Quantum Networks for Concentrating Entanglement

If two parties, Alice and Bob, share some number, n, of partially entangled pairs of qubits, then it is possible for them to concentrate these pairs into some smaller number of maximally entangled states. We present a simplified version of the algorithm for such entanglement concentration, and we describe efficient networks for implementing these operations.Comment: 15 pages, 2 figure

Substituting a qubit for an arbitrarily large number of classical bits

We show that a qubit can be used to substitute for an arbitrarily large number of classical bits. We consider a physical system S interacting locally with a classical field phi(x) as it travels directly from point A to point B. The field has the property that its integrated value is an integer multiple of some constant. The problem is to determine whether the integer is odd or even. This task can be performed perfectly if S is a qubit. On the otherhand, if S is a classical system then we show that it must carry an arbitrarily large amount of classical information. We identify the physical reason for such a huge quantum advantage, and show that it also implies a large difference between the size of quantum and classical memories necessary for some computations. We also present a simple proof that no finite amount of one-way classical communication can perfectly simulate the effect of quantum entanglement.Comment: 8 pages, LaTeX, no figures. v2: added result on entanglement simulation with classical communication; v3: minor correction to main proof, change of title, added referenc

Violating the Shannon capacity of metric graphs with entanglement

The Shannon capacity of a graph G is the maximum asymptotic rate at which messages can be sent with zero probability of error through a noisy channel with confusability graph G. This extensively studied graph parameter disregards the fact that on atomic scales, Nature behaves in line with quantum mechanics. Entanglement, arguably the most counterintuitive feature of the theory, turns out to be a useful resource for communication across noisy channels. Recently, Leung, Mancinska, Matthews, Ozols and Roy [Comm. Math. Phys. 311, 2012] presented two examples of graphs whose Shannon capacity is strictly less than the capacity attainable if the sender and receiver have entangled quantum systems. Here we give new, possibly infinite, families of graphs for which the entangled capacity exceeds the Shannon capacity.Comment: 15 pages, 2 figure

Quantum Property Testing

A language L has a property tester if there exists a probabilistic algorithm that given an input x only asks a small number of bits of x and distinguishes the cases as to whether x is in L and x has large Hamming distance from all y in L. We define a similar notion of quantum property testing and show that there exist languages with quantum property testers but no good classical testers. We also show there exist languages which require a large number of queries even for quantumly testing

Nondeterministic Instance Complexity and Proof Systems with Advice

Motivated by strong Karp-Lipton collapse results in bounded arithmetic, Cook and Krajíček [1] have recently introduced the notion of propositional proof systems with advice. In this paper we investigate the following question: Given a language L , do there exist polynomially bounded proof systems with advice for L ? Depending on the complexity of the underlying language L and the amount and type of the advice used by the proof system, we obtain different characterizations for this problem. In particular, we show that the above question is tightly linked with the question whether L has small nondeterministic instance complexity

One-qubit fingerprinting schemes

Fingerprinting is a technique in communication complexity in which two parties (Alice and Bob) with large data sets send short messages to a third party (a referee), who attempts to compute some function of the larger data sets. For the equality function, the referee attempts to determine whether Alice's data and Bob's data are the same. In this paper, we consider the extreme scenario of performing fingerprinting where Alice and Bob both send either one bit (classically) or one qubit (in the quantum regime) messages to the referee for the equality problem. Restrictive bounds are demonstrated for the error probability of one-bit fingerprinting schemes, and show that it is easy to construct one-qubit fingerprinting schemes which can outperform any one-bit fingerprinting scheme. The author hopes that this analysis will provide results useful for performing physical experiments, which may help to advance implementations for more general quantum communication protocols.Comment: 9 pages; Fixed some typos; changed order of bibliographical reference

Improved Quantum Communication Complexity Bounds for Disjointness and Equality

We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and non-deterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound being n/2. We show this by improving a general bound for non-deterministic protocols of de Wolf. We also give an O(sqrt{n}c^{log^* n})-qubit bounded-error protocol for disjointness, modifying and improving the earlier O(sqrt{n}log n) protocol of Buhrman, Cleve, and Wigderson, and prove an Omega(sqrt{n}) lower bound for a large class of protocols that includes the BCW-protocol as well as our new protocol.Comment: 11 pages LaTe

Super-activation of quantum non-locality

In this paper we show that quantum non-locality can be super-activated. That is, one can obtain violations of Bell inequalities by tensorizing a local state with itself. Moreover, previous results suggest that such Bell violations can be very large.Comment: v2: Refs added. Same results, v3: Minor corrections. Close to the published versio

Matrix product states and the quantum max-flow/min-cut conjectures

In this note we discuss the geometry of matrix product states with periodic boundary conditions and provide three infinite sequences of examples where the quantum max-flow is strictly less than the quantum min-cut. In the first we fix the underlying graph to be a 4-cycle and verify a prediction of Hastings that inequality occurs for infinitely many bond dimensions. In the second we generalize this result to a 2d-cycle. In the third we show that the 2d-cycle with periodic boundary conditions gives inequality for all d when all bond dimensions equal two, namely a gap of at least 2^{d-2} between the quantum max-flow and the quantum min-cut.Comment: 12 pages, 3 figures - Final version accepted for publication on J. Math. Phy

Tight Noise Thresholds for Quantum Computation with Perfect Stabilizer Operations

We study how much noise can be tolerated by a universal gate set before it loses its quantum-computational power. Specifically we look at circuits with perfect stabilizer operations in addition to imperfect non-stabilizer gates. We prove that for all unitary single-qubit gates there exists a tight depolarizing noise threshold that determines whether the gate enables universal quantum computation or if the gate can be simulated by a mixture of Clifford gates. This exact threshold is determined by the Clifford polytope spanned by the 24 single-qubit Clifford gates. The result is in contrast to the situation wherein non-stabilizer qubit states are used; the thresholds in that case are not currently known to be tight.Comment: 4 pages, 2 figure