Non-Markovian quantum processes: Complete framework and efficient characterization

Felix A. Pollock, César Rodríguez-Rosario, Thomas Frauenheim, Mauro Paternostro, and Kavan Modi

Phys. Rev. A 97, 012127 – Published 25 January 2018

Abstract

Currently, there is no systematic way to describe a quantum process with memory solely in terms of experimentally accessible quantities. However, recent technological advances mean we have control over systems at scales where memory effects are non-negligible. The lack of such an operational description has hindered advances in understanding physical, chemical, and biological processes, where often unjustified theoretical assumptions are made to render a dynamical description tractable. This has led to theories plagued with unphysical results and no consensus on what a quantum Markov (memoryless) process is. Here, we develop a universal framework to characterize arbitrary non-Markovian quantum processes. We show how a multitime non-Markovian process can be reconstructed experimentally, and that it has a natural representation as a many-body quantum state, where temporal correlations are mapped to spatial ones. Moreover, this state is expected to have an efficient matrix-product-operator form in many cases. Our framework constitutes a systematic tool for the effective description of memory-bearing open-system evolutions.

Received 18 October 2017

DOI:https://doi.org/10.1103/PhysRevA.97.012127

Physics Subject Headings (PhySH)

Open quantum systems & decoherence Quantum formalism

General Physics

Authors & Affiliations

Felix A. Pollock^1,*, César Rodríguez-Rosario², Thomas Frauenheim², Mauro Paternostro³, and Kavan Modi^1,†

¹School of Physics & Astronomy, Monash University, Clayton, Victoria 3800, Australia
²Bremen Center for Computational Materials Science, University of Bremen, Am Fallturm 1, D-28359 Bremen, Germany
³School of Mathematics and Physics, Queens University, Belfast BT7 1NN, United Kingdom

^*felix.pollock@monash.edu
^†kavan.modi@monash.edu

Operational Markov Condition for Quantum Processes

Felix A. Pollock, César Rodríguez-Rosario, Thomas Frauenheim, Mauro Paternostro, and Kavan Modi

Phys. Rev. Lett. 120, 040405 (2018)

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Vol. 97, Iss. 1 — January 2018

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Figure 1
(a) The conventional approach to open quantum dynamics attempts to relate the states of the system ( $S$ ) at different times by considering system-environment ( $S − E$ ) unitary dynamics and averaging out the state of the environment $(E)$ . The averaging of the unknown variables is denoted by the red line. This cuts through $S − E$ correlations, leading to issues that have hindered progress in understanding and characterizing non-Markovian dynamics. (b) The operational framework relates the operations an experimentalist can perform on $S$ , denoted by $A_{k - 1 : 0} = {A_{k - 1}, \dots, A_{0}}$ , to the state of $S$ at a later time. The red line here cuts between the objects the experimentalist can control and those that they cannot. (c) This leads to the description of a quantum stochastic process as a mapping, encapsulated in the process tensor $T_{k : 0}$ , from the set of control operations to the output state $ρ_{k}$ of $S$ . The process tensor contains all the information about the $S − E$ initial state and interactions that can be inferred from the system's dynamics alone.
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Figure 2
Quantum circuit to simulate the process tensor. Any process tensor can be simulated by the quantum circuit above. For each time step an ancilla of dimension $d_{j} \geq {(d_{S} \prod_{n = 0}^{j - 1} d_{n})}^{2}$ and prepared in a state $η_{j}$ is introduced. The unitary at each step can be decomposed as $U_{j : j - 1} = V_{j} W_{j : j - 1}$ , where $W_{j : j - 1}$ acts on the system and all previous ancillas and $V_{j}$ acts on all subsystems including the new ancilla. See Sec. pp2-s2 of Appendix pp2 for a detailed proof of the converse statement of Theorem 2.
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Figure 3
The conventional picture of open dynamics is fully contained in the process tensor. When the initial system-environment state is uncorrelated (i.e., $ρ_{0}^{S E} = ρ_{0}^{S} \otimes ρ_{0}^{E}$ ), the picture of evolution according to a CPTP map can be recovered by acting with the identity map $I$ (doing nothing) at all time steps but the first. The initial state is simply given by $ρ_{0} = A_{0} [ρ_{0}^{S}]$ .
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Figure 4
A process tensor acting on a correlated operation. Both the process tensor and the sequence of control operations can be represented as quantum combs [15]. The density operator at time step $k$ results from their contraction. Any correlated CPTP operation acting on the system can be implemented by interacting the system repeatedly with an ancillary system $A$ , such that $S A$ unitaries $V_{j}$ are applied at each time step. Any correlated CP operation can be implemented with a further measurement with the correct outcome [38].
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Figure 5
Full-process tomography. In a convenient, but not unique, scheme for full tomography, the system is measured at each time step and then freshly prepared. That is, the preparation at step $k$ is independent of the previous measurements and preparations. A linear combination of measurements and preparations, each chosen from a set that linearly spans the operator space, is sufficient to span the space of control operations. Having statistics for all possible measurements and preparations at all times is sufficient to construct $T_{k : 0}$ .
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Figure 6
Generalized Choi-Jamiołkowski isomorphism. This quantum circuit prepares the state that represents the process tensor element by element. The resources required are $k$ maximally entangled pairs of ancillas of dimension $d$ , that is, $k {log}_{2} (d)$ ebits. Correlations between pairs of ancillas in $Υ_{k : 0}$ correspond directly to the memory inherent in the non-Markovian evolution. Any desired element of this state can be sampled using the techniques of quantum state tomography. See Appendix pp3 for details.
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