arXiv subjects

Mark Hillery

Publications and source records attributed to Mark Hillery.

At least 37 records · Page 2Linked to original sources

arXiv2021

Average concurrence and entanglement swapping

We study the role of average concurrence in entanglement swapping in quantum networks. We begin with qubit pure states, and there is a very simple rule governing the propagation of average concurrence in multiple swaps. We look at examples of mixed qubit states, and find the relation for pure states gives an upper bound on what is possible with mixed states. We then move on to qudits, where we make use of the I-concurrence. Here the situation is not as simple as for qubits, but in some cases relatively straightforward results can be obtained.

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arXiv2020

Constructing a qubit POVM from quantum data

Given an ensemble of qubits, which we are told consists of a mixture of two pure states, one with probability $η_{0}$ and one with probability $η_{1}$, we want to find a POVM that will discriminate between the two states by measuring the qubits. We do not know the states, and for any given qubit, we do not know which of the two states it is in. This can be viewed as learning a POVM from quantum data. Once found, the POVM can be used to separate the remaining qubits in the ensemble into two groups, corresponding to the two states present in the ensemble. In order to find the POVM, we need more information about the possible states. We examine several cases. First, we suppose that we know that the Bloch vectors of the states lie in the x-z plane and their \emph{a priori} probabilities are equal. We next keep the restriction to the x-z plane, but allow the \emph{a priori} probabilities to be different. Finally, we consider the case in which the Bloch vectors of the states have the same z component.

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arXiv2020

Finding more than one path through a simple maze with a quantum walk

We study quantum walks through chains consisting of two and three star graphs. The first star has a distinguished vertex labelled START and the last has one labelled END. There are multiple paths between these two vertices, and the object is to find these paths. We show that a quantum walk can do this with a quantum speedup.

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arXiv2020

A group-theoretic approach to elimination measurements of qubit sequences

Most measurements are designed to tell you which of several alternatives have occurred, but it is also possible to make measurements that eliminate possibilities and tell you an alternative that did not occur. Measurements of this type have proven useful in quantum foundations and in quantum cryptography. Here we show how group theory can be used to design such measurements. After some general considerations, we focus on the case of measurements on two-qubit states that eliminate one state. We then move on to construct measurements that eliminate two three-qubit states and four four-qubit states. A condition that constrains the construction of elimination measurements is then presented. Finally, in an appendix, we briefly consider the case of elimination measurements with failure probabilities and an elimination measurement on $n$-qubit states.

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arXiv2020

Wave-particle duality relations based on entropic bounds for which-way information

We present wave-particle duality relations involving the relative entropy coherence measure, which plays a prominent role in the resource theory of coherence. The main input in these relations is an entropic bound for the which-way information, which we derive in this letter. We show that this latter crucially depends on the choice of the measurement strategy to obtain the path information. In particular, we present results for two strategies: zero-error identification of the path-detector states, which never produces an error but sometimes fails to return a conclusive answer, and a mixed strategy where both errors and failure are allowed.

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arXiv2020

Unambiguous quantum state elimination for qubit sequences

Quantum state elimination measurements tell us what states a quantum system does not have. This is different from state discrimination, where one tries to determine what the state of a quantum system is, rather than what it is not. Apart from being of fundamental interest, quantum state elimination may find uses in quantum communication and quantum cryptography. We consider unambiguous quantum state elimination for two or more qubits, where each qubit can be in one of two possible states. Optimal measurements for eliminating one and two states out of four two-qubit states are given. We also prove that if we want to maximise the average number of eliminated overall N-qubit states, then individual measurements on each qubit are optimal.

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arXiv2019

Extracting unambiguous information from a single qubit by sequential observers

In a recent paper [Phys. Rev. Lett. 111, 100501 (2013)], a scheme was proposed where subsequent observers can extract unambiguous information about the initial state of a qubit, with finite joint probability of success. Here, we generalize the problem for arbitrary preparation probabilities (arbitrary priors). We discuss two different schemes: one where only the joint probability of success is maximized and another where, in addition, the joint probability of failure is also minimized. We also derive the mutual information for these schemes and show that there are some parameter regions for the scheme without minimizing the joint failure probability where, even though the joint success probability is maximum, no information is actually transmitted by Alice.

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arXiv2018

A quantum controlled quantum eraser

We show that the protocol known as quantum state separation can be used to transfer information between the phase and path of a particle in an interferometer. When applied to a quantum eraser, this allows us to erase some, but not all, of the path information. We can control how much path information we wish to erase.

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arXiv2018

A generalized wave-particle duality relation for finite groups

Wave-particle duality relations express the fact that knowledge about the path a particle took suppresses information about its wave-like properties, in particular, its ability to generate an interference pattern. Recently, duality relations in which the wave-like properties are quantified by using measures of quantum coherence have been proposed. Quantum coherence can be generalized to a property called group asymmetry. Here we derive a generalized duality relation involving group asymmetry, which is closely related to the success probability of discriminating between the actions of the elements of a group. The second quantity in the duality relation, the one generalizing which-path information, is related to information about the irreducible representations that make up the group representation.

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arXiv2017

Finding paths in tree graphs with a quantum walk

In this paper, we analyze the potential for new types of searches using the formalism of scattering random walks on Quantum Computers. Given a particular type of graph consisting of nodes and connections, a "Tree Maze", we would like to find a selected final node as quickly as possible, faster than any classical search algorithm. We show that this can be done using a quantum random walk, both exactly through numerical calculations as well as analytically using eigenvectors and eigenvalues of the quantum system.

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arXiv2017

Finding paths with quantum walks or quantum walking through a maze

We show that it is possible to use a quantum walk to find a path from one marked vertex to another. In the specific case of $M$ stars connected in a chain, one can find the path from the first star to the last one in $O(M\sqrt{N})$ steps, where $N$ is the number of spokes of each star. First we provide an analytical result showing that by starting in a phase-modulated highly superposed initial state we can find the path in $O(M\sqrt{N}\log M)$ steps. Next, we improve this efficiency by showing that the recovery of the path can also be performed by a series of successive searches when we start at the last known position and search for the next connection in $O(\sqrt{N})$ steps leading to the overall efficiency of $O(M\sqrt{N})$. For this result we use the analytical solution that can be obtained for a ring of stars of double the length of the chain.

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arXiv2017

Duality games and operational duality relations

We give operational meaning to wave-particle duality in terms of discrimination games. Duality arises as a constraint on the probability of winning these games. The games are played with the aid of an n-port interferometer, and involve 3 parties, Alice and Bob, who cooperate, and the House, who supervises the game. In one game called ways they attempt to determine the path of a particle in the interferometer. In another, called phases, they attempt to determine which set of known phases have been applied to the different paths. The House determines which game is to be played by flipping a coin. We find a tight wave-particle duality relation that allows us to relate the probabilities of winning these games, and use it to find an upper bound on the probability of winning the combined game. This procedure allows us to describe wave-particle duality in terms of discrimination probabilities.

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arXiv2017

Sequential discrimination of qudits by multiple observers

We discuss a scheme in which sequential state-discrimination measurements are performed on qudits to determine the quantum state in which they were initially prepared. The qudits belong to a set of nonorthogonal quantum states and hence cannot be distinguished with certainty. Unambiguous state discrimination allows error-free measurements at the expense of occasionally failing to give a conclusive answer about the state of the qudit. Qudits have the potential to carry more information per transmission than qubits. We considered the situation in which Alice sends one of N qudits, where the dimension of the qudits is also N. We look at two cases, one in which the states all have the same overlap and one in which the qudits are divided into two sets, with qudits in different sets having different overlaps. We also study the robustness of our scheme against a simple eavesdropping attack and found that by using qudits rather than qubits, there is a greater probability that an eavesdropper will introduce errors and be detected.

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arXiv2016

Retrodiction of a sequence of measurement results in qubit interferometers

We study how well we can retrodict results of measurements made on a quantum system if we can make measurements on its final state. We know what measurements were made, but not their results. An initial examination shows that we can gain anywhere from no information to perfect information about the results of previous measurements, depending on the measurements and the initial state of the system. The case of two two-outcome measurements, the second of which is a projective measurement, is examined in some detail. We then look at a model of a qubit interferometer in which measurements are made in order to determine the path the qubit followed. The measurement made on the final state of the qubit depends on the information about previous measurement results that we are trying to determine. One can attempt to find the result of just one of the measurements, all of them, or find a measurement sequence that was not realized. We study all three possibilities.

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arXiv2016

Coherence as a resource in decision problems: The Deutsch-Jozsa algorithm and a variation

That superpositions of states can be useful for performing tasks in quantum systems has been known since the early days of quantum information, but only recently has quantitative theory of quantum coherence been proposed. Here we apply that theory to an analysis of the Deutsch-Jozsa algorithm, which depends on quantum coherence for its operation. The Deutsch-Jozsa algorithm solves a decision problem, and we focus on a probabilistic version of that problem, comparing probability of being correct for both classical and quantum procedures. In addition, we study a related decision problem in which the quantum procedure has one-sided error while the classical procedure has two-sided error. The role of coherence on the quantum success probabilities in both of these problems is examined.

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arXiv2015

Relations between coherence and path information

We find two relations between coherence and path-information in a multi-path interferometer. The first builds on earlier results for the two-path interferometer, which used minimum-error state discrimination between detector states to provide the path information. For visibility, which was used in the two-path case, we substitute a recently defined $l_{1}$ measure of quantum coherence. The second is an entropic relation in which the path information is characterized by the mutual information between the detector states and the outcome of the measurement performed on them, and the coherence measure is one based on relative entropy.

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arXiv2015

Bell inequalities from group actions: Three parties and non-Abelian groups

In a previous publication, we showed how group actions can be used to generate Bell inequalities. The group action yields a set of measurement probabilities whose sum is the basic element in the inequality. The sum has an upper bound if the probabilities are a result of a local, realistic theory, but this bound can be violated if the probabilities come from quantum mechanics. In our first paper, we considered the case of only two parties making the measurements and single-generator groups. Here we show that the method can be extended to three parties, and it can also be extended to non-Abelian groups. We discuss the resulting inequalities in terms of nonlocal games.

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arXiv2014

Bell inequalities from group actions of single-generator groups

We study a method of generating Bell inequalities by using group actions of single-generator abelian groups. Two parties, Alice and Bob, each make one of M possible measurements on a system, with each measurement having K possible outcomes. The probabilities for the outcomes of these measurements are P(a_j = k, b_{j'}=k'), where j,j' are in the set {1,2,... M} and k,k' are in the set {0,1,... K-1}. The sums of some subsets of these probabilities have upper bounds when the probabilities result from a local, realistic theory that can be violated if the probabilities come from quantum mechanics. In our case the subsets of probabilities are generated by a group action, in particular, a representation of a single-generator group acting on product states in a tensor-product Hilbert space. We show how this works for several cases, including M=2, K=3, and general M, K=2. We also discuss the resulting inequalities in terms of nonlocal games.

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