28 research outputs found
Aspects of mutually unbiased bases in odd prime power dimensions
We rephrase the Wootters-Fields construction [Ann. Phys., {\bf 191}, 363
(1989)] of a full set of mutually unbiased bases in a complex vector space of
dimensions , where is an odd prime, in terms of the character
vectors of the cyclic group of order . This form may be useful in
explicitly writing down mutually unbiased bases for .Comment: 3 pages, latex, no figure
Singlet states and the estimation of eigenstates and eigenvalues of an unknown Controlled-U gate
We consider several problems that involve finding the eigenvalues and
generating the eigenstates of unknown unitary gates. We first examine
Controlled-U gates that act on qubits, and assume that we know the eigenvalues.
It is then shown how to use singlet states to produce qubits in the eigenstates
of the gate. We then remove the assumption that we know the eigenvalues and
show how to both find the eigenvalues and produce qubits in the eigenstates.
Finally, we look at the case where the unitary operator acts on qutrits and has
eigenvalues of 1 and -1, where the eigenvalue 1 is doubly degenerate. The
eigenstates are unknown. We are able to use a singlet state to produce a qutrit
in the eigenstate corresponding to the -1 eigenvalue.Comment: Latex, 10 pages, no figure
Optimal Conclusive Discrimination of Two Non-orthogonal Pure Product Multipartite States Locally
We consider one copy of a quantum system prepared in one of two
non-orthogonal pure product states of multipartite distributed among separated
parties. We show that there exist protocols which obtain optimal probability in
the sense of conclusive discrimination by means of local operations and
classical communications(LOCC) as good as by global operations. Also, we show a
protocol which minimezes the average number of local operations. Our result
implies that two product pure multipartite states might not have the non-local
property though more than two can have.Comment: revtex, 3 pages, no figur
Reduction Theorems for Optimal Unambiguous State Discrimination of Density Matrices
We present reduction theorems for the problem of optimal unambiguous state
discrimination (USD) of two general density matrices. We show that this problem
can be reduced to that of two density matrices that have the same rank and
are described in a Hilbert space of dimensions . We also show how to use
the reduction theorems to discriminate unambiguously between N mixed states (N
\ge 2).Comment: 6 pages, 1 figur
Optimally Conclusive Discrimination of Non-orthogonal Entangled States Locally
We consider one copy of a quantum system prepared with equal prior
probability in one of two non-orthogonal entangled states of multipartite
distributed among separated parties. We demonstrate that these two states can
be optimally distinguished in the sense of conclusive discrimination by local
operations and classical communications(LOCC) alone. And this proves strictly
the conjecture that Virmani et.al. [8] confirmed numerically and analytically.
Generally, the optimal protocol requires local POVM operations which are
explicitly constructed. The result manifests that the distinguishable
information is obtained only and completely at the last operation and all prior
ones give no information about that state.Comment: 4 pages, no figure, revtex. few typos correcte
The Frobenius formalism in Galois quantum systems
Quantum systems in which the position and momentum take values in the ring
and which are described with -dimensional Hilbert space, are
considered. When is the power of a prime, the position and momentum take
values in the Galois field , the position-momentum phase space is
a finite geometry and the corresponding `Galois quantum systems' have stronger
properties. The study of these systems uses ideas from the subject of field
extension in the context of quantum mechanics. The Frobenius automorphism in
Galois fields leads to Frobenius subspaces and Frobenius transformations in
Galois quantum systems. Links between the Frobenius formalism and Riemann
surfaces, are discussed
The minimum-error discrimination via Helstrom family of ensembles and Convex Optimization
Using the convex optimization method and Helstrom family of ensembles
introduced in Ref. [1], we have discussed optimal ambiguous discrimination in
qubit systems. We have analyzed the problem of the optimal discrimination of N
known quantum states and have obtained maximum success probability and optimal
measurement for N known quantum states with equiprobable prior probabilities
and equidistant from center of the Bloch ball, not all of which are on the one
half of the Bloch ball and all of the conjugate states are pure. An exact
solution has also been given for arbitrary three known quantum states. The
given examples which use our method include: 1. Diagonal N mixed states; 2. N
equiprobable states and equidistant from center of the Bloch ball which their
corresponding Bloch vectors are inclined at the equal angle from z axis; 3.
Three mirror-symmetric states; 4. States that have been prepared with equal
prior probabilities on vertices of a Platonic solid.
Keywords: minimum-error discrimination, success probability, measurement,
POVM elements, Helstrom family of ensembles, convex optimization, conjugate
states PACS Nos: 03.67.Hk, 03.65.TaComment: 15 page
Quantum Gambling Using Two Nonorthogonal States
We give a (remote) quantum gambling scheme that makes use of the fact that
quantum nonorthogonal states cannot be distinguished with certainty. In the
proposed scheme, two participants Alice and Bob can be regarded as playing a
game of making guesses on identities of quantum states that are in one of two
given nonorthogonal states: if Bob makes a correct (an incorrect) guess on the
identity of a quantum state that Alice has sent, he wins (loses). It is shown
that the proposed scheme is secure against the nonentanglement attack. It can
also be shown heuristically that the scheme is secure in the case of the
entanglement attack.Comment: no essential correction, 4 pages, RevTe
Mixed quantum state detection with inconclusive results
We consider the problem of designing an optimal quantum detector with a fixed
rate of inconclusive results that maximizes the probability of correct
detection, when distinguishing between a collection of mixed quantum states. We
develop a sufficient condition for the scaled inverse measurement to maximize
the probability of correct detection for the case in which the rate of
inconclusive results exceeds a certain threshold. Using this condition we
derive the optimal measurement for linearly independent pure-state sets, and
for mixed-state sets with a broad class of symmetries. Specifically, we
consider geometrically uniform (GU) state sets and compound geometrically
uniform (CGU) state sets with generators that satisfy a certain constraint.
We then show that the optimal measurements corresponding to GU and CGU state
sets with arbitrary generators are also GU and CGU respectively, with
generators that can be computed very efficiently in polynomial time within any
desired accuracy by solving a semidefinite programming problem.Comment: Submitted to Phys. Rev.
Greenberger-Horne-Zeilinger paradoxes for N quNits
In this paper we show the series of Greenberger-Horne-Zeilinger paradoxes for
N maximally entangled N-dimensional quantum systems.Comment: 6 page