10 research outputs found
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
Testing for a pure state with local operations and classical communication
We examine the problem of using local operations and classical communication
(LOCC) to distinguish a known pure state from an unknown (possibly mixed)
state, bounding the error probability from above and below. We study the
asymptotic rate of detecting multiple copies of the pure state and show that,
if the overlap of the two states is great enough, then they can be
distinguished asymptotically as well with LOCC as with global measurements;
otherwise, the maximal Schmidt coefficient of the pure state is sufficient to
determine the asymptotic error rate.Comment: 11 pages, 2 figures. Published version with small revisions and
expanded title
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
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
Discrimination of quantum states under locality constraints in the many-copy setting
We study the discrimination of a pair of orthogonal quantum states in the
many-copy setting. This is not a problem when arbitrary quantum measurements
are allowed, as then the states can be distinguished perfectly even with one
copy. However, it becomes highly nontrivial when we consider states of a
multipartite system and locality constraints are imposed. We hence focus on the
restricted families of measurements such as local operation and classical
communication (LOCC), separable operations (SEP), and the
positive-partial-transpose operations (PPT) in this paper.
We first study asymptotic discrimination of an arbitrary multipartite
entangled pure state against its orthogonal complement using LOCC/SEP/PPT
measurements. We prove that the incurred optimal average error probability
always decays exponentially in the number of copies, by proving upper and lower
bounds on the exponent. In the special case of discriminating a maximally
entangled state against its orthogonal complement, we determine the explicit
expression for the optimal average error probability and the optimal trade-off
between the type-I and type-II errors, thus establishing the associated
Chernoff, Stein, Hoeffding, and the strong converse exponents. Our technique is
based on the idea of using PPT operations to approximate LOCC.
Then, we show an infinite separation between SEP and PPT operations by
providing a pair of states constructed from an unextendible product basis
(UPB): they can be distinguished perfectly by PPT measurements, while the
optimal error probability using SEP measurements admits an exponential lower
bound. On the technical side, we prove this result by providing a quantitative
version of the well-known statement that the tensor product of UPBs is UPB.Comment: Comments are welcom
Entanglement Theory and the Quantum Simulation of Many-Body Physics
In this thesis we present new results relevant to two important problems in
quantum information science: the development of a theory of entanglement and
the exploration of the use of controlled quantum systems to the simulation of
quantum many-body phenomena.
In the first part we introduce a new approach to the study of entanglement by
considering its manipulation under operations not capable of generating
entanglement and show there is a total order for multipartite quantum states in
this framework. We also present new results on hypothesis testing of correlated
sources and give further evidence on the existence of NPPT bound entanglement.
In the second part, we study the potential as well as the limitations of a
quantum computer for calculating properties of many-body systems. First we
analyse the usefulness of quantum computation to calculate additive
approximations to partition functions and spectral densities of local
Hamiltonians. We then show that the determination of ground state energies of
local Hamiltonians with an inverse polynomial spectral gap is QCMA-hard.
In the third and last part, we approach the problem of quantum simulating
many-body systems from a more pragmatic point of view. We analyze the
realization of paradigmatic condensed matter Hamiltonians in arrays of coupled
microcavities, such as the Bose-Hubbard and the anisotropic Heisenberg models,
and discuss the feasibility of an experimental realization with
state-of-the-art current technology.Comment: 230 pages. PhD thesis, Imperial College London. Chapters 6, 7 and 8
contain unpublished materia