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Quantum Communication and Complexity
 Theoretical Computer Science
, 2000
"... In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We sur ..."
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In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We survey the main results of the young area of quantum communication complexity: its relation to teleportation and dense coding, the main examples of fast quantum communication protocols, lower bounds, and some applications. 1 Introduction The area of communication complexity deals with the following type of problem. There are two separated parties, called Alice and Bob. Alice receives some input x 2 X, Bob receives some y 2 Y , and together they want to compute some function f(x; y). As the value f(x; y) will generally depend on both x and y, neither Alice nor Bob will have sufficient information to do the computation by themselves, so they will have to communicate in order to achieve their go...
Interaction in Quantum Communication and the Complexity of Set Disjointness
, 2001
"... One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much fewer, even exponentially fewer, qubits than possible ..."
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Cited by 33 (6 self)
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One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much fewer, even exponentially fewer, qubits than possible classically [1], [2], [3]. Moreover, some of these methods have a very simple structurethey involve only few message exchanges between the communicating parties. We consider the question as to whether every classical protocol may be transformed to a \simpler" quantum protocolone that has similar eciency, but uses fewer message exchanges.
The power of unentanglement
, 2008
"... The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than o ..."
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Cited by 27 (3 self)
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The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. • We give a protocol by which a verifier can be convinced that a 3Sat formula of size n is satisfiable, with constant soundness, given Õ ( √ n) unentangled quantum witnesses with O (log n) qubits each. Our protocol relies on Dinur’s version of the PCP Theorem and is inherently nonrelativizing. • We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. • We give evidence that QMA(2) ⊆ PSPACE, by showing that this would follow from “strong amplification ” of QMA(2) protocols. • We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with one.
von Neumann entropy penalization and low rank matrix approximation.
, 2010
"... Abstract We study a problem of estimation of a Hermitian nonnegatively definite matrix ρ of unit trace (for instance, a density matrix of a quantum system) based on n i.i.d. measurements (X 1 , Y 1 ), . . . , (X n , Y n ), where {X j } being random i.i.d. Hermitian matrices and {ξ j } being i.i.d. ..."
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Cited by 19 (2 self)
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Abstract We study a problem of estimation of a Hermitian nonnegatively definite matrix ρ of unit trace (for instance, a density matrix of a quantum system) based on n i.i.d. measurements (X 1 , Y 1 ), . . . , (X n , Y n ), where {X j } being random i.i.d. Hermitian matrices and {ξ j } being i.i.d. random variables with E(ξ j X j ) = 0. The estimator is considered, where S is the set of all nonnegatively definite Hermitian m × m matrices of trace 1. The goal is to derive oracle inequalities showing how the estimation error depends on the accuracy of approximation of the unknown state ρ by lowrank matrices.
Properties of classical and quantum JensenShannon divergence
 Phys. Rev. A 2009
"... The JensenShannon divergence (JSD) is a symmetrized and smoothed version of the all important divergence measure of information theory, the KullbackLeibler divergence. It defines a true metric – precisely, it is the square of a metric. We prove a stronger result for a family of divergence measures ..."
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Cited by 14 (0 self)
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The JensenShannon divergence (JSD) is a symmetrized and smoothed version of the all important divergence measure of information theory, the KullbackLeibler divergence. It defines a true metric – precisely, it is the square of a metric. We prove a stronger result for a family of divergence measures based on the Tsallis entropy, that includes the JSD. Furthermore we elaborate on details of geometric properties of the JSD. Analogously, the quantum JensenShannon divergence (QJSD) is a symmetrized version of the quantum relative entropy that has recently been considered as a distance measure for quantum states. We prove for a new family of distance measures for states, including the QJSD, that each member is the square of a metric for all qubits, strengthening recent results by Lamberti et al. We also discuss geometric properties of the QJSD. In analogy to Lin’s generalization of the JSD, we also define the general QJSD for a weighting of any number of states and discuss interpretations of both quantities. 1
New results in the simultaneous message passing model. Unpublished, available at arXiv:0902.3056
, 2009
"... Abstract—Consider the following Simultaneous Message Passing (SMP) model for computing a relation f ⊆ X ×Y×Z. In this model Alice, on input x ∈ X and Bob, on input y ∈ Y, send one message each to a third party Referee who then outputs a z ∈ Z such that (x, y, z) ∈ f. We first show optimal Direct su ..."
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Cited by 6 (3 self)
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Abstract—Consider the following Simultaneous Message Passing (SMP) model for computing a relation f ⊆ X ×Y×Z. In this model Alice, on input x ∈ X and Bob, on input y ∈ Y, send one message each to a third party Referee who then outputs a z ∈ Z such that (x, y, z) ∈ f. We first show optimal Direct sum results for all relations f in this model, both in the quantum and classical settings, in the situation where we allow shared resources (shared entanglement in quantum protocols and public coins in classical protocols) between Alice and Referee and Bob and Referee and no shared resource between Alice and Bob. This implies that, in this model, the communication required to compute k simultaneous instances of f, with constant success overall, is at least ktimes the communication required to compute one instance with constant success. This in particular implies an earlier Direct sum result, shown by Chakrabarti, Shi, Wirth and Yao [CSWY01] for the Equality function (and a class of other socalled robust functions), in the classical SMP model with no shared resources between any parties. Furthermore we investigate the gap between the SMP model and the oneway model in communication complexity and exhibit a partial function that is exponentially more expensive in the former if quantum communication with entanglement is allowed, compared to the latter even in the deterministic case.
The complexity of joint computation
, 2012
"... Joint computation is the ubiquitous scenario in which a computer is presented with not one, but many computational tasks to perform. A fundamental question arises: when can we cleverly combine computations, to perform them with greater efficiency or reliability than by tackling them separately? This ..."
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Joint computation is the ubiquitous scenario in which a computer is presented with not one, but many computational tasks to perform. A fundamental question arises: when can we cleverly combine computations, to perform them with greater efficiency or reliability than by tackling them separately? This thesis investigates the power and, especially, the limits of efficient joint computation, in several computational models: query algorithms, circuits, and Turing machines. We significantly improve and extend past results on limits to efficient joint computation for multiple independent tasks; identify barriers to progress towards better circuit lower bounds for multipleoutput operators; and begin an original line of inquiry into the complexity of joint computation. In more detail, we make contributions in the following areas: Improved direct product theorems for randomized query complexity: The "direct product problem" seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every Tquery algorithm has success proba