514 research outputs found
Unicity conditions for low-rank matrix recovery
Low-rank matrix recovery addresses the problem of recovering an unknown
low-rank matrix from few linear measurements. Nuclear-norm minimization is a
tractible approach with a recent surge of strong theoretical backing. Analagous
to the theory of compressed sensing, these results have required random
measurements. For example, m >= Cnr Gaussian measurements are sufficient to
recover any rank-r n x n matrix with high probability. In this paper we address
the theoretical question of how many measurements are needed via any method
whatsoever --- tractible or not. We show that for a family of random
measurement ensembles, m >= 4nr - 4r^2 measurements are sufficient to guarantee
that no rank-2r matrix lies in the null space of the measurement operator with
probability one. This is a necessary and sufficient condition to ensure uniform
recovery of all rank-r matrices by rank minimization. Furthermore, this value
of precisely matches the dimension of the manifold of all rank-2r matrices.
We also prove that for a fixed rank-r matrix, m >= 2nr - r^2 + 1 random
measurements are enough to guarantee recovery using rank minimization. These
results give a benchmark to which we may compare the efficacy of nuclear-norm
minimization
Optimal quantum detectors for unambiguous detection of mixed states
We consider the problem of designing an optimal quantum detector that
distinguishes unambiguously between a collection of mixed quantum states. Using
arguments of duality in vector space optimization, we derive necessary and
sufficient conditions for an optimal measurement that maximizes the probability
of correct detection. We show that the previous optimal measurements that were
derived for certain special cases satisfy these optimality conditions. We then
consider state sets with strong symmetry properties, and show that the optimal
measurement operators for distinguishing between these states share the same
symmetries, and can be computed very efficiently by solving a reduced size
semidefinite program.Comment: Submitted to Phys. Rev.
Quantum Detection with Unknown States
We address the problem of distinguishing among a finite collection of quantum
states, when the states are not entirely known. For completely specified
states, necessary and sufficient conditions on a quantum measurement minimizing
the probability of a detection error have been derived. In this work, we assume
that each of the states in our collection is a mixture of a known state and an
unknown state. We investigate two criteria for optimality. The first is
minimization of the worst-case probability of a detection error. For the second
we assume a probability distribution on the unknown states, and minimize of the
expected probability of a detection error.
We find that under both criteria, the optimal detectors are equivalent to the
optimal detectors of an ``effective ensemble''. In the worst-case, the
effective ensemble is comprised of the known states with altered prior
probabilities, and in the average case it is made up of altered states with the
original prior probabilities.Comment: Refereed version. Improved numerical examples and figures. A few
typos fixe
Distinguishing mixed quantum states: Minimum-error discrimination versus optimum unambiguous discrimination
We consider two different optimized measurement strategies for the
discrimination of nonorthogonal quantum states. The first is conclusive
discrimination with a minimum probability of inferring an erroneous result, and
the second is unambiguous, i. e. error-free, discrimination with a minimum
probability of getting an inconclusive outcome, where the measurement fails to
give a definite answer. For distinguishing between two mixed quantum states, we
investigate the relation between the minimum error probability achievable in
conclusive discrimination, and the minimum failure probability that can be
reached in unambiguous discrimination of the same two states. The latter turns
out to be at least twice as large as the former for any two given states. As an
example, we treat the case that the state of the quantum system is known to be,
with arbitrary prior probability, either a given pure state, or a uniform
statistical mixture of any number of mutually orthogonal states. For this case
we derive an analytical result for the minimum probability of error and perform
a quantitative comparison to the minimum failure probability.Comment: Replaced by final version, accepted for publication in Phys. Rev. A.
Revtex4, 6 pages, 3 figure
Uniqueness conditions for low-rank matrix recovery
Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear measurements. Nuclear-norm minimization is a tractable approach with a recent surge of strong theoretical backing. Analagous to the theory of compressed sensing, these results have required random measurements. For example, m ≥ Cnr Gaussian measurements are sufficient to recover any rank-r n x n matrix with high probability. In this paper we address the theoretical question of how many measurements are needed via any method whatsoever - tractable or not. We show that for a family of random measurement ensembles, m ≥ 4nr-4r^2 measurements are sufficient to guarantee that no rank-2r matrix lies in the null space of the measurement operator with probability one. This is a necessary and sufficient condition to ensure uniform recovery of all rank-r matrices by rank minimization. Furthermore, this value of m precisely matches the dimension of the manifold of all rank-2r matrices. We also prove that for a fixed rank-r matrix, m ≥ 2nr – r^2 + 1 random measurements are enough to guarantee recovery using rank minimization. These results give a benchmark to which we may compare the efficacy of nuclear-norm minimization
Minimum-error discrimination between symmetric mixed quantum states
We provide a solution of finding optimal measurement strategy for
distinguishing between symmetric mixed quantum states. It is assumed that the
matrix elements of at least one of the symmetric quantum states are all real
and nonnegative in the basis of the eigenstates of the symmetry operator.Comment: 10 page
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