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 $N=p^r$, where $p$ is an odd prime, in terms of the character vectors of the cyclic group $G$ of order $p$. This form may be useful in explicitly writing down mutually unbiased bases for $N=p^r$.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 $n$ and are described in a Hilbert space of dimensions $2n$. 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 ${\cal Z}_d$ and which are described with $d$-dimensional Hilbert space, are considered. When $d$ is the power of a prime, the position and momentum take values in the Galois field $GF(p^ \ell)$, 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