QIC Abstracts

 Vol.13 No.7&8, July 1, 2013

Research Articles:

Multiaccess quantum communication and product higher rank numerical range (pp0541-0566)
Maciej Demianowicz, Pawel Horodecki, and Karol Zyczkowski
In the present paper we initiate the study of the product higher rank numerical range. The latter, being a variant of the higher rank numerical range [M.--D. Choi {\it et al.}, Rep. Math. Phys. {\bf 58}, 77 (2006); Lin. Alg. Appl. {\bf 418}, 828 (2006)], is a natural tool for studying a construction of quantum error correction codes for multiple access channels. We review properties of this set and relate it to other numerical ranges, which were recently introduced in the literature. Further, the concept is applied to the construction of codes for bi--unitary two--access channels with a hermitian noise model. Analytical techniques for both outerbounding the product higher rank numerical range and determining its exact shape are developed for this case. Finally, the reverse problem of constructing a noise model for a given product range is considered.

No-broadcasting of non-signalling boxes via operations which transform local boxes into local ones (pp0567-0582)
P. Joshi, Andrzej Grudka, Karol Horodecki, Michal Horodecki, Pawel Horodecki, and Ryszard Horodecki
We deal with families of probability distributions satisfying non-signalling condition, called non-signalling boxes and consider a class of operations that transform local boxes into local ones (the one that admit LHV model). We prove that any operation from this class cannot broadcast a bipartite non-local box with 2 binary inputs and outputs. We consider a function called anti-Robustness which can not decrease under these operations. The proof reduces to showing that anti-Robustness would decrease after broadcasting.

Quantum fingerprints that keep secrets (pp0583-0606)
Dmitry Gavinsky and Tsuyoshi Ito
We introduce a new type of cryptographic primitive that we call a \e{hiding fingerprinting scheme}.
A (quantum) fingerprinting scheme maps a binary string of length $n$ to $d$ (qu)bits, typically $d\ll n$, such that given any string $y$ and a fingerprint of $x$, one can decide with high accuracy whether $x=y$. It can be seen that a classical fingerprint of $x$ that guarantees error $\le\eps$ necessarily reveals \asOm{\Min{n,\log(1/\eps)}} bits of information about $x$. We call a scheme \e{hiding} if it reveals \aso{\Min{n,\log(1/\eps)}} bits; accordingly, no classical scheme is hiding. }{We construct quantum hiding fingerprinting schemes. Our schemes are computationally efficient and their hiding properties are shown to be optimal.

Fast and efficient exact synthesis of single-qubit unitaries generated by Clifford and T gates (pp0607-0630)
Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca
In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$, in the single-qubit case. We report an efficient synthesis algorithm, with an exact optimality guarantee on the number of Hadamard gates used. We conjecture that the equivalence of the sets of unitaries implementable by circuits over the Clifford and T library and unitaries over the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$ holds in the $n$-qubit case.

Efficient quantum circuits for binary elliptic curve arithmetic: reducing $T$-gate complexity (pp0631-0644)
Brittanney Amento, Martin Rotteler, and Rainer Steinwandt
Elliptic curves over finite fields ${\mathbb F}_{2^n}$ play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with affine or projective coordinates. In this paper we show that changing the curve representation allows a substantial reduction in the number of $T$-gates needed to implement the curve arithmetic. As a tool, we present a quantum circuit for computing multiplicative inverses in $\mathbb F_{2^n}$ in depth $\bigO(n\log_2 n)$ using a polynomial basis representation, which may be of independent interest.

Most robust and fragile two-qubit entangled states under deploarizing channels (pp0645-0660)
Chao-Qian Pang, Fu-Lin Zhang, Yue Jiang, Mai-Lin Liang, and Jing-Ling Chen
For a two-qubit system under local depolarizing channels, the most robust and most fragile states are derived for a given concurrence or negativity. For the one-sided channel, the pure states are proved to be the most robust ones, with the aid of the evolution equation for entanglement given by Konrad \emph{et al.} [Nat. Phys. 4, 99 (2008)].  Based on a generalization of the evolution equation for entanglement, we classify the ansatz states in our investigation by the amount of robustness, and consequently derive the most fragile states. For the two-sided channel, the pure states are the most robust for a fixed concurrence. Under the uniform channel, the most fragile states have the minimal negativity when the concurrence is given in the region $[1/2,1]$. For a given negativity, the most robust states are the ones with the maximal concurrence, and the most fragile ones are the pure states with minimum of concurrence. When the entanglement approaches zero, the most fragile states under general nonuniform channels tend to the ones in the uniform channel. Influences on robustness by entanglement, degree of mixture, and asymmetry between the two qubits are discussed through numerical calculations.  It turns out that the concurrence and negativity are major factors for the robustness. When they are fixed, the impact of the mixedness becomes obvious. In the nonuniform channels, the most fragile states are closely correlated with the asymmetry, while the most robust ones with the degree of mixture.

Limit theorems for the interference terms of discrete-time quantum walks on the line (pp0661-0671)
Takuya Machida
The probability distributions of discrete-time quantum walks have been often investigated, and many interesting properties of them have been discovered. The probability that the walker can be find at a position is defined by diagonal elements of the density matrix. On the other hand, although off-diagonal parts of the density matrices have an important role to quantify quantumness, they have not received attention in quantum walks. We focus on the off-diagonal parts of the density matrices for discrete-time quantum walks on the line and derive limit theorems for them.

Galois automorphisms of a symmetric measurement (pp0672-0720)
D.M. Appleby, Hulya Yadsan-Appleby, and Gerhard Zauner
Symmetric Informationally Complete Positive Operator Valued Measures (usually referred to as SIC-POVMs or simply as SICs) have been constructed in every dimension $\le 67$. However, a proof that they exist in every finite dimension has yet to be constructed. In this paper we examine the Galois group of SICs covariant with respect to the Weyl-Heisenberg group (or WH SICs as we refer to them). The great majority (though not all) of the known examples are of this type. Scott and Grassl have noted that every known exact WH SIC is expressible in radicals (except for dimension $3$ which is exceptional in this and several other respects), which means that the corresponding Galois group is solvable. They have also calculated the Galois group for most known exact examples. The purpose of this paper is to take the analysis of Scott and Grassl further. We first prove a number of theorems regarding the structure of the Galois group and the relation between it and the extended Clifford group. We then examine the Galois group for the known exact fiducials and on the basis of this we propose a list of nine conjectures concerning its structure. These conjectures represent a considerable strengthening of the theorems we have actually been able to prove. Finally we generalize the concept of an anti-unitary to the concept of a $g$-unitary, and show that every WH SIC fiducial is an eigenvector of a family of $g$-unitaries (apart from dimension 3).

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