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Register a new userConstrunctions of LOCC indistinguishable set of generalized Bell states
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In this paper, we mainly consider the local indistinguishability of the set of mutually orthogonal bipartite generalized Bell states (GBSs). We construct small sets of GBSs with cardinality smaller than $d$ which are not distinguished by one-way local operations and classical communication (1-LOCC) in $dotimes d$. The constructions, based on linear system and Vandermonde matrix, is simple and effective. The results give a unified upper bound for the minimum cardinality of 1-LOCC indistinguishable set of GBSs, and greatly improve previous results in [Zhang emph{et al.}, Phys. Rev. A 91, 012329 (2015); Wang emph{et al.}, Quantum Inf. Process. 15, 1661 (2016)]. The case that $d$ is odd of the results also shows that the set of 4 GBSs in $5otimes 5$ in [Fan, Phys. Rev. A 75, 014305 (2007)] is indeed a 1-LOCC indistinguishable set which can not be distinguished by Fans method.
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In general, for a bipartite quantum system $mathbb{C}^{d}otimesmathbb{C}^{d}$ and an integer $k$ such that $4leq kle d$,there are few necessary and sufficient conditions for local discrimination of sets of $k$ generalized Bell states (GBSs) and it is difficult to locally distinguish $k$-GBS sets.In this paper, we consider the local discrimination of GBS sets and the purpose is to completely solve the problem of local discrimination of GBS sets in some bipartite quantum systems,specifically, we show some necessary and sufficient conditions for local discrimination of GBS sets by which the local discrimination of GBS sets can be quickly determined.Firstly some sufficient conditions are given, these sufficient conditions are practical and effective.Fan$^{,}$s and Wang et al.$^{,}$s results (Phys Rev Lett 92:177905, 2004: Phys Rev A 99:022307, 2019) can be deduced as special cases of these conditions.Secondly in $mathbb{C}^{4}otimesmathbb{C}^{4}$, a necessary and sufficient condition for local discrimination of GBS sets is provided,all locally indistinguishable 4-GBS sets are found,and then we can quickly determine the local discriminability of an arbitrary GBS set.In $mathbb{C}^{5}otimesmathbb{C}^{5}$, a concise necessary and sufficient condition for one-way local discrimination of GBS sets is obtained,which gives an affirmative answer to the case $d=5$ of the problem proposed by Wang et al. (Phys Rev A 99:022307, 2019).
The arithmetic problem of factoring an integer $N$ can be translated into the physics of a quantum device, a result that supports Polyas and Hilberts conjecture to prove Riemanns hypothesis. The energies of this system, being univocally related to the factors of $N$, are the eigenvalues of a bounded Hamiltonian. Here we solve the quantum conditions and show that the histogram of the discrete energies, provided by the spectrum of the system, should be interpreted in number theory as the relative probability for a prime to be a factor candidate of $N$. This is equivalent to a quantum sieve that is demonstrated to require only $ o(log sqrt N)^3$ energy measurements to solve the problem, recovering Shors complexity result. Hence, the outcome can be seen as a probability map that a pair of primes solve the given factorization problem. Furthermore, we show that a possible embodiment of this quantum simulator corresponds to two entangled particles in a Penning trap. The possibility to build the simulator experimentally is studied in detail. The results show that factoring numbers, many orders of magnitude larger than those computed with experimentally available quantum computers, is achievable using typical parameters in Penning traps.
In this paper, we study the local unitary classification for pairs (triples) of generalized Bell states, based on the local unitary equivalence of two sets. In detail, we firstly introduce some general unitary operators which give us more local unitary equivalent sets besides Clifford operators. And then we present two necessary conditions for local unitary equivalent sets which can be used to examine the local inequivalence. Following this approach, we completely classify all of pairs in $dotimes d$ quantum system into $prod_{j=1}^{n} (k_{j} + 1) $ LU-inequivalent pairs when the prime factorization of $d=prod_{j=1}^{n}p_j^{k_j}$. Moreover, all of triples in $p^alphaotimes p^alpha$ quantum system for prime $p$ can be partitioned into $frac{(alpha + 3)}{6}p^{alpha} + O(alpha p^{alpha-1})$ LU-inequivalent triples, especially, when $alpha=2$ and $p>2$, there are exactly $lfloor frac{5}{6}p^{2}rfloor + lfloor frac{p-2}{6}+(-1)^{lfloorfrac{p}{3}rfloor}frac{p}{3}rfloor + 3$ LU-inequivalent triples.
We consider a notion of relative homology (and cohomology) for surfaces with two types of boundaries. Using this tool, we study a generalization of Kitaevs code based on surfaces with mixed boundaries. This construction includes both Bravyi and Kitaevs and Freedman and Meyers extension of Kitaevs toric code. We argue that our generalization offers a denser storage of quantum information. In a planar architecture, we obtain a three-fold overhead reduction over the standard architecture consisting of a punctured square lattice.
Quantum discord Q is a function of density matrix elements. The domain of such a function in the case of two-qubit system with X density matrix may consist of three subdomains at most: two ones where the quantum discord is expressed in closed analytical forms (Q_{pi/2} and Q_0) and an intermediate subdomain for which, to extract the quantum discord Q_theta, it is required to solve in general numerically a one-dimensional minimization problem to find the optimal measurement angle thetain(0,pi/2). Hence the quantum discord is given by a piecewise-analytic-numerical formula Q=min{Q_{pi/2}, Q_theta, Q_0}. Equations for determining the boundaries between these subdomains are obtained. The boundaries consist of bifurcation points. The Q_{theta} subdomains are discovered in the generalized Horodecki states, in the dynamical phase flip channel model, in the anisotropic spin systems at thermal equilibrium, in the heteronuclear dimers in an external magnetic field. We found that transitions between Q_{theta} subdomain and Q_{pi/2} and Q_0 ones occur suddenly but continuously and smoothly, i.e., nonanalyticity is hidden and can be observed in higher derivatives of discord function.
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