6,090 research outputs found
Concepts of quantum non-Markovianity: a hierarchy
Markovian approximation is a widely-employed idea in descriptions of the
dynamics of open quantum systems (OQSs). Although it is usually claimed to be a
concept inspired by classical Markovianity, the term quantum Markovianity is
used inconsistently and often unrigorously in the literature. In this report we
compare the descriptions of classical stochastic processes and quantum
stochastic processes (as arising in OQSs), and show that there are inherent
differences that lead to the non-trivial problem of characterizing quantum
non-Markovianity. Rather than proposing a single definition of quantum
Markovianity, we study a host of Markov-related concepts in the quantum regime.
Some of these concepts have long been used in quantum theory, such as quantum
white noise, factorization approximation, divisibility, Lindblad master
equation, etc.. Others are first proposed in this report, including those we
call past-future independence, no (quantum) information backflow, and
composability. All of these concepts are defined under a unified framework,
which allows us to rigorously build hierarchy relations among them. With
various examples, we argue that the current most often used definitions of
quantum Markovianity in the literature do not fully capture the memoryless
property of OQSs. In fact, quantum non-Markovianity is highly
context-dependent. The results in this report, summarized as a hierarchy
figure, bring clarity to the nature of quantum non-Markovianity.Comment: Clarifications and references added; discussion of the related
classical hierarchy significantly improved. To appear in Physics Report
On the dynamics of initially correlated open quantum systems: theory and applications
We show that the dynamics of any open quantum system that is initially
correlated with its environment can be described by a set of (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
characterised using only local operations on the system, via a generalisation
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 centres, 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.Comment: 19 pages. Comments welcom
Extracting dynamical equations from experimental data is NP-hard
The behavior of any physical system is governed by its underlying dynamical
equations. Much of physics is concerned with discovering these dynamical
equations and understanding their consequences. In this work, we show that,
remarkably, identifying the underlying dynamical equation from any amount of
experimental data, however precise, is a provably computationally hard problem
(it is NP-hard), both for classical and quantum mechanical systems. As a
by-product of this work, we give complexity-theoretic answers to both the
quantum and classical embedding problems, two long-standing open problems in
mathematics (the classical problem, in particular, dating back over 70 years).Comment: For mathematical details, see arXiv:0908.2128[math-ph]. v2: final
version, accepted in Phys. Rev. Let
Open Quantum Dynamics: Complete Positivity and Entanglement
We review the standard treatment of open quantum systems in relation to
quantum entanglement, analyzing, in particular, the behaviour of bipartite
systems immersed in a same environment. We first focus upon the notion of
complete positivity, a physically motivated algebraic constraint on the quantum
dynamics, in relation to quantum entanglement, i.e. the existence of
statistical correlations which can not be accounted for by classical
probability. We then study the entanglement power of heat baths versus their
decohering properties, a topic of increasing importance in the framework of the
fast developing fields of quantum information, communication and computation.
The presentation is self contained and, through several examples, it offers a
detailed survey of the physics and of the most relevant and used techniques
relative to both quantum open system dynamics and quantum entanglement.Comment: LaTex, 77 page
Quantum Markov Channels for Qubits
We examine stochastic maps in the context of quantum optics. Making use of
the master equation, the damping basis, and the Bloch picture we calculate a
non-unital, completely positive, trace-preserving map with unequal damping
eigenvalues. This results in what we call the squeezed vacuum channel. A
geometrical picture of the effect of stochastic noise on the set of pure state
qubit density operators is provided. Finally, we study the capacity of the
squeezed vacuum channel to transmit quantum information and to distribute EPR
states.Comment: 18 pages, 4 figure
Response functions as quantifiers of non-Markovianity
Quantum non-Markovianity is crucially related to the study of dynamical maps,
which are usually derived for initially factorized system-bath states. We here
demonstrate that linear response theory also provides a way to derive dynamical
maps, but for initially correlated (and in general entangled) states.
Importantly, these maps are always time-translational invariant and allow for a
much simpler quantification of non-Markovianity compared to previous
approaches. We apply our theory to the Caldeira-Leggett model, for which our
quantifier is valid beyond linear response and can be expressed analytically.
We find that a classical Brownian particle coupled to an Ohmic bath can already
exhibit non-Markovian behaviour, a phenomenon related to the initial state
preparation procedure. Furthermore, for a peaked spectral density we
demonstrate that there is no monotonic relation between our quantifier and the
system-bath coupling strength, the sharpness of the peak or the resonance
frequency in the bath.Comment: Updated version with additional information. 5 + 5 pages incl. 2
figures. Comments are welcom
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