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Quantum Channels with Memory

Tomáš Rybár

Published Nov 30, 2012• 4 citations

Abstract

Abstract Quantum memory channels represent a very general, yet simple and comprehensible model for causal processes. As such they have attracted considerable research interest, mostly aimed on their transfer capabilities and structure properties. Most notably it was shown that memory channels can be implemented via physically naturally motivated collision models. We also define the concept of repeatable channels and show that only unital channels can be implemented repeatably with pure memory channels. In the special case of qubit channels we also show that every unital qubit channel has a repeatable implementation. We also briefly explore the possibilities of stroboscopical simulation of channels and show that all random unitary channels can be stroboscopically simulated. Particularly in qubit case, all indivisible qubit channels are also random unitary, hence for qubit all indivisible channels can be stroboscopically simulated. Memory channels also naturally capture the framework of correlated experiments. We develop methods to gather and interpret data obtained in such setting and in detail examine the two qubit case. We also show that for control unitary interactions the measured data will never contradict a simple unitary evolution. Thus no memory effects can be spotted then.

DOI: 10.2478/v10155-012-0004-3

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A sequence of controlled collisions between a quantum system and its environment (composed of a set of quantum objects) naturally simulates (with arbitrary precision) any Markovian quantum dynamics of the system under consideration. In this paper we propose and study the problem of simulation of an arbitrary quantum channel via collision models. We show that a correlated environment is capable to simulate non-Markovian evolutions leading to any indivisible qubit channel. In particular, we derive the corresponding master equation generating a continuous time non-Markovian dynamics implementing the universal NOT gate being an example of the most non-Markovian quantum channels.

Quantum memory channels are attracting growing interest, motivated by both realistic possibilities of transferring information by means of quantum hardware and inadequacies of the memoryless approximation. In fact, subsequent uses of the same quantum transmission resource can be significantly correlated. In this paper we review two Hamiltonian models describing memory effects in a purely dephasing spin-boson channel and in a channel with damping visualized by a micromaser system, respectively. In both cases, we show that the quantum information transmission rates are higher than in the memoryless limit.

From the point of view of communication theory, a quantum channel can be defined as a physical process mapping from initial to final state in the form of CPTP (Completely positive trace preserving) map. Quantum information theory provide a theoretical framework and some mathematical methods to accomplish this. In order to preserve information, we proposed a scheme for the protection of two-qubit state through various noise channel, where successive uses of noise channels are correlated. These successive uses of channels are called memory. Different types of noise channels with memory have been discussed in details. We compare the dynamics of entanglement of two-qubit state under different noise channel and it has been seen that one can preserve two-qubit steady state quantum correlation under noise channels with memory. The study of memory effect in quantum channel will emerge as an interesting phenomena to trace back information from the surrounding which is considered as a resource in quantum information task. However, we also define some mores measures of entanglement and their preservation. We hope that our new findings may help to study the two-qubit entanglement structure.

Quantum memories are key ingredients for future quantum networks. Developing methods for their certification is therefore an important, yet challenging, problem. In this work, we present a general framework for certifying quantum memories and quantifying their performance in an operational and efficient manner, taking advantage of developments in the theory of quantum channels. We start by giving a pedagogical introduction to these concepts and tools and illustrate their natural application to quantum memories. Specifically, we quantify the fidelity of a quantum memory via the notion of average channel fidelity, which can be estimated precisely in an experiment via a finite set of probe states forming a so-called

2

-design. We then present a number of witnessing techniques for certifying this quantity, and show that it gives access to other important properties of the channel, such as its dimensionality and its capacity to transmit quantum information. We consider mostly a simple prepare-and-measure scenario, but discuss also the case of an entanglement-based setup, as well as a semi-device-independent approach. Finally, we apply these methods to a solid-state quantum memory experiment, certifying an average channel fidelity of 98.6%. Our certification methods can be readily applied to other quantum devices, such as fibers, transducers and gates.

We present a general model for quantum channels with memory and show that it is sufficiently general to encompass all causal automata: any quantum process in which outputs up to some time

t

do not depend on inputs at times ${t}^{\ensuremath{'}}>t$ can be decomposed into a concatenated memory channel. We then examine and present different physical setups in which channels with memory may be operated for the transfer of (private) classical and quantum information. These include setups in which either the receiver or a malicious third party have control of the initializing memory. We introduce classical and quantum channel capacities for these settings and give several examples to show that they may or may not coincide. Entropic upper bounds on the various channel capacities are given. For forgetful quantum channels, in which the effect of the initializing memory dies out as time increases, coding theorems are presented to show that these bounds may be saturated. Forgetful quantum channels are shown to be open and dense in the set of quantum memory channels.