723 research outputs found
Minimum guesswork discrimination between quantum states
© Rinton Press. Error probability is a popular and well-studied optimization criterion in discriminating non-orthogonal quantum states. It captures the threat from an adversary who can only query the actual state once. However, when the adversary is able to use a brute-force strategy to query the state, discrimination measurement with minimum error probability does not necessarily minimize the number of queries to get the actual state. In light of this, we take Massey’s guesswork as the underlying optimization criterion and study the problem of minimum guesswork discrimination. We show that this problem can be reduced to a semidefinite programming problem. Necessary and sufficient conditions when a measurement achieves minimum guesswork are presented. We also reveal the relation between minimum guesswork and minimum error probability. We show that the two criteria generally disagree with each other, except for the special case with two states. Both upper and lower information-theoretic bounds on minimum guesswork are given. For geometrically uniform quantum states, we provide sufficient conditions when a measurement achieves minimum guesswork. Moreover, we give the necessary and sufficient condition under which making no measurement at all would be the optimal strategy
Guesswork of a quantum ensemble
The guesswork of a quantum ensemble quantifies the minimum number of guesses
needed in average to correctly guess an unknown message encoded in the states
of the ensemble. Here, we derive sufficient conditions under which the
computation of the minimum guesswork can be recast as a discrete problem. We
show that such conditions are always satisfied for qubit ensembles with uniform
probability distribution, thus settling the problem in that case. As
applications, we compute the guesswork for any qubit regular-polygonal and
regular-polyhedral ensembles.Comment: 4 pages, 1 figure, 1 tabl
Generic local distinguishability and completely entangled subspaces
A subspace of a multipartite Hilbert space is completely entangled if it
contains no product states. Such subspaces can be large with a known maximum
size, S, approaching the full dimension of the system, D. We show that almost
all subspaces with dimension less than or equal to S are completely entangled,
and then use this fact to prove that n random pure quantum states are
unambiguously locally distinguishable if and only if n does not exceed D-S.
This condition holds for almost all sets of states of all multipartite systems,
and reveals something surprising. The criterion is identical for separable and
for nonseparable states: entanglement makes no difference.Comment: 12 page
Guessing Revisited: A Large Deviations Approach
The problem of guessing a random string is revisited. A close relation
between guessing and compression is first established. Then it is shown that if
the sequence of distributions of the information spectrum satisfies the large
deviation property with a certain rate function, then the limiting guessing
exponent exists and is a scalar multiple of the Legendre-Fenchel dual of the
rate function. Other sufficient conditions related to certain continuity
properties of the information spectrum are briefly discussed. This approach
highlights the importance of the information spectrum in determining the
limiting guessing exponent. All known prior results are then re-derived as
example applications of our unifying approach.Comment: 16 pages, to appear in IEEE Transaction on Information Theor
Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels
This thesis combines two parallel research directions: an exploration into the
continuity properties of certain entropic quantities, and an investigation
into a simple class of physical systems whose time evolution
is given by the repeated application of a quantum channel.
In the first part of the thesis, we present a general technique for
establishing local and uniform continuity bounds for Schur concave functions;
that is, for real-valued functions which are decreasing in the majorization
pre-order. Continuity bounds provide a quantitative measure of robustness,
addressing the following question: If there is some uncertainty or error in
the input, how much uncertainty is there in the output? Our technique uses a
particular relationship between majorization and the trace distance between
quantum states (or total variation distance, in the case of probability
distributions). Namely, the majorization pre-order attains a maximum and a
minimum over ε-balls in this distance. By tracing the path of the
majorization-minimizer as a function of the distance ε, we obtain the
path of ``majorization flow’’. An analysis of the derivatives of Schur
concave functions along this path immediately yields tight continuity bounds
for such functions.
In this way, we find a new proof of the Audenaert-Fannes continuity bound for
the von Neumann entropy, and the necessary and sufficient conditions for its
saturation, in a universal framework which extends to the other functions,
including the Rényi and Tsallis entropies. In particular, we prove a novel
uniform continuity bound for the α-Rényi entropy with α > 1 with
much improved dependence on the dimension of the underlying system and the
parameter α compared to previously known bounds. We show that this
framework can also be used to provide continuity bounds for other Schur
concave functions, such as the number of connected components of a certain
random graph model as a function of the underlying probability distribution,
and the number of distinct realizations of a random variable in some fixed
number of independent trials as a function of the underlying probability mass
function. The former has been used in modeling the spread of epidemics, while
the latter has been studied in the context of estimating measures of
biodiversity from observations; in these contexts, our continuity bounds
provide quantitative estimates of robustness to noise or data collection
errors.
In the second part, we consider repeated interaction systems, in which a
system of interest interacts with a sequence of probes, i.e. environmental
systems, one at a time. The state of the system after each interaction is
related to the state of the system before the interaction by the so-called
reduced dynamics, which is described by the action of a quantum channel. When
each probe and the way it interacts with the system is identical, the reduced
dynamics at each step is identical. In this scenario, under the additional
assumption that the reduced dynamics satisfies a faithfulness property, we
characterize which repeated interaction systems break any initially-present
entanglement between the system and an untouched reference, after finitely
many steps. In this case, the reduced dynamics is said to be eventually
entanglement-breaking. This investigation helps improve our
understanding of which kinds of noisy time evolution destroy entanglement.
When the probes and their interactions with the system are slowly-varying
(i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality
between the entropy change of the system and the energy change of the probes,
in the limit in which the number of steps tends to infinity and both the
difference between consecutive probes and the difference between their
interactions vanishes. This analysis proceeds at a fine-grained level by means
of a two-time measurement protocol, in which the energy of the probes is
measured before and after each interaction. The quantities of interest are
then studied as random variables on the space of outcomes of the energy
measurements of the probes, providing a deeper insight into the interrelations
between energy and entropy in this setting.Cantab Capital Institute for the Mathematics of Informatio
Quantum surveillance and 'shared secrets'. A biometric step too far? CEPS Liberty and Security in Europe, July 2010
It is no longer sensible to regard biometrics as having neutral socio-economic, legal and political impacts. Newer generation biometrics are fluid and include behavioural and emotional data that can be combined with other data. Therefore, a range of issues needs to be reviewed in light of the increasing privatisation of ‘security’ that escapes effective, democratic parliamentary and regulatory control and oversight at national, international and EU levels, argues Juliet Lodge, Professor and co-Director of the Jean Monnet European Centre of Excellence at the University of Leeds, U
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