221 research outputs found
Positive contraction mappings for classical and quantum Schrodinger systems
The classical Schrodinger bridge seeks the most likely probability law for a
diffusion process, in path space, that matches marginals at two end points in
time; the likelihood is quantified by the relative entropy between the sought
law and a prior, and the law dictates a controlled path that abides by the
specified marginals. Schrodinger proved that the optimal steering of the
density between the two end points is effected by a multiplicative functional
transformation of the prior; this transformation represents an automorphism on
the space of probability measures and has since been studied by Fortet,
Beurling and others. A similar question can be raised for processes evolving in
a discrete time and space as well as for processes defined over non-commutative
probability spaces. The present paper builds on earlier work by Pavon and
Ticozzi and begins with the problem of steering a Markov chain between given
marginals. Our approach is based on the Hilbert metric and leads to an
alternative proof which, however, is constructive. More specifically, we show
that the solution to the Schrodinger bridge is provided by the fixed point of a
contractive map. We approach in a similar manner the steering of a quantum
system across a quantum channel. We are able to establish existence of quantum
transitions that are multiplicative functional transformations of a given Kraus
map, but only for the case of uniform marginals. As in the Markov chain case,
and for uniform density matrices, the solution of the quantum bridge can be
constructed from the fixed point of a certain contractive map. For arbitrary
marginal densities, extensive numerical simulations indicate that iteration of
a similar map leads to fixed points from which we can construct a quantum
bridge. For this general case, however, a proof of convergence remains elusive.Comment: 27 page
A complete family of separability criteria
We introduce a new family of separability criteria that are based on the
existence of extensions of a bipartite quantum state to a larger number
of parties satisfying certain symmetry properties. It can be easily shown that
all separable states have the required extensions, so the non-existence of such
an extension for a particular state implies that the state is entangled. One of
the main advantages of this approach is that searching for the extension can be
cast as a convex optimization problem known as a semidefinite program (SDP).
Whenever an extension does not exist, the dual optimization constructs an
explicit entanglement witness for the particular state. These separability
tests can be ordered in a hierarchical structure whose first step corresponds
to the well-known Positive Partial Transpose (Peres-Horodecki) criterion, and
each test in the hierarchy is at least as powerful as the preceding one. This
hierarchy is complete, in the sense that any entangled state is guaranteed to
fail a test at some finite point in the hierarchy, thus showing it is
entangled. The entanglement witnesses corresponding to each step of the
hierarchy have well-defined and very interesting algebraic properties that in
turn allow for a characterization of the interior of the set of positive maps.
Coupled with some recent results on the computational complexity of the
separability problem, which has been shown to be NP-hard, this hierarchy of
tests gives a complete and also computationally and theoretically appealing
characterization of mixed bipartite entangled states.Comment: 21 pages. Expanded introduction. References added, typos corrected.
Accepted for publication in Physical Review
Matrix Product Density Operators: when do they have a local parent Hamiltonian?
We study whether one can write a Matrix Product Density Operator (MPDO) as
the Gibbs state of a quasi-local parent Hamiltonian. We conjecture this is the
case for generic MPDO and give supporting evidences. To investigate the
locality of the parent Hamiltonian, we take the approach of checking whether
the quantum conditional mutual information decays exponentially. The MPDO we
consider are constructed from a chain of 1-input/2-output (`Y-shaped')
completely-positive maps, i.e. the MPDO have a local purification. We derive an
upper bound on the conditional mutual information for bistochastic channels and
strictly positive channels, and show that it decays exponentially if the
correctable algebra of the channel is trivial. We also introduce a conjecture
on a quantum data processing inequality that implies the exponential decay of
the conditional mutual information for every Y-shaped channel with trivial
correctable algebra. We additionally investigate a close but nonequivalent
cousin: MPDO measured in a local basis. We provide sufficient conditions for
the exponential decay of the conditional mutual information of the measured
states, and numerically confirmed they are generically true for certain random
MPDO.Comment: Added Github code for Propostion III.6; added few names in
acknowledgement after discussion with them about DPI for CM
Strict Positivity and -Majorization
Motivated by quantum thermodynamics we first investigate the notion of strict
positivity, that is, linear maps which map positive definite states to
something positive definite again. We show that strict positivity is decided by
the action on any full-rank state, and that the image of non-strictly positive
maps lives inside a lower-dimensional subalgebra. This implies that the
distance of such maps to the identity channel is lower bounded by one.
The notion of strict positivity comes in handy when generalizing the
majorization ordering on real vectors with respect to a positive vector to
majorization on square matrices with respect to a positive definite matrix .
For the two-dimensional case we give a characterization of this ordering via
finitely many trace norm inequalities and, moreover, investigate some of its
order properties. In particular it admits a unique minimal and a maximal
element. The latter is unique as well if and only if minimal eigenvalue of
has multiplicity one.Comment: Supersedes arXiv:2003.0416
Quantum entanglement
All our former experience with application of quantum theory seems to say:
{\it what is predicted by quantum formalism must occur in laboratory}. But the
essence of quantum formalism - entanglement, recognized by Einstein, Podolsky,
Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a
new resource as real as energy.
This holistic property of compound quantum systems, which involves
nonclassical correlations between subsystems, is a potential for many quantum
processes, including ``canonical'' ones: quantum cryptography, quantum
teleportation and dense coding. However, it appeared that this new resource is
very complex and difficult to detect. Being usually fragile to environment, it
is robust against conceptual and mathematical tools, the task of which is to
decipher its rich structure.
This article reviews basic aspects of entanglement including its
characterization, detection, distillation and quantifying. In particular, the
authors discuss various manifestations of entanglement via Bell inequalities,
entropic inequalities, entanglement witnesses, quantum cryptography and point
out some interrelations. They also discuss a basic role of entanglement in
quantum communication within distant labs paradigm and stress some
peculiarities such as irreversibility of entanglement manipulations including
its extremal form - bound entanglement phenomenon. A basic role of entanglement
witnesses in detection of entanglement is emphasized.Comment: 110 pages, 3 figures, ReVTex4, Improved (slightly extended)
presentation, updated references, minor changes, submitted to Rev. Mod. Phys
The Principle of Locality. Effectiveness, fate and challenges
The Special Theory of Relativity and Quantum Mechanics merge in the key
principle of Quantum Field Theory, the Principle of Locality. We review some
examples of its ``unreasonable effectiveness'' (which shows up best in the
formulation of Quantum Field Theory in terms of operator algebras of local
observables) in digging out the roots of Global Gauge Invariance in the
structure of the local observable quantities alone, at least for purely massive
theories; but to deal with the Principle of Local Gauge Invariance is still a
problem in this frame. This problem emerges also if one attempts to figure out
the fate of the Principle of Locality in theories describing the gravitational
forces between elementary particles as well. Spacetime should then acquire a
quantum structure at the Planck scale, and the Principle of Locality is lost.
It is a crucial open problem to unravel a replacement in such theories which is
equally mathematically sharp and reduces to the Principle of Locality at larger
scales. Besides exploring its fate, many challenges for the Principle of
Locality remain; among them, the analysis of Superselection Structure and
Statistics also in presence of massless particles, and to give a precise
mathematical formulation to the Measurement Process in local and relativistic
terms; for which we outline a qualitative scenario which avoids the EPR
Paradox.Comment: 36 pages. Survey partially based on a talk delivered at the Meeting
"Algebraic Quantum Field Theory: 50 years", Goettingen, July 29-31, 2009, in
honor of Detlev Buchholz. Submitted to Journal of Mathematical Physic
Linear maps as sufficient criteria for entanglement depth and compatibility in many-body systems
Physical transformations are described by linear maps that are completely
positive and trace preserving (CPTP). However, maps that are positive (P) but
not completely positive (CP) are instrumental to derive
separability/entanglement criteria. Moreover, the properties of such maps can
be linked to entanglement properties of the states they detect. Here, we extend
the results presented in [Phys. Rev A 93, 042335 (2016)], where sufficient
separability criteria for bipartite systems were derived. In particular, we
analyze the entanglement depth of an -qubit system by proposing linear maps
that, when applied to any state, result in a bi-separable state for the
partitions, i.e., -entanglement depth. Furthermore, we derive
criteria to detect arbitrary -entanglement depth tailored to states in
close vicinity of the completely depolarized state (the normalized identity
matrix). We also provide separability (or - entanglement depth) conditions
in the symmetric sector, including for diagonal states. Finally, we suggest how
similar map techniques can be used to derive sufficient conditions for a set of
expectation values to be compatible with separable states or
local-hidden-variable theories. We dedicate this paper to the memory of the
late Andrzej Kossakowski, our spiritual and intellectual mentor in the field of
linear maps.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1512.0827
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