Results 1  10
of
14
Quantum Proofs for Classical Theorems
, 2009
"... Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (nonquantum) areas. In this paper we survey these results and the quantum toolbox they use. ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (nonquantum) areas. In this paper we survey these results and the quantum toolbox they use.
Unboundederror oneway classical and quantum communication complexity
 Proc. 34th ICALP, Lecture Notes in Comput. Sci. 4596
, 2007
"... Abstract. This paper studies the gap between classical oneway communication complexity C(f) and its quantum counterpart Q(f), under the unboundederror setting, i.e., it is enough that the success probability is strictly greater than 1/2. It is proved that for any (total or partial) Boolean functio ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
Abstract. This paper studies the gap between classical oneway communication complexity C(f) and its quantum counterpart Q(f), under the unboundederror setting, i.e., it is enough that the success probability is strictly greater than 1/2. It is proved that for any (total or partial) Boolean function f, Q(f) = ⌈C(f)/2⌉, i.e., the former is exactly (without an error of even ±1) one half as large as the latter. The result has an application to obtaining (again an exact) bound for the existence of (m, n, p)QRAC which is the nqubit random access coding that can recover any one of m original bits with success probability ≥ p. We can prove that (m, n,> 1/2)QRAC exists if and only if m ≤ 2 2n − 1. 1
Minimumerror discrimination of quantum states: New bounds and comparison
, 2009
"... The minimumerror probability of ambiguous discrimination for two quantum states is the wellknown Helstrom limit presented in 1976. Since then, it has been thought of as an intractable problem to obtain the minimumerror probability for ambiguously discriminating arbitrary m quantum states. In this ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
The minimumerror probability of ambiguous discrimination for two quantum states is the wellknown Helstrom limit presented in 1976. Since then, it has been thought of as an intractable problem to obtain the minimumerror probability for ambiguously discriminating arbitrary m quantum states. In this paper, we obtain a new lower bound on the minimumerror probability for ambiguous discrimination and compare this bound with six other bounds in the literature. Moreover, we show that the bound between ambiguous and unambiguous discrimination does not extend to ensembles of more than two states. Specifically, the main technical contributions are described as follows: (1) We derive a new lower bound on the minimumerror probability for ambiguous discrimination among arbitrary m mixed quantum states with given prior probabilities, and we present a necessary and sufficient condition to show that this lower bound is attainable. (2) We compare this new lower bound with six other bounds in the literature in detail, and, in some cases, this bound is optimal. (3) It is known that if m = 2, the optimal inconclusive probability of unambiguous discrimination QU and the minimumerror probability of ambiguous discrimination QE between arbitrary given m mixed quantum states have the relationship QU ≥ 2QE. In this paper, we show that, however, if m> 2, the relationship QU ≥ 2QE may not hold again in general, and there may be no supremum of QU/QE for more than two states, which may also reflect an essential difference between discrimination for twostates and multistates. (4) A number of examples are constructed.
Unbounded Error Quantum Query Complexity
, 2007
"... This work studies the quantum query complexity of Boolean functions in a scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded error quantum query complexity is exa ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
This work studies the quantum query complexity of Boolean functions in a scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, we show that the “blackbox ” approach to convert quantum query algorithms into communication protocols by BuhrmanCleveWigderson [STOC’98] is optimal even in the unbounded error setting. We also study a setting related to the unbounded error model, called the weakly unbounded error setting, where the cost of a query algorithm is given by q + log(1/2(p − 1/2)), where q is the number of queries made and p> 1/2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight Θ(log n) separation between quantum and classical query complexity in the weakly unbounded error setting for a partial Boolean function. We also show the asymptotic equivalence between them for some wellstudied total Boolean functions. 1
Accessible versus Holevo information for a binary random variable
, 2006
"... The accessible information Iacc(E) of an ensemble E is the maximum mutual information between a random variable encoded into quantum states, and the probabilistic outcome of a quantum measurement of the encoding. Accessible information is extremely difficult to characterize analytically; even bounds ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
The accessible information Iacc(E) of an ensemble E is the maximum mutual information between a random variable encoded into quantum states, and the probabilistic outcome of a quantum measurement of the encoding. Accessible information is extremely difficult to characterize analytically; even bounds on it are hard to place. The celebrated Holevo bound states that accessible information cannot exceed χ(E), the quantum mutual information between the random variable and its encoding. However, for general ensembles, the gap between the Iacc(E) and χ(E) may be arbitrarily large. We consider the special case of a binary random variable, which often serves as a stepping stone towards other results in information theory and communication complexity. We give explicit lower bounds on the the accessible information Iacc(E) of an ensemble E ∆ = {(p, ρ0), (1−p, ρ1)}, with 0 ≤ p ≤ 1, as functions of p and χ(E). The bounds are incomparable in the sense that they surpass each other in different parameter regimes. Our bounds arise by measuring the ensemble according to a complete orthogonal measurement that preserves the fidelity of the states ρ0, ρ1. As an intermediate step, therefore, we give new relations between the two quantities Iacc(E), χ(E) and the fidelity B(ρ0, ρ1). 1
MultiSource Randomness Extractors Against Quantum Side Information, and their Applications
, 2014
"... We study the problem of constructing multisource extractors in the quantum setting, which extract almost uniform random bits against an adversary who collects quantum side information from several initially independent classical random sources. This is a natural generalization of the two much studi ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
We study the problem of constructing multisource extractors in the quantum setting, which extract almost uniform random bits against an adversary who collects quantum side information from several initially independent classical random sources. This is a natural generalization of the two much studied problems of seeded randomness extraction against quantum side information, and classical independent source extractors. With new challenges such as potential entanglement in the side information, it is not a prior clear under what conditions do quantum multisource extractors exist; the only previous work in this setting is [19], where the classical innerproduct twosource extractors of [7] and [10] are shown to be quantum secure in the restricted Independent Adversary (IA) Model and entangled Bounded Storage (BS) Model. In this paper we propose a new model called General Entangled (GE) Adversary Model, which allows arbitrary entanglement in the side information and subsumes both the IA model and the BS model. We proceed to show how to construct GEsecure quantum multisource extractors. To that end, we propose another model called Onesided Adversary (OA) Model, which is weaker than all the above models. Somewhat surprisingly, we establish an equivalence between strong
Nearoptimal extractors against quantum storage
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 133 (2009)
, 2009
"... We give nearoptimal constructions of extractors secure against quantum boundedstorage adversaries. One instantiation gives the first such extractor to achieve an output length Θ(K − b), where K is the source’s entropy and b the adversary’s storage, depending linearly on the adversary’s amount of s ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We give nearoptimal constructions of extractors secure against quantum boundedstorage adversaries. One instantiation gives the first such extractor to achieve an output length Θ(K − b), where K is the source’s entropy and b the adversary’s storage, depending linearly on the adversary’s amount of storage, together with a polylogarithmic seed length. Another instantiation achieves a logarithmic key length, with a slightly smaller output length Θ((K − b)/K γ) for any γ> 0. In contrast, the previous best construction [Ts09] could only extract (K/b) 1/15 bits. Our construction follows Trevisan’s general reconstruction paradigm [Tre01], and in fact our proof of security shows that essentially all extractors constructed using this paradigm are secure against quantum storage, with optimal parameters. Our argument is based on bounds for a generalization of quantum random access codes, which we call quantum functional access codes. This is crucial as it lets us avoid the local listdecoding algorithm central to the approach in [Ts09], which was the source of the multiplicative overhead. Some of our constructions have the additional advantage that every bit of the output is a function of only a polylogarithmic number of bits from the source, which is crucial for some cryptographic applications.
Research Article Entropic Lower Bound for Distinguishability of Quantum States
"... which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For a system randomly prepared in a number of quantum states, we present a lower bound for the distinguishability of the quantum states, that is, the success probability of det ..."
Abstract
 Add to MetaCart
(Show Context)
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For a system randomly prepared in a number of quantum states, we present a lower bound for the distinguishability of the quantum states, that is, the success probability of determining the states in the form of entropy. When the states are all pure, acquiring the entropic lower bound requires only the density operator and the number of the possible states.This entropic bound shows a relation between the von Neumann entropy and the distinguishability. 1.