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arXiv:1407.4690

Dealing with ignorance: universal discrimination, learning and quantum correlations

Gael Sentís

Published Jul 17, 2014• 0 citations
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Abstract

The problem of discriminating the state of a quantum system among a number of hypothetical states is usually addressed under the assumption that one has perfect knowledge of the possible states of the system. In this thesis, I analyze the role of the prior information available in facing such problems, and consider scenarios where the information regarding the possible states is incomplete. In front of a complete ignorance of the possible states' identity, I discuss a quantum "programmable" discrimination machine for qubit states that accepts this information as input programs using a quantum encoding, rather than as a classical description. The optimal performance of these machines is studied for general qubit states when several copies are provided, in the schemes of unambiguous, minimum-error, and error-margin discrimination. Then, this type of automation in discrimination tasks is taken further. By realizing a programmable machine as a device that is trained through quantum information to perform a specific task, I propose a quantum "learning" machine for classifying qubit states that does not require a quantum memory to store the qubit programs and, nevertheless, performs as good as quantum mechanics permits. Such learning machine thus allows for several optimal uses with no need for retraining. A similar learning scheme is also discussed for coherent states of light. I present it in the context of the readout of a classical memory by means of classically correlated coherent signals, when these are produced by an imperfect source. I show that, in this case, the retrieval of information stored in the memory can be carried out more accurately when fully general quantum measurements are used. Finally, as a transversal topic, I propose an efficient algorithmic way of decomposing any quantum measurement into convex combinations of simpler (extremal) measurements.

DOI
10.48550/arxiv.1407.4690

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The problem of discriminating the state of a quantum system among a number of hypothetical states is usually addressed under the assumption that one has perfect knowledge of the possible states of the system. In this thesis, I analyze the role of the prior information available in facing such problems, and consider scenarios where the information regarding the possible states is incomplete. In front of a complete ignorance of the possible states' identity, I discuss a quantum "programmable" discrimination machine for qubit states that accepts this information as input programs using a quantum encoding, rather than as a classical description. The optimal performance of these machines is studied for general qubit states when several copies are provided, in the schemes of unambiguous, minimum-error, and error-margin discrimination. Then, this type of automation in discrimination tasks is taken further. By realizing a programmable machine as a device that is trained through quantum information to perform a specific task, I propose a quantum "learning" machine for classifying qubit states that does not require a quantum memory to store the qubit programs and, nevertheless, performs as good as quantum mechanics permits. Such learning machine thus allows for several optimal uses with no need for retraining. A similar learning scheme is also discussed for coherent states of light. I present it in the context of the readout of a classical memory by means of classically correlated coherent signals, when these are produced by an imperfect source. I show that, in this case, the retrieval of information stored in the memory can be carried out more accurately when fully general quantum measurements are used. Finally, as a transversal topic, I propose an efficient algorithmic way of decomposing any quantum measurement into convex combinations of simpler (extremal) measurements.
We develop a quantum learning scheme for binary discrimination of coherent states of light. This is a problem of technological relevance for the reading of information stored in a digital memory. In our setting, a coherent light source is used to illuminate a memory cell and retrieve its encoded bit by determining the quantum state of the reflected signal. We consider a situation where the amplitude of the states produced by the source is not fully known, but instead this information is encoded in a large training set comprising many copies of the same coherent state. We show that an optimal global measurement, performed jointly over the signal and the training set, provides higher successful identification rates than any learning strategy based on first estimating the unknown amplitude by means of Gaussian measurements on the training set, followed by an adaptive discrimination procedure on the signal. By considering a simplified variant of the problem, we argue that this is the case even for non-Gaussian estimation measurements. Our results show that, even in absence of entanglement, collective quantum measurements yield an enhancement in the readout of classical information, which is particularly relevant in the operating regime of low-energy signals.
Quantum state discrimination is a fundamental primitive in quantum statistics where one has to correctly identify the state of a system that is in one of two possible known states. A programmable discrimination machine performs this task when the pair of possible states is not a priori known but instead the two possible states are provided through two respective program ports. We study optimal programmable discrimination machines for general qubit states when several copies of states are available in the data or program ports. Two scenarios are considered: One in which the purity of the possible states is a priori known, and the fully universal one where the machine operates over generic mixed states of unknown purity. We find analytical results for both the unambiguous and minimum error discrimination strategies. This allows us to calculate the asymptotic performance of programmable discrimination machines when a large number of copies are provided and to recover the standard state discrimination and state comparison values as different limiting cases.
We consider the problem of correctly classifying a given quantum two-level system (qubit) which is known to be in one of two equally probable quantum states. We assume that this task should be performed by a quantum machine which does not have at its disposal a complete classical description of the two template states, but can only have partial prior information about their level of purity and mutual overlap. Moreover, similarly to the classical supervised learning paradigm, we assume that the machine can be trained by n qubits prepared in the first template state and by n more qubits prepared in the second template state. In this situation, we are interested in the optimal process which correctly classifies the input qubit with the largest probability allowed by quantum mechanics. The problem is studied in its full generality for a number of different prior information scenarios and for an arbitrary size n of the training data. Finite size corrections around the asymptotic limit n → ∞ are derived. When the states are assumed to be pure, with known overlap, the problem is also solved in the case of d-level systems.
In quantum information and quantum computing, the carrier of information is some quantum system and information is encoded in its state. After processing the state in the quantum processor, the information has to be read out. Clearly, this task is equivalent to determining the final state of the system. We begin by briefly reviewing various possible state discrimination strategies that are optimal with respect to some reasonable criteria and report on recent advances in the unambiguous discrimination of mixed quantum states. This strategy has been successfully applied to devise a class of novel probabilistic quantum algorithms and has been demonstrated experimentally, using a linear optical implementation via generalized interferometers. In the second part we present a scheme for communication via completely unknown quantum states. In this context we discuss programmable quantum state discriminators that are universal, i.e. perform optimally on average, independently of the actual states used for the communication scheme. We conclude with a discussion of possible experimental implementations of the proposed device.
A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the minimum error rate allowed by quantum mechanics for any size of the training set. This result is shown to be robust under (an arbitrary amount of) noise and under (statistical) variations in the composition of the training set, provided it is large enough. This machine can be used an arbitrary number of times without retraining. Its required classical memory grows only logarithmically with the number of training qubits, while its excess risk decreases as the inverse of this number, and twice as fast as the excess risk of an "estimate-and-discriminate" machine, which estimates the states of the training qubits and classifies the data qubit with a discrimination protocol tailored to the obtained estimates.
We describe a class of programmable devices that can discriminate between two quantum states. We consider two cases. In the first, both states are unknown. One copy of each of the unknown states is provided as an input, or program, for the two program registers, and the data state, which is guaranteed to be prepared in one of the program states, is fed into the data register of the device. This device will then tell us, in an optimal way, which of the templates stored in the program registers the data state matches. In the second case, we know one of the states while the other is unknown. One copy of the unknown state is fed into the single program register, and the data state which is guaranteed to be prepared in either the program state or the known state, is fed into the data register. The device will then tell us, again optimally, whether the data state matches the template or is the known state. We determine two types of optimal devices. The first performs discrimination with minimum error, and the second performs optimum unambiguous discrimination. In all cases we first treat the simpler problem of only one copy of the data state and then generalize the treatment to nnn copies. In comparison to other works we find that providing n>1n>1n>1 copies of the data state yields higher success probabilities than providing n>1n>1n>1 copies of the program states.