arXiv subjects

Mark Hillery

Publications and source records attributed to Mark Hillery.

At least 55 records · Page 3Linked to original sources

arXiv2014

Finding structural anomalies in star graphs: A general approach

We develop a general theory for a quantum-walk search on a star graph. A star graph has N edges each of which is attached to a central vertex. A graph G is attached to one of these edges, and we would like to find out to which edge it is attached. This is done by means of a quantum walk, a quantum version of a random walk. This walk contains O(\sqrt{N}) steps, which represents a speedup over a classical search, which would require O(N) steps. The overall graph, star plus G, is divided into two parts, and we find that for a quantum speedup to occur, the eigenvalues associated with these two parts in the N goes to infinity limit must be the same. Our theory tells us how the initial state of the walk should be chosen, and how many steps the walk must make in order to find G.

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arXiv2013

Extracting information from a qubit by multiple observers: Toward a theory of sequential state discrimination

We discuss sequential unambiguous state-discrimination measurements performed on the same qubit. Alice prepares a qubit in one of two possible states. The qubit is first sent to Bob, who measures it, and then on to Charlie, who also measures it. The object in both cases is to determine which state Alice sent. In an unambiguous state discrimination measurement, we never make a mistake, i.e. misidentify the state, but the measurement may fail, in which case we gain no information about which state was sent. We find that there is a nonzero probability for both Bob and Charlie to identify the state, and we maximize this probability. The probability that Charlie's measurement succeeds depends on how much information about the state Alice sent is left in the qubit after Bob's measurement, and this information can be quantified by the overlap between the two possible states in which Bob's measurement leaves the qubit. This paper is a first step toward developing a theory of nondestructive sequential quantum measurements, which could be useful in quantum communication schemes.

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arXiv2013

Maximum quantum violations of a class of Bell inequalities

We study a class of Bell inequalities and find their maximum quantum violation. These inequalities involve n parties, two measurements per party, with each measurement having two outcomes. The n=2 case corresponds to the CH inequality. We use the method of Jordan bases to find the maximum quantum violations. Results are found for the cases n=2 through n=7.

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arXiv2012

Quantum walks as a probe of structural anomalies in graphs

We study how quantum walks can be used to find structural anomalies in graphs via several examples. Two of our examples are based on star graphs, graphs with a single central vertex to which the other vertices, which we call external vertices, are connected by edges. In the basic star graph, these are the only edges. If we now connect a subset of the external vertices to form a complete subgraph, a quantum walk can be used to find these vertices with a quantum speedup. Thus, under some circumstances, a quantum walk can be used to locate where the connectivity of a network changes. We also look at the case of two stars connected at one of their external vertices. A quantum walk can find the vertex shared by both graphs, again with a quantum speedup. This provides an example of using a quantum walk in order to find where two networks are connected. Finally, we use a quantum walk on a complete bipartite graph to find an extra edge that destroys the bipartite nature of the graph.

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arXiv2011

Quantum tests for the linearity and permutation invariance of Boolean functions

The goal in function property testing is to determine whether a black-box Boolean function has a certain property or is epsilon-far from having that property. The performance of the algorithm is judged by how many calls need to be made to the black box in order to determine, with high probability, which of the two alternatives is the case. Here we present two quantum algorithms, the first to determine whether the function is linear and the second to determine whether it is symmetric (invariant under permutations of the arguments). Both require O(epsilon^{-2/3}) calls to the oracle, which is better than known classical algorithms. In addition, in the case of linearity testing, if the function is linear, the quantum algorithm identifies which linear function it is. The linearity test combines the Bernstein-Vazirani algorithm and amplitude amplification, while the test to determine whether a function is symmetric uses projective measurements and amplitude amplification.

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arXiv2011

Decision problems with quantum black boxes

We examine how to distinguish between unitary operators, when the exact form of the possible operators is not known. Instead we are supplied with "programs" in the form of unitary transforms, which can be used as references for identifying the unknown unitary transform. All unitary transforms should be used as few times as possible. This situation is analoguous to programmable state discrimination. One difference, however, is that the quantum state to which we apply the unitary transforms may be entangled, leading to a richer variety of possible strategies. By suitable selection of an input state and generalized measurement of the output state, both unambiguous and minimum-error discrimination can be achieved. Pairwise comparison of operators, comparing each transform to be identified with a program transform, is often a useful strategy. There are, however, situations in which more complicated strategies perform better. This is the case especially when the number of allowed applications of program operations is different from the number of the transforms to be identified.

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arXiv2011

Toward protocols for quantum-ensured privacy and secure voting

We present a number of schemes that use quantum mechanics to preserve privacy, in particular, we show that entangled quantum states can be useful in maintaining privacy. We further develop our original proposal [see Phys. Lett. A 349, 75 (2006)] for protecting privacy in voting, and examine its security under certain types of attacks, in particular dishonest voters and external eavesdroppers. A variation of these quantum-based schemes can be used for multi-party function evaluation. We consider functions corresponding to group multiplication of $N$ group elements, with each element chosen by a different party. We show how quantum mechanics can be useful in maintaining the privacy of the choices group elements.

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arXiv2011

Searches on star graphs and equivalent oracle problems

We examine a search on a graph among a number of different kinds of objects (vertices), one of which we want to find. In a standard graph search, all of the vertices are the same, except for one, the marked vertex, and that is the one we wish to find. We examine the case in which the unmarked vertices can be of different types, so the background against which the search is done is not uniform. We find that the search can still be successful, but the probability of success is lower than in the uniform background case, and that probability decreases with the number of types of unmarked vertices. We also show how the graph searches can be rephrased as equivalent oracle problems.

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arXiv2010

Finding structural anomalies in graphs by means of quantum walks

We explore the possibility of using quantum walks on graphs to find structural anomalies, such as extra edges or loops, on a graph. We focus our attention on star graphs, whose edges are like spokes coming out of a central hub. If there are $N$ spokes, we show that a quantum walk can find an extra edge connecting two of the spokes or a spoke with a loop on it in $O(\sqrt{N})$ steps. We initially find that if all of the spokes have loops except one, the walk will not find the spoke without a loop, but this can be fixed if we choose the phase with which the particle is reflected from the vertex without the loop. Consequently, quantum walks can, under some circumstances, be used to find structural anomalies in graphs.

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arXiv2010

Quantum algorithms for testing Boolean functions

We discuss quantum algorithms, based on the Bernstein-Vazirani algorithm, for finding which variables a Boolean function depends on. There are 2^n possible linear Boolean functions of n variables; given a linear Boolean function, the Bernstein-Vazirani quantum algorithm can deterministically identify which one of these Boolean functions we are given using just one single function query. The same quantum algorithm can also be used to learn which input variables other types of Boolean functions depend on, with a success probability that depends on the form of the Boolean function that is tested, but does not depend on the total number of input variables. We also outline a procedure to futher amplify the success probability, based on another quantum algorithm, the Grover search.

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arXiv2010

Application of entanglement conditions to spin systems

There have been numerous studies of entanglement in spin systems. These have usually focussed on examining the entanglement between individual spins or determining whether the state of the system is completely separable. Here we present conditions that allow us to determine whether blocks of spins are entangled. We show that sometimes these conditions can detect entanglement better than conditions involving individual spins. We apply these conditions to study entanglement in spin wave states, both when there are only a few magnons present and also at finite temperature.

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arXiv2010

Conditions for entanglement in multipartite systems

We introduce two entanglement conditions that take the form of inequalities involving expectation values of operators. These conditions are sufficient conditions for entanglement, that is if they are satisfied the state is entangled, but if they are not, one can say nothing about the entanglement of the state. These conditions are quite flexible, because the operators in them are not specified, and they are particularly useful in detecting multipartite entanglement. We explore the range of utility of these conditions by considering a number of examples of entangled states, and seeing under what conditions entanglement in them can be detected by the inequalities presented here.

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arXiv2009

Detecting entanglement with non-hermitian operators

We derive several entanglement conditions employing non-hermitian operators. We start with two conditions that were derived previously for field mode operators, and use them to derive conditions that can be used to show the existence of field-atom entanglement and entanglement between groups of atoms. The original conditions can be strengthened by making them invariant under certain sets of local unitary transformations, such as Gaussian operations. We then apply these conditions to several examples, such as the Dicke model. We conclude with a short discussion of how local uncertainty relations with non-hermitian operators can be used to derive entanglement conditions.

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arXiv2009

Searching via walking: How to find a marked subgraph of a graph using quantum walks

We show how a quantum walk can be used to find a marked edge or a marked complete subgraph of a complete graph. We employ a version of a quantum walk, the scattering walk, which lends itself to experimental implementation. The edges are marked by adding elements to them that impart a specific phase shift to the particle as it enters or leaves the edge. If the complete graph has N vertices and the subgraph has K vertices, the particle becomes localized on the subgraph in O(N/K) steps. This leads to a quantum search that is quadratically faster than a corresponding classical search. We show how to implement the quantum walk using a quantum circuit and a quantum oracle, which allows us to specify the resource needed for a quantitative comparison of the efficiency of classical and quantum searches -- the number of oracle calls.

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arXiv2009

Quantum Machines

We discuss quantum information processing machines. We start with single purpose machines that either redistribute quantum information or identify quantum states. We then move on to machines that can perform a number of functions, with the function they perform being determined by a program, which is itself a quantum state. Examples of both deterministic and probabilistic programmable machines are given, and we conclude with a discussion of the utility of quantum programs.

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arXiv2009

Quantum searches on highly symmetric graphs

We study scattering quantum walks on highly symmetric graphs and use the walks to solve search problems on these graphs. The particle making the walk resides on the edges of the graph, and at each time step scatters at the vertices. All of the vertices have the same scattering properties except for a subset of special vertices. The object of the search is to find a special vertex. A quantum circuit implementation of these walks is presented in which the set of special vertices is specified by a quantum oracle. We consider the complete graph, a complete bipartite graph, and an $M$-partite graph. In all cases, the dimension of the Hilbert space in which the time evolution of the walk takes place is small (between three and six), so the walks can be completely analyzed analytically. Such dimensional reduction is due to the fact that these graphs have large automorphism groups. We find the usual quadratic quantum speedups in all cases considered.

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arXiv2009

An Introduction to the Quantum Theory of Nonlinear Optics

This article is provides an introduction to the quantum theory of optics in nonlinear dielectric media. We begin with a short summary of the classical theory of nonlinear optics, that is nonlinear optics done with classical fields. We then discuss the canonical formalism for fields and its quantization. This is applied to quantizing the electromagnetic field in free space. The definition of a nonclassical state of the electromagnetic field is presented, and several examples are examined. This is followed by a brief introduction to entanglement in the context of field modes. The next task is the quantization of the electromagnetic field in an inhomogeneous, linear dielectric medium. Before going on to field quantization in nonlinear media, we discuss a number of commonly employed phenomenological models for quantum nonlinear optical processes. We then quantize the field in both nondispersive and dispersive nonlinear media. Flaws in the most commonly used methods of accomplishing this task are pointed out and discussed. Once the quantization has been completed, it is used to study a multimode theory of parametric down conversion and the propagation of quantum solitons.

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arXiv2009

Unambiguous identification of coherent states II: Multiple resources

We consider unambiguous identification of coherent states of electromagnetic field. In particular, we study possible generalizations of an optical setup proposed in M. Sedlák {\it et al.}, Phys. Rev. A {\bf 76}, 022326 (2007). We show how the unambiguous identification of coherent states can be performed in a general case when multiple copies of unknown and the reference states are available. We also investigate whether reference states after the measurement can be "recovered" and further used for subsequent unambiguous identification tasks. We show that in spite of the fact that the recovered reference states are disturbed by measurements they can be repeatedly used for unambiguous identifications. We analyze the role of various imperfections in preparation of the unknown and the reference coherent states on the performance of our unambiguous identification setup.

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