a protocol where control over intial correlations between polarization and frequency gives full freedom to implement any dephasing dynamics in the open system polarization, thus dictating how its information content changes in time. We study the two-qubit memory effects of dephasing dynamics and compare them to the effect of such noise in the transmission efficiency in superdense coding. Optimal noise configurations in different situations are found. Finally, we characterize the families of initial states for which the decay of quantum correlations depends or does not depend on the noise location.
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.