Minimum guesswork discrimination between quantum states

© Rinton Press. Error probability is a popular and well-studied optimization criterion in discriminating non-orthogonal quantum states. It captures the threat from an adversary who can only query the actual state once. However, when the adversary is able to use a brute-force strategy to query the state, discrimination measurement with minimum error probability does not necessarily minimize the number of queries to get the actual state. In light of this, we take Massey’s guesswork as the underlying optimization criterion and study the problem of minimum guesswork discrimination. We show that this problem can be reduced to a semidefinite programming problem. Necessary and sufficient conditions when a measurement achieves minimum guesswork are presented. We also reveal the relation between minimum guesswork and minimum error probability. We show that the two criteria generally disagree with each other, except for the special case with two states. Both upper and lower information-theoretic bounds on minimum guesswork are given. For geometrically uniform quantum states, we provide sufficient conditions when a measurement achieves minimum guesswork. Moreover, we give the necessary and sufficient condition under which making no measurement at all would be the optimal strategy

Quantum computation, quantum theory and AI

The main purpose of this paper is to examine some (potential) applications of quantum computation in AI and to review the interplay between quantum theory and AI. For the readers who are not familiar with quantum computation, a brief introduction to it is provided, and a famous but simple quantum algorithm is introduced so that they can appreciate the power of quantum computation. Also, a (quite personal) survey of quantum computation is presented in order to give the readers a (unbalanced) panorama of the field. The author hopes that this paper will be a useful map for AI researchers who are going to explore further and deeper connections between AI and quantum computation as well as quantum theory although some parts of the map are very rough and other parts are empty, and waiting for the readers to fill in. © 2009 Elsevier B.V. All rights reserved

Optimization Algorithms in Wireless and Quantum Communications

Since the first communication systems were developed, the scientific community has been witnessing attempts to increase the amount of information that can be transmitted. In the last 10--15 years there has been a tremendous amount of research towards developing multi-antenna systems which would hopefully provide high-data-rate transmissions. However, increasing the overall amount of transmitted information increases the complexity of the necessary signal processing. A large portion of this thesis deals with several important issues in signal processing of multi-antenna systems. In almost every particular case the goal is to develop a technique/algorithm so that the overall complexity of the signal processing is significantly decreased. In the first part of the thesis a very important problem of signal detection in MIMO (multiple-input multiple-output) systems is considered. The problem is analyzed in two different scenarios: when the transmission medium (channel) 1) is known and 2) is unknown at the receiver. The former case is often called coherent and the later non-coherent MIMO detection. Both cases usually amount to solving highly complex NP-hard combinatorial optimization problems. For the coherent case we develop a significant improvement of the traditional sphere decoder algorithm commonly used for this type of detection. An interesting connection between the new improved algorithm and the H-infinity estimation theory is established, and the performance improvement over the standard sphere decoder is demonstrated. For the non-coherent case we develop a counterpart to the standard sphere decoder, the so-called out-sphere decoder. The complexity of the algorithm is viewed as a random variable; its expected value is analyzed and shown to be significantly smaller than the one of the overall exhaustive search. In the non-coherent case, in addition to the complexity analysis of the exact out-sphere decoder, we analyze the performance loss of a suboptimal technique. We show that only a moderate loss of a few dbs in power required at the transmitter will occur if a polynomial algorithm based on the semi-definite relaxation is used in place of any exact technique (which of course is not known to be polynomial). In the second part of the thesis we consider a few problems that arise in wireless broadcast channels. Namely, we consider the problem of the information symbol vector design at the transmitter. A polynomial linear precoding technique is constructed. It enables achieving data rates very close to the ones achieved with DPC (dirty paper coding) technique. Additionally, for another suboptimal polynomial scheme (the so-called nulling and cancelling), we show that it asymptotically achieves the same data rate as the optimal, exponentially complex, DPC. In the last part of the thesis we consider a quantum counterpart of the signal detection from classical communication. In quantum systems the signals are quantum states and the quantum detection problem amounts to designing measurement operators which have to satisfy certain quantum mechanics laws. A specific type of quantum detection called unambiguous detection, which has numerous applications including quantum filtering, has recently attracted a lot of attention in the research community. We develop a general framework for numerically solving this problem using the tools from the convex optimization theory. Furthermore, in the special case where the two quantum states are of rank 2, we construct an explicit analytical solution for the measurement operators. At the end we would like to emphasize that the contribution of this thesis goes beyond the specific problems mentioned here. Most algorithmic optimization techniques developed in this paper are generally applicable. While it is a fact that our results were originally motivated by wireless and quantum communications applications, we believe that the developed techniques will find applications in many different areas where similar optimization problems appear.</p

Quantum-inspired classification based on quantum state discrimination

We present quantum-inspired algorithms for classification tasks inspired by the problem of quantum state discrimination. By construction, these algorithms can perform multiclass classification, prevent overfitting, and generate probability outputs. While they could be implemented on a quantum computer, we focus here on classical implementations of such algorithms. The training of these classifiers involves Semi-Definite Programming. We also present a relaxation of these classifiers that utilizes Linear Programming (but that can no longer be interpreted as a quantum measurement). Additionally, we consider a classifier based on the Pretty Good Measurement (PGM) and show how to implement it using an analogue of the so-called Kernel Trick, which allows us to study its performance on any number of copies of the input state. We evaluate these classifiers on the MNIST and MNIST-1D datasets and find that the PGM generally outperforms the other quantum-inspired classifiers and performs comparably to standard classifiers.Comment: 19 pages, 4 figure

Unambiguous State Discrimination for Quantum Communications

Il problema della discriminazione tra stati non ortogonali è onnipresente nelle informazioni quantistiche e nel calcolo quantistico. L'approccio con Unambiguous State Discrimination (USD), introduce risultati inconcludenti per ottenere una perfetta discriminazione di stato e mira a massimizzare la probabilità di risultati conclusivi.La seguente tesi tratta USD per le comunicazioni quantistiche e verrà confrontata con i sistemi quantistici e classici.ope

Quantum statistical inference and communication

This thesis studies the limits on the performances of inference tasks with quantum data and quantum operations. Our results can be divided in two main parts. In the first part, we study how to infer relative properties of sets of quantum states, given a certain amount of copies of the states. We investigate the performance of optimal inference strategies according to several figures of merit which quantifies the precision of the inference. Since we are not interested in obtaining a complete reconstruction of the states, optimal strategies do not require to perform quantum tomography. In particular, we address the following problems: - We evaluate the asymptotic error probabilities of optimal learning machines for quantum state discrimination. Here, a machine receives a number of copies of a pair of unknown states, which can be seen as training data, together with a test system which is initialized in one of the states of the pair with equal probability. The goal is to implement a measurement to discriminate in which state the test system is, minimizing the error probability. We analyze the optimal strategies for a number of different settings, differing on the prior incomplete information on the states available to the agent. - We evaluate the limits on the precision of the estimation of the overlap between two unknown pure states, given N and M copies of each state. We find an asymptotic expansion of a Fisher information associated with the estimation problem, which gives a lower bound on the mean square error of any estimator. We compute the minimum average mean square error for random pure states, and we evaluate the effect of depolarizing noise on qubit states. We compare the performance of the optimal estimation strategy with the performances of other intuitive strategies, such as the swap test and measurements based on estimating the states. - We evaluate how many samples from a collection of N d-dimensional states are necessary to understand with high probability if the collection is made of identical states or they differ more than a threshold according to a motivated closeness measure. The access to copies of the states in the collection is given as follows: each time the agent ask for a copy of the states, the agent receives one of the states with some fixed probability, together with a different label for each state in the collection. We prove that the problem can be solved with O(pNd=2) copies, and that this scaling is optimal up to a constant independent on d;N; . In the second part, we study optimal classical and quantum communication rates for several physically motivated noise models. - The quantum and private capacities of most realistic channels cannot be evaluated from their regularized expressions. We design several degradable extensions for notable channels, obtaining upper bounds on the quantum and private capacities of the original channels. We obtain sufficient conditions for the degradability of flagged extensions of channels which are convex combination of other channels. These sufficient conditions are easy to verify and simplify the construction of degradable extensions. - We consider the problem of transmitting classical information with continuous variable systems and an energy constraint, when it is impossible to maintain a shared reference frame and in presence of losses. At variance with phase-insensitive noise models, we show that, in some regimes, squeezing improves the communication rates with respect to coherent state sources and with respect to sources producing up to two-photon Fock states. We give upper and lower bounds on the optimal coherent state rate and show that using part of the energy to repeatedly restore a phase reference is strictly suboptimal for high energies

Quantum elimination measurements

If an initial state is prepared from a known set, then the aim of a quantum state elimination measurement is to rule out a subset of the possible initial states. We use semi-definite programming to find either bounds or exact results on the success probabilities of certain elimination measurements. In conjunction we use an analytic approach to find optimal measurements. We obtain optimal measurements for unambiguous elimination in a two-qubit case where each qubit is in one of two possible states. We also show how it might be possible to use our elimination measurements in a QKD protocol. In addition we prove that the best method to eliminate the highest average number of states for sequences of qubits with each qubit in one of two possible states is individual unambiguous measurements. Furthermore we show the method of decomposing a unitary matrix into beamsplitter-like operations found by Reck et al. and apply this to our elimination measurement to realise a way of experimental implementation. In the final chapter we look at joint measurements and find the optimal probe state that we would use to minimise the uncertainty in our estimation of the sharpness of a measurement between two observables.Engineering and Physical Sciences Research Council (EPSRC) funding

Linear optics and photodetection achieve near-optimal unambiguous coherent state discrimination

Coherent states of the quantum electromagnetic field, the quantum description of ideal laser light, are prime candidates as information carriers for optical communications. A large body of literature exists on their quantum-limited estimation and discrimination. However, very little is known about the practical realizations of receivers for unambiguous state discrimination (USD) of coherent states. Here we fill this gap and outline a theory of USD with receivers that are allowed to employ: passive multimode linear optics, phase-space displacements, auxiliary vacuum modes, and on-off photon detection. Our results indicate that, in some regimes, these currently-available optical components are typically sufficient to achieve near-optimal unambiguous discrimination of multiple, multimode coherent states.Comment: 18 pages, 10 figures, and 2 tables. Appendices included. Additional references added. Comments welcome

On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback

© 1963-2012 IEEE. We initiate the study of zero-error communication via quantum channels when the receiver and the sender have at their disposal a noiseless feedback channel of unlimited quantum capacity, generalizing Shannon's zero-error communication theory with instantaneous feedback. We first show that this capacity is only a function of the linear span of Choi-Kraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub non-commutative bipartite graph. Then, we go on to show that the feedback-assisted capacity is non-zero (allowing for a constant amount of activating noiseless communication) if and only if the non-commutative bipartite graph is non-trivial, and give a number of equivalent characterizations. This result involves a far-reaching extension of the conclusive exclusion of quantum states. We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede. We demonstrate that this bound to have many good properties, including being additive and given by a minimax formula. We also prove a coding theorem showing that this quantity is the entanglement-assisted capacity against an adversarially chosen channel from the set of all channels with the same Choi-Kraus span, which can also be interpreted as the feedback-assisted unambiguous capacity. The proof relies on a generalization of the Postselection Lemma (de Finetti reduction) that allows to reflect additional constraints, and which we believe to be of independent interest. This capacity is a relaxation of the feedback-assisted zero-error capacity; however, we have to leave open the question of whether they coincide in general. We illustrate our ideas with a number of examples, including classical-quantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions

Semidefinite programming relaxations for quantum correlations

Semidefinite programs are convex optimisation problems involving a linear objective function and a domain of positive semidefinite matrices. Over the last two decades, they have become an indispensable tool in quantum information science. Many otherwise intractable fundamental and applied problems can be successfully approached by means of relaxation to a semidefinite program. Here, we review such methodology in the context of quantum correlations. We discuss how the core idea of semidefinite relaxations can be adapted for a variety of research topics in quantum correlations, including nonlocality, quantum communication, quantum networks, entanglement, and quantum cryptography.Comment: To be submitted to Reviews of Modern Physic