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Salman Beigi

Publications and source records attributed to Salman Beigi.

At least 55 records · Page 3Linked to original sources
arXiv2016

On the Duality of Additivity and Tensorization

A function is said to be additive if, similar to mutual information, expands by a factor of $n$, when evaluated on $n$ i.i.d. repetitions of a source or channel. On the other hand, a function is said to satisfy the tensorization property if it remains unchanged when evaluated on i.i.d. repetitions. Additive rate regions are of fundamental importance in network information theory, serving as capacity regions or upper bounds thereof. Tensorizing measures of correlation have also found applications in distributed source and channel coding problems as well as the distribution simulation problem. Prior to our work only two measures of correlation, namely the hypercontractivity ribbon and maximal correlation (and their derivatives), were known to have the tensorization property. In this paper, we provide a general framework to obtain a region with the tensorization property from any additive rate region. We observe that hypercontractivity ribbon indeed comes from the dual of the rate region of the Gray-Wyner source coding problem, and generalize it to the multipartite case. Then we define other measures of correlation with similar properties from other source coding problems. We also present some applications of our results.

cs.IT↗
arXiv2016

Phi-Entropic Measures of Correlation

A measure of correlation is said to have the tensorization property if it is unchanged when computed for i.i.d.\ copies. More precisely, a measure of correlation between two random variables $(X, Y)$ denoted by $ρ(X, Y)$, has the tensorization property if $ρ(X^n, Y^n)=ρ(X, Y)$ where $(X^n, Y^n)$ is $n$ i.i.d.\ copies of $(X, Y)$.Two well-known examples of such measures are the maximal correlation and the hypercontractivity ribbon (HC~ribbon). We show that the maximal correlation and HC ribbons are special cases of $Φ$-ribbon, defined in this paper for any function $Φ$ from a class of convex functions ($Φ$-ribbon reduces to HC~ribbon and the maximal correlation for special choices of $Φ$). Any $Φ$-ribbon is shown to be a measures of correlation with the tensorization property. We show that the $Φ$-ribbon also characterizes the $Φ$-strong data processing inequality constant introduced by Raginsky. We further study the $Φ$-ribbon for the choice of $Φ(t)=t^2$ and introduce an equivalent characterization of this ribbon.

cs.IT↗
arXiv2015

Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm

We develop the notions of hypercontractivity (HC) and the log-Sobolev (LS) inequality for completely bounded norms of one-parameter semigroups of super-operators acting on matrix algebras. We prove the equivalence of the completely bounded versions of HC and LS under suitable hypotheses. We also prove a version of the Gross Lemma which allows LS at general $q$ to be deduced from LS at $q=2$.

math-ph↗
arXiv2014

Deterministic Randomness Extraction from Generalized and Distributed Santha-Vazirani Sources

A Santha-Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for non-binary sequences. We show that unlike the binary case, deterministic randomness extraction in the generalized case is sometimes possible. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions coincide in "non-degenerate" cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs $(a_1, b_1), (a_2, b_2), \ldots$ distributed between two parties, where the first party receives $a_i$'s and the second one receives $b_i$'s. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (non-distributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from i.i.d. sources to adversarial sources.

cs.CC↗
arXiv2014

The Value of Help Bits in Randomized and Average-Case Complexity

"Help bits" are some limited trusted information about an instance or instances of a computational problem that may reduce the computational complexity of solving that instance or instances. In this paper, we study the value of help bits in the settings of randomized and average-case complexity. Amir, Beigel, and Gasarch (1990) show that for constant $k$, if $k$ instances of a decision problem can be efficiently solved using less than $k$ bits of help, then the problem is in P/poly. We extend this result to the setting of randomized computation: We show that the decision problem is in P/poly if using $\ell$ help bits, $k$ instances of the problem can be efficiently solved with probability greater than $2^{\ell-k}$. The same result holds if using less than $k(1 - h(α))$ help bits (where $h(\cdot)$ is the binary entropy function), we can efficiently solve $(1-α)$ fraction of the instances correctly with non-vanishing probability. We also extend these two results to non-constant but logarithmic $k$. In this case however, instead of showing that the problem is in P/poly we show that it satisfies "$k$-membership comparability," a notion known to be related to solving $k$ instances using less than $k$ bits of help. Next we consider the setting of average-case complexity: Assume that we can solve $k$ instances of a decision problem using some help bits whose entropy is less than $k$ when the $k$ instances are drawn independently from a particular distribution. Then we can efficiently solve an instance drawn from that distribution with probability better than $1/2$. Finally, we show that in the case where $k$ is super-logarithmic, assuming $k$-membership comparability of a decision problem, one cannot prove that the problem is in P/poly by a "black-box proof."

cs.CC↗
arXiv2014

Maximal Entanglement - A New Measure of Entanglement

Maximal correlation is a measure of correlation for bipartite distributions. This measure has two intriguing features: (1) it is monotone under local stochastic maps; (2) it gives the same number when computed on i.i.d. copies of a pair of random variables. This measure of correlation has recently been generalized for bipartite quantum states, for which the same properties have been proved. In this paper, based on maximal correlation, we define a new measure of entanglement which we call maximal entanglement. We show that this measure of entanglement is faithful (is zero on separable states and positive on entangled states), is monotone under local quantum operations, and gives the same number when computed on tensor powers of a bipartite state.

quant-ph↗
arXiv2013

Sandwiched Rényi Divergence Satisfies Data Processing Inequality

Sandwiched (quantum) $α$-Rényi divergence has been recently defined in the independent works of Wilde et al. (arXiv:1306.1586) and Müller-Lennert et al (arXiv:1306.3142v1). This new quantum divergence has already found applications in quantum information theory. Here we further investigate properties of this new quantum divergence. In particular we show that sandwiched $α$-Rényi divergence satisfies the data processing inequality for all values of $α> 1$. Moreover we prove that $α$-Holevo information, a variant of Holevo information defined in terms of sandwiched $α$-Rényi divergence, is super-additive. Our results are based on Hölder's inequality, the Riesz-Thorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.

quant-ph↗
arXiv2013

On Dimension Bounds for Auxiliary Quantum Systems

Expressions of several capacity regions in quantum information theory involve an optimization over auxiliary quantum registers. Evaluating such expressions requires bounds on the dimension of the Hilbert space of these auxiliary registers, for which no non-trivial technique is known; we lack a quantum analog of the Carathéodory theorem. In this paper, we develop a new non-Carathéodory-type tool for evaluating expressions involving a single quantum auxiliary register and several classical random variables. As we show, such expressions appear in problems of entanglement-assisted Gray-Wyner and entanglement-assisted channel simulation, where the question of whether entanglement helps in these settings is related to that of evaluating expressions with a single quantum auxiliary register. To evaluate such expressions, we argue that developing a quantum analog of the Carathéodory theorem requires a better understanding of a notion which we call ``quantum conditioning." We then proceed by proving a few results about quantum conditioning, one of which is that quantum conditioning is strictly richer than the usual classical conditioning.

quant-ph↗
arXiv2013

Symmetries of Codeword Stabilized Quantum Codes

Symmetry is at the heart of coding theory. Codes with symmetry, especially cyclic codes, play an essential role in both theory and practical applications of classical error-correcting codes. Here we examine symmetry properties for codeword stabilized (CWS) quantum codes, which is the most general framework for constructing quantum error-correcting codes known to date. A CWS code Q can be represented by a self-dual additive code S and a classical code C, i.,e., Q=(S,C), however this representation is in general not unique. We show that for any CWS code Q with certain permutation symmetry, one can always find a self-dual additive code S with the same permutation symmetry as Q such that Q=(S,C). As many good CWS codes have been found by starting from a chosen S, this ensures that when trying to find CWS codes with certain permutation symmetry, the choice of S with the same symmetry will suffice. A key step for this result is a new canonical representation for CWS codes, which is given in terms of a unique decomposition as union stabilizer codes. For CWS codes, so far mainly the standard form (G,C) has been considered, where G is a graph state. We analyze the symmetry of the corresponding graph of G, which in general cannot possess the same permutation symmetry as Q. We show that it is indeed the case for the toric code on a square lattice with translational symmetry, even if its encoding graph can be chosen to be translational invariant.

quant-ph↗
arXiv2013

Information Causality is a Special Point in the Dual of the Gray-Wyner Region

Information Causality contributes to the program of deriving fundamentals of quantum theory from information theoretic principles. It puts restrictions on the amount of information learned by a party (Bob) from the other party (Alice) in a one-way communication scenario: Bob receives an index b, and after a one-way communication from Alice, tries to recover the b-th bit of Alice's input. In this manuscript we note that Information Causality is indeed a point in the dual of the Gray-Wyner region in the case where Alice's input bits are chosen independently at random. The main motivation of this work was to study this connection in the case of correlated input bits. Our approach to forge this connection was to propose a new postulate called Accessibility of Mutual Information (AMI) on the underling physical theory. This approach, explained in the first version of this paper, fails since AMI is not satisfied by the quantum theory.

quant-ph↗
arXiv2012

Indistinguishable Chargeon-Fluxion Pairs in the Quantum Double of Finite Groups

We consider the category of finite dimensional representations of the quantum double of a finite group as a modular tensor category. We study auto-equivalences of this category whose induced permutations on the set of simple objects (particles) are of the special form of PJ, where J sends every particle to its charge conjugation and P is a transposition of a chargeon-fluxion pair. We prove that if the underlying group is the semidirect product of the additive and multiplicative groups of a finite field, then such an auto-equivalence exists. In particular, we show that for S_3 (the permutation group over three letters) there is a chargeon and a fluxion which are not distinguishable. Conversely, by considering such permutations as modular invariants, we show that a transposition of a chargeon-fluxion pair forms a modular invariant if and only if the corresponding group is isomorphic to the semidirect product of the additive and multiplicative groups of a finite near-field.

quant-ph↗
arXiv2011

Classification of the phases of 1D spin chains with commuting Hamiltonians

We consider the class of spin Hamiltonians on a 1D chain with periodic boundary conditions that are (i) translational invariant, (ii) commuting and (iii) scale invariant, where by the latter we mean that the ground state degeneracy is independent of the system size. We correspond a directed graph to a Hamiltonian of this form and show that the structure of its ground space can be read from the cycles of the graph. We show that the ground state degeneracy is the only parameter that distinguishes the phases of these Hamiltonians. Our main tool in this paper is the idea of Bravyi and Vyalyi (2005) in using the representation theory of finite dimensional C^*-algebras to study commuting Hamiltonians.

quant-ph↗
arXiv2011

Simplified instantaneous non-local quantum computation with applications to position-based cryptography

Instantaneous measurements of non-local observables between space-like separated regions can be performed without violating causality. This feat relies on the use of entanglement. Here we propose novel protocols for this task and the related problem of multipartite quantum computation with local operations and a single round of classical communication. Compared to previously known techniques, our protocols reduce the entanglement consumption by an exponential amount. We also prove a linear lower bound on the amount of entanglement required for the implementation of a certain non-local measurement. These results relate to position-based cryptography: an amount of entanglement scaling exponentially in the number of communicated qubits is sufficient to render any such scheme insecure. Furthermore, we show that certain schemes are secure under the assumption that the adversary has less entanglement than a given linear bound and is restricted to classical communication.

quant-ph↗
arXiv2011

A Lower Bound on the Value of Entangled Binary Games

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, w_q is its entangled value, and w_{sdp} is the optimum value of the corresponding semidefinite program, then 0.68w_{sdp} < w_q <= w_{sdp}.

quant-ph↗
arXiv2011

Quantum interactive proofs with short messages

This paper considers three variants of quantum interactive proof systems in which short (meaning logarithmic-length) messages are exchanged between the prover and verifier. The first variant is one in which the verifier sends a short message to the prover, and the prover responds with an ordinary, or polynomial-length, message; the second variant is one in which any number of messages can be exchanged, but where the combined length of all the messages is logarithmic; and the third variant is one in which the verifier sends polynomially many random bits to the prover, who responds with a short quantum message. We prove that in all of these cases the short messages can be eliminated without changing the power of the model, so the first variant has the expressive power of QMA and the second and third variants have the expressive power of BQP. These facts are proved through the use of quantum state tomography, along with the finite quantum de Finetti theorem for the first variant.

quant-ph↗
arXiv2011

Approximating the Set of Separable States Using the Positive Partial Transpose Test

The positive partial transpose test is one of the main criteria for detecting entanglement, and the set of states with positive partial transpose is considered as an approximation of the set of separable states. However, we do not know to what extent this criterion, as well as the approximation, are efficient. In this paper, we show that the positive partial transpose test gives no bound on the distance of a density matrix from separable states. More precisely, we prove that, as the dimension of the space tends to infinity, the maximum trace distance of a positive partial transpose state from separable states tends to 1. Using similar techniques, we show that the same result holds for other well-known separability criteria such as reduction criterion, majorization criterion and symmetric extension criterion. We also bring an evidence that the sets of positive partial transpose states and separable states have totally different shapes.

quant-ph↗
arXiv2011

Entanglement-assisted zero-error capacity is upper bounded by the Lovasz theta function

The zero-error capacity of a classical channel is expressed in terms of the independence number of some graph and its tensor powers. This quantity is hard to compute even for small graphs such as the cycle of length seven, so upper bounds such as the Lovasz theta function play an important role in zero-error communication. In this paper, we show that the Lovasz theta function is an upper bound on the zero-error capacity even in the presence of entanglement between the sender and receiver.

quant-ph↗