QIC Abstracts

 Vol.10 No.11&12 November 1, 2010

Research Articles:

Are quantum correlations symmetric? (pp0901-0910)
Karol Horodecki, Michal Horodecki, and Pawel Horodecki
We provide operational definition of asymmetry of entanglement: An entangled state contains asymmetric entanglement if its subsystems can not be exchanged (swapped) by means of local operations and classical communication. We show that in general states have asymmetric entanglement. This allows to construct nonsymmetric measure of entanglement, and a parameter that reports asymmetry of entanglement contents of quantum state. We propose asymptotic measure of asymmetry of entanglement, and show that states for which it is nonzero, contain necessarily bound entanglement.

A lower bound on the value of entangled binary games (pp0911-0924)
Salman Beigi
A two-player one-round binary game consists of two cooperative players who each replies by one bit to a message that he receives privately; they win the game if both questions and answers satisfy some predetermined property. A game is called entangled if the players are allowed to share a priori entanglement. It is well-known that the maximum winning probability (value) of entangled XOR-games (binary games in which the predetermined property depends only on the XOR of the two output bits) can be computed by a semidefinite program. In this paper we extend this result in the following sense; if a binary game is uniform, meaning that in an optimal strategy the marginal distributions of the output of each player are uniform, then its entangled value can be efficiently computed by a semidefinite program. We also introduce a lower bound on the entangled value of a general two-player one-round game; this bound depends on the size of the output set of each player and can be computed by a semidefinite program. In particular, we show that if the game is binary, $\omega_q$ is its entangled value, and $\omega_{sdp}$ is the optimum value of the corresponding semidefinite program, then $0.68\,\omega_{sdp} < \omega_q \leq \omega_{sdp}$.

When is there a multipartite maximum entangled state? (pp0925-0935)
Runyao Duan and Yaoyun Shi 
For a multipartite quantum system of the dimension $d_1\otimes d_2\otimes\cdots\otimes d_n$, where $d_1\ge d_2\ge\cdots\ge d_n\ge2$, is there an entangled state {\em maximum} in the sense that all other states in the system can be obtained from the state through local quantum operations and classical communications (LOCC)? When $d_1\ge\Pi_{i=2}^n d_i$, such state exists. We show that this condition is also necessary. Our proof, somewhat surprisingly, uses results from algebraic complexity theory.

High-fidelity universal quantum gates (pp0936-0946)
Ran Li and Frank Gaitan
Twisted rapid passage is a type of non-adiabatic rapid passage that generates controllable quantum interference effects that were first observed experimentally in $2003$. It is shown that twisted rapid passage sweeps can be used to implement a universal set of quantum gates $\calGU$ that operate with high-fidelity. The gate set $\calGU$ consists of the Hadamard and NOT gates, together with variants of the phase, $\pi /8$, and controlled-phase gates. For each gate $g$ in $\calGU$, sweep parameter values are provided which simulations indicate will produce a unitary operation that approximates $g$ with error probability$P_{e} < 10^{-4}$. Note that \textit{all\/} gates in $\calGU$ are implemented using a \textit{single family\/} of control-field, and the error probability for each gate falls below the rough-and-ready estimate for the accuracy threshold $P_{a}\sim 10^{-4}$.

Quantum entanglement of Dirac field in background of an asymptotically flat static black hole (pp0947-0955)
Jieci Wang, Qiyuan Pan, Songbai Chen, and Jiliang Jing
The entanglement of the Dirac field in the asymptotically flat black hole is investigated. Unlike the bosonic case in which the initial entanglement vanishes in the limit of infinite Hawking temperature, in this case the entanglement achieves a nonvanishing minimum values, which shows that the entanglement is never completely destroyed when black hole evaporates completely. Another interesting result is that the mutual information in this limit equals to just half of its own initial value, which may be an universal property for any fields.

New classes of TQC associated with self-dual, quasi self-dual and denser tessellations (pp0956-0970)
C. D. Albuquerque, R. Palazzo Jr., and E. B. Silva
In this paper we present six classes of topological quantum codes (TQC) on compact surfaces with genus $g\ge 2$. These codes are derived from self-dual, quasi self-dual and denser tessellations associated with embeddings of self-dual complete graphs and complete bipartite graphs on the corresponding compact surfaces. The majority of the new classes has the self-dual tessellations as their algebraic and geometric supporting mathematical structures. Every code achieves minimum distance 3 and its encoding rate is such that $\frac{k}{n} \rightarrow 1$ as $n \rightarrow \infty$, except for the one case where $\frac{k}{n} \rightarrow \frac{1}{3}$ as $n \rightarrow \infty$.

No-cloning theorem for a single POVM (pp0971-0980)
Alexey E. Rastegin
Cloning of statistics of general quantum measurement is discussed. The presented approach is connected with the known concept of observable cloning, but differs in some essential respects. The reasons are illustrated within some variety of the B92 protocol. As it is shown, there exist pairs of states such that the perfect cloning of given POVM is not possible. We discuss some properties of these intolerant sets. An example allowing the perfect cloning is presented as well.

On Steane's  enlargement of Calderbank-Shor-Steane codes (pp0981-0993)
Hachiro Fujita
In this paper we present the Hilbert space representation and the decoding of the enlarged Calderbank-Shor-Steane code constructed by Steane. We also give a generalization of the code.

Geometric measure of quantum discord under decoherence (pp0994-1003)
Xiao-Ming Lu, Zhengjun Xi, Zhe Sun, and Xiaoguang Wang
The dynamics of a geometric measure of the quantum discord (GMQD) under decoherence is investigated. We show that the GMQD of a two-qubit state can be alternatively obtained through the singular values of a $3\times4$ matrix whose elements are the expectation values of Pauli matrices of the two qubits. By using Heisenberg picture, the analytic results of the GMQD is obtained for three typical kinds of the quantum decoherence channels. We compare the dynamics of the GMQD with that of the quantum discord and of entanglement. We show that a sudden change in the decay rate of the GMQD does not always imply that of the quantum discord, and vice versa. We also give a general analysis on the sudden change in behavior and find that at least for the Bell diagonal states, the sudden changes in decay rates of the GMQD and that of the quantum discord occur simultaneously.

Limit theorems for quantum walks with memory (pp1004-1017)
Norio Konno and Takuya Machida
Recently Mc Gettrick introduced and studied a discrete-time 2-state quantum walk (QW) with a memory in one dimension. He gave an expression for the amplitude of the QW by path counting method. Moreover he showed that the return probability of the walk is more than 1/2 for any even time. In this paper, we compute the stationary distribution by considering the walk as a 4-state QW without memory. Our result is consistent with his claim. In addition, we obtain the weak limit theorem of the rescaled QW. This behavior is strikingly different from the corresponding classical random walk and the usual 2-state QW without memory as his numerical simulations suggested.

Relationship between the n-tangle and the residual entanglement of even n qubits (pp1018-1028)
Xiangrong Li and Dafa Li
We show that for even $n$ qubits,$\ n$-tangle is the square of the SLOCC polynomial invariant of degree 2. We find that the $n$-tangle is not the residual entanglement for any even $n\geq 4$\ qubits. We give a necessary and sufficient condition for the vanishing of the concurrence $C_{1(2...n)}$. The condition implies that the concurrence $C_{1(2...n)}$ is always positive for any entangled states while the $n$-tangle vanishes for some entangled states. We argue that for even $n$\ qubits, the concurrence $ C_{1(2...n)}$\ is equal to or greater than the $n$-tangle.\ Further, we reveal that the residual entanglement is a partial measure for product states of any $n$ qubits while the $n$-tangle is multiplicative for some product states.

Symmetric states: local unitary equivalence via stabilizers (pp1029-1041)
Curt D. Cenci, David W. Lyons, Laura M. Snyder, and Scott N. Walck
We classify local unitary equivalence classes of symmetric states via a classification of their local unitary stabilizer subgroups. For states whose local unitary stabilizer groups have a positive number of continuous degrees of freedom, the classification is exhaustive. We show that local unitary stabilizer groups with no continuous degrees of freedom are isomorphic to finite subgroups of the rotation group $SO(3)$, and give examples of states with discrete stabilizers.

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