1,433 research outputs found
Hilbert's projective metric in quantum information theory
We introduce and apply Hilbert's projective metric in the context of quantum
information theory. The metric is induced by convex cones such as the sets of
positive, separable or PPT operators. It provides bounds on measures for
statistical distinguishability of quantum states and on the decrease of
entanglement under LOCC protocols or other cone-preserving operations. The
results are formulated in terms of general cones and base norms and lead to
contractivity bounds for quantum channels, for instance improving Ruskai's
trace-norm contraction inequality. A new duality between distinguishability
measures and base norms is provided. For two given pairs of quantum states we
show that the contraction of Hilbert's projective metric is necessary and
sufficient for the existence of a probabilistic quantum operation that maps one
pair onto the other. Inequalities between Hilbert's projective metric and the
Chernoff bound, the fidelity and various norms are proven.Comment: 32 pages including 3 appendices and 3 figures; v2: minor changes,
published versio
On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states
Motivated by the recent discovery of a quantum Chernoff theorem for
asymptotic state discrimination, we investigate the distinguishability of two
bipartite mixed states under the constraint of local operations and classical
communication (LOCC), in the limit of many copies. While for two pure states a
result of Walgate et al. shows that LOCC is just as powerful as global
measurements, data hiding states (DiVincenzo et al.) show that locality can
impose severe restrictions on the distinguishability of even orthogonal states.
Here we determine the optimal error probability and measurement to discriminate
many copies of particular data hiding states (extremal d x d Werner states) by
a linear programming approach. Surprisingly, the single-copy optimal
measurement remains optimal for n copies, in the sense that the best strategy
is measuring each copy separately, followed by a simple classical decision
rule. We also put a lower bound on the bias with which states can be
distinguished by separable operations.Comment: 11 pages; v2: Journal version; Minor errors fixed in Section I
Asymptotic State Discrimination and a Strict Hierarchy in Distinguishability Norms
In this paper, we consider the problem of discriminating quantum states by
local operations and classical communication (LOCC) when an arbitrarily small
amount of error is permitted. This paradigm is known as asymptotic state
discrimination, and we derive necessary conditions for when two multipartite
states of any size can be discriminated perfectly by asymptotic LOCC. We use
this new criterion to prove a gap in the LOCC and separable distinguishability
norms. We then turn to the operational advantage of using two-way classical
communication over one-way communication in LOCC processing. With a simple
two-qubit product state ensemble, we demonstrate a strict majorization of the
two-way LOCC norm over the one-way norm.Comment: Corrected errors from the previous draft. Close to publication for
Quantum State Local Distinguishability via Convex Optimization
Entanglement and nonlocality play a fundamental role in quantum computing. To understand the interplay between these phenomena, researchers have considered the model of local operations and classical communication, or LOCC for short, which is a restricted subset of all possible operations that can be performed on a multipartite quantum system. The task of distinguishing states from a set that is known a priori to all parties is one of the most basic problems among those used to test the power of LOCC protocols, and it has direct applications to quantum data hiding, secret sharing and quantum channel capacity. The focus of this thesis is on state distinguishability problems for classes of quantum operations that are more powerful than LOCC, yet more restricted than global operations, namely the classes of separable and positive-partial-transpose (PPT) measurements. We
build a framework based on convex optimization (on cone programming, in particular) to study such problems. Compared to previous approaches to the problem, the method described in this thesis provides precise numerical bounds and quantitative analytic results. By combining the duality theory of cone programming with the channel-state duality, we also establish a novel connection between the state distinguishability problem and the study of positive linear maps, which is a topic of independent interest in quantum information theory.
We apply our framework to several questions that were left open in previous works regarding the distinguishability of maximally entangled states and unextendable product sets. First, we exhibit small sets of orthogonal maximally entangled states in that are not perfectly distinguishable by LOCC. As a consequence of this, we show a gap between the power of PPT and separable measurements for the task of distinguishing sets consisting only of maximally entangled states. Furthermore, we prove tight bounds on the entanglement cost that is necessary to distinguish any sets of Bell states, thus showing that quantum teleportation is optimal for this task. Finally, we provide an easily checkable characterization of when an unextendable product set is perfectly discriminated by separable measurements, along with the first known example of an unextendable product set that cannot be perfectly discriminated by separable measurements
Tight bounds on the distinguishability of quantum states under separable measurements
One of the many interesting features of quantum nonlocality is that the
states of a multipartite quantum system cannot always be distinguished as well
by local measurements as they can when all quantum measurements are allowed. In
this work, we characterize the distinguishability of sets of multipartite
quantum states when restricted to separable measurements -- those which contain
the class of local measurements but nevertheless are free of entanglement
between the component systems. We consider two quantities: The separable
fidelity -- a truly quantum quantity -- which measures how well we can "clone"
the input state, and the classical probability of success, which simply gives
the optimal probability of identifying the state correctly.
We obtain lower and upper bounds on the separable fidelity and give several
examples in the bipartite and multipartite settings where these bounds are
optimal. Moreover the optimal values in these cases can be attained by local
measurements. We further show that for distinguishing orthogonal states under
separable measurements, a strategy that maximizes the probability of success is
also optimal for separable fidelity. We point out that the equality of fidelity
and success probability does not depend on an using optimal strategy, only on
the orthogonality of the states. To illustrate this, we present an example
where two sets (one consisting of orthogonal states, and the other
non-orthogonal states) are shown to have the same separable fidelity even
though the success probabilities are different.Comment: 19 pages; published versio
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