41 research outputs found
Classical and Quantum Parts of the Quantum Dynamics: the Discrete-Time Case
In the study of open quantum systems modeled by a unitary evolution of a
bipartite Hilbert space, we address the question of which parts of the
environment can be said to have a "classical action" on the system, in the
sense of acting as a classical stochastic process. Our method relies on the
definition of the Environment Algebra, a relevant von Neumann algebra of the
environment. With this algebra we define the classical parts of the environment
and prove a decomposition between a maximal classical part and a quantum part.
Then we investigate what other information can be obtained via this algebra,
which leads us to define a more pertinent algebra: the Environment Action
Algebra. This second algebra is linked to the minimal Stinespring
representations induced by the unitary evolution on the system. Finally in
finite dimension we give a characterization of both algebras in terms of the
spectrum of a certain completely positive map acting on the states of the
environment
Approximate tensorization of the relative entropy for noncommuting conditional expectations
In this paper, we derive a new generalisation of the strong subadditivity of
the entropy to the setting of general conditional expectations onto arbitrary
finite-dimensional von Neumann algebras. The latter inequality, which we call
approximate tensorization of the relative entropy, can be expressed as a lower
bound for the sum of relative entropies between a given density and its
respective projections onto two intersecting von Neumann algebras in terms of
the relative entropy between the same density and its projection onto an
algebra in the intersection, up to multiplicative and additive constants. In
particular, our inequality reduces to the so-called quasi-factorization of the
entropy for commuting algebras, which is a key step in modern proofs of the
logarithmic Sobolev inequality for classical lattice spin systems. We also
provide estimates on the constants in terms of conditions of clustering of
correlations in the setting of quantum lattice spin systems. Along the way, we
show the equivalence between conditional expectations arising from Petz
recovery maps and those of general Davies semigroups.Comment: 31 page
Characterization of equivariant maps and application to entanglement detection
We study equivariant linear maps between finite-dimensional matrix algebras,
as introduced by Bhat. These maps satisfy an algebraic property which makes it
easy to study their positivity or k-positivity. They are therefore particularly
suitable for applications to entanglement detection in quantum information
theory. We characterize their Choi matrices. In particular, we focus on a
subfamily that we call (a, b)-unitarily equivariant. They can be seen as both a
generalization of maps invariant under unitary conjugation as studied by Bhat
and as a generalization of the equivariant maps studied by Collins et al. Using
representation theory, we fully compute them and study their graphical
representation, and show that they are basically enough to study all
equivariant maps. We finally apply them to the problem of entanglement
detection and prove that they form a sufficient (infinite) family of positive
maps to detect all k-entangled density matrices.Comment: 16 pages, 4 figure
Group transference techniques for the estimation of the decoherence times and capacities of quantum Markov semigroups
Capacities of quantum channels and decoherence times both quantify the extent
to which quantum information can withstand degradation by interactions with its
environment. However, calculating capacities directly is known to be
intractable in general. Much recent work has focused on upper bounding certain
capacities in terms of more tractable quantities such as specific norms from
operator theory. In the meantime, there has also been substantial recent
progress on estimating decoherence times with techniques from analysis and
geometry, even though many hard questions remain open. In this article, we
introduce a class of continuous-time quantum channels that we called
transferred channels, which are built through representation theory from a
classical Markov kernel defined on a compact group. We study two subclasses of
such kernels: H\"ormander systems on compact Lie-groups and Markov chains on
finite groups. Examples of transferred channels include the depolarizing
channel, the dephasing channel, and collective decoherence channels acting on
qubits. Some of the estimates presented are new, such as those for channels
that randomly swap subsystems. We then extend tools developed in earlier work
by Gao, Junge and LaRacuente to transfer estimates of the classical Markov
kernel to the transferred channels and study in this way different
non-commutative functional inequalities. The main contribution of this article
is the application of this transference principle to the estimation of various
capacities as well as estimation of entanglement breaking times, defined as the
first time for which the channel becomes entanglement breaking. Moreover, our
estimates hold for non-ergodic channels such as the collective decoherence
channels, an important scenario that has been overlooked so far because of a
lack of techniques.Comment: 35 pages, 2 figures. Close to published versio
Rapid thermalization of spin chain commuting Hamiltonians
We prove that spin chains weakly coupled to a large heat bath thermalize rapidly at any temperature for finite-range, translation-invariant commuting Hamiltonians, reaching equilibrium in a time which scales logarithmically with the system size. From a physical point of view, our result rigorously establishes the absence of dissipative phase transitions for Davies evolutions over translation-invariant spin chains. The result has also implications in the understanding of Symmetry Protected Topological phases for open quantum systems
Entropy decay for Davies semigroups of a one dimensional quantum lattice
Given a finite-range, translation-invariant commuting system Hamiltonians on a spin chain, we show that the Davies semigroup describing the reduced dynamics resulting from the joint Hamiltonian evolution of a spin chain weakly coupled to a large heat bath thermalizes rapidly at any temperature. More precisely, we prove that the relative entropy between any evolved state and the equilibrium Gibbs state contracts exponentially fast with an exponent that scales logarithmically with the length of the chain. Our theorem extends a seminal result of Holley and Stroock [40] to the quantum setting, up to a logarithmic overhead, as well as provides an exponential improvement over the non-closure of the gap proved by Brandao and Kastoryano [43]. This has wide-ranging applications to the study of many-body in and out-of-equilibrium quantum systems. Our proof relies upon a recently derived strong decay of correlations for Gibbs states of one dimensional, translation-invariant local Hamiltonians, and tools from the theory of operator spaces.Depto. de AnĂĄlisis MatemĂĄtico y MatemĂĄtica AplicadaFac. de Ciencias MatemĂĄticasFALSEunpu