262 research outputs found
Designing Optimal Quantum Detectors Via Semidefinite Programming
We consider the problem of designing an optimal quantum detector to minimize
the probability of a detection error when distinguishing between a collection
of quantum states, represented by a set of density operators. We show that the
design of the optimal detector can be formulated as a semidefinite programming
problem. Based on this formulation, we derive a set of necessary and sufficient
conditions for an optimal quantum measurement. We then show that the optimal
measurement can be found by solving a standard (convex) semidefinite program
followed by the solution of a set of linear equations or, at worst, a standard
linear programming problem. By exploiting the many well-known algorithms for
solving semidefinite programs, which are guaranteed to converge to the global
optimum, the optimal measurement can be computed very efficiently in polynomial
time.
Using the semidefinite programming formulation, we also show that the rank of
each optimal measurement operator is no larger than the rank of the
corresponding density operator. In particular, if the quantum state ensemble is
a pure-state ensemble consisting of (not necessarily independent) rank-one
density operators, then we show that the optimal measurement is a pure-state
measurement consisting of rank-one measurement operators.Comment: Submitted to IEEE Transactions on Information Theor
Two-sided estimates of minimum-error distinguishability of mixed quantum states via generalized Holevo-Curlander bounds
We prove a concise factor-of-2 estimate for the failure rate of optimally
distinguishing an arbitrary ensemble of mixed quantum states, generalizing work
of Holevo [Theor. Probab. Appl. 23, 411 (1978)] and Curlander [Ph.D. Thesis,
MIT, 1979]. A modification to the minimal principle of Cocha and Poor
[Proceedings of the 6th International Conference on Quantum Communication,
Measurement, and Computing (Rinton, Princeton, NJ, 2003)] is used to derive a
suboptimal measurement which has an error rate within a factor of 2 of the
optimal by construction. This measurement is quadratically weighted and has
appeared as the first iterate of a sequence of measurements proposed by Jezek
et al. [Phys. Rev. A 65, 060301 (2002)]. Unlike the so-called pretty good
measurement, it coincides with Holevo's asymptotically optimal measurement in
the case of nonequiprobable pure states. A quadratically weighted version of
the measurement bound by Barnum and Knill [J. Math. Phys. 43, 2097 (2002)] is
proven. Bounds on the distinguishability of syndromes in the sense of
Schumacher and Westmoreland [Phys. Rev. A 56, 131 (1997)] appear as a
corollary. An appendix relates our bounds to the trace-Jensen inequality.Comment: It was not realized at the time of publication that the lower bound
of Theorem 10 has a simple generalization using matrix monotonicity (See [J.
Math. Phys. 50, 062102]). Furthermore, this generalization is a trivial
variation of a previously-obtained bound of Ogawa and Nagaoka [IEEE Trans.
Inf. Theory 45, 2486-2489 (1999)], which had been overlooked by the autho
Theory of Quantum Pulse Position Modulation and Related Numerical Problems
The paper deals with quantum pulse position modulation (PPM), both in the
absence (pure states) and in the presence (mixed states) of thermal noise,
using the Glauber representation of coherent laser radiation. The objective is
to find optimal (or suboptimal) measurement operators and to evaluate the
corresponding error probability. For PPM, the correct formulation of quantum
states is given by the tensorial product of m identical Hilbert spaces, where m
is the PPM order. The presence of mixed states, due to thermal noise, generates
an optimization problem involving matrices of huge dimensions, which already
for 4-PPM, are of the order of ten thousand. To overcome this computational
complexity, the currently available methods of quantum detection, which are
based on explicit results, convex linear programming and square root
measurement, are compared to find the computationally less expensive one. In
this paper a fundamental role is played by the geometrically uniform symmetry
of the quantum PPM format. The evaluation of error probability confirms the
vast superiority of the quantum detection over its classical counterpart.Comment: 10 pages, 7 figures, accepted for publication in IEEE Trans. on
Communication
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
A Semidefinite Programming Approach to Optimal Unambiguous Discrimination of Quantum States
In this paper we consider the problem of unambiguous discrimination between a
set of linearly independent pure quantum states. We show that the design of the
optimal measurement that minimizes the probability of an inconclusive result
can be formulated as a semidefinite programming problem. Based on this
formulation, we develop a set of necessary and sufficient conditions for an
optimal quantum measurement. We show that the optimal measurement can be
computed very efficiently in polynomial time by exploiting the many well-known
algorithms for solving semidefinite programs, which are guaranteed to converge
to the global optimum.
Using the general conditions for optimality, we derive necessary and
sufficient conditions so that the measurement that results in an equal
probability of an inconclusive result for each one of the quantum states is
optimal. We refer to this measurement as the equal-probability measurement
(EPM). We then show that for any state set, the prior probabilities of the
states can be chosen such that the EPM is optimal.
Finally, we consider state sets with strong symmetry properties and equal
prior probabilities for which the EPM is optimal. We first consider
geometrically uniform state sets that are defined over a group of unitary
matrices and are generated by a single generating vector. We then consider
compound geometrically uniform state sets which are generated by a group of
unitary matrices using multiple generating vectors, where the generating
vectors satisfy a certain (weighted) norm constraint.Comment: To appear in IEEE Transactions on Information Theor
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