One advantage of quantum algorithms over classical computation is the possibility to spread out, process, analyse and extract information in multipartite configurations in coherent superpositions of classical states. This will be discussed in terms of quantum state identification problems based on a proper partitioning of mutually orthogonal sets of states. The question arises whether or not it is possible to encode equibalanced decision problems into quantum systems, so that a single invocation of a filter used for state discrimination suffices to obtain the result.
One advantage of quantum algorithms over classical computation is the possibility to spread out, process, analyse and extract information in multipartite configurations in coherent superpositions of classical states. This will be discussed in terms of quantum state identification problems based on a proper partitioning of mutually orthogonal sets of states. The question arises whether or not it is possible to encode equibalanced decision problems into quantum systems, so that a single invocation of a filter used for state discrimination suffices to obtain the result.
In quantum information and quantum computing, the carrier of information is some quantum system and information is encoded in its state. The state, however, is not an observable in quantum mechanics, thus a fundamental problem arises: after processing the information it has to be read out or, in other words, the state of the system must be determined. When the possible target states are orthogonal, this is a relatively simple task, provided the set of possible target states is known. But when the possible target states are not orthogonal their discrimination is far from being trivial even if their set is known. Thus the problem of discriminating among non-orthogonal states is ubiquitous in quantum information and quantum computing, underlying all communication and computing schemes. It is the subject of this talk to review recent theoretical and experimental advances in this field.
Quantum state discrimination underlies various applications in quantum information processing tasks. It essentially describes the distinguishability of quantum systems in different states, and the general process of extracting classical information from quantum systems. It is also useful in quantum information applications, such as the characterization of mutual information in cryptographic protocols, or as a technique for deriving fundamental theorems on quantum foundations. It has deep connections to physical principles such as relativistic causality. Quantum state discrimination traces a long history of several decades, starting with the early attempts to formalize information processing of physical systems such as optical communication with photons. Nevertheless, in most cases, the problems of finding optimal strategies of quantum state discrimination remain unsolved, and related applications are valid in some limited cases only. The present review aims to provide an overview on quantum state discrimination, covering some recent progress, and addressing applications in some selected areas. This review serves to strengthen the link between results in quantum state discrimination and quantum information applications, by showing the ways in which the fundamental results are exploited in applications and vice versa.
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.
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.
The quantum discrimination problem is an important topic of quantum information theory. Discriminating among a set of M non-orthogonal quantum states means choosing among one of them based on a quantum measurement. In order to accomplish this task one must choose a measure of information to rank the potential solutions. When the Shannon mutual information is considered as such a measure one has an accessible information problem. The accessible information problem consists on the maximization of the mutual information over all sorts of quantum measurements. Analytical solutions for this optimization problem are not available for the general case. Even numerical solutions are hard to find because of the nonlinear nature of the mutual information function. On the other hand, genetic algorithms work pretty well with "misbehaved" functions. Based on the previous statements, we propose solving the accessible information problem via genetic algorithms.
Quantum information processing is appealing due its ability to solve certain problems quantitatively faster than classical information processing. Most quantum algorithms have been studied in discretely parameterized systems, but many quantum systems are continuously parameterized. The field of quantum optics in particular has sophisticated techniques for manipulating continuously parameterized quantum states of light, but the lack of a code-state formalism has hindered the study of quantum algorithms in these systems. To address this situation, a code-state formalism for the solution of oracle decision problems in continuously-parameterized quantum systems is developed.
Quantum information processing is appealing due its ability to solve certain problems quantitatively faster than classical information processing. Most quantum algorithms have been studied in discretely parameterized systems, but many quantum systems are continuously parameterized. The field of quantum optics in particular has sophisticated techniques for manipulating continuously parameterized quantum states of light, but the lack of a code-state formalism has hindered the study of quantum algorithms in these systems. To address this situation, a code-state formalism for the solution of oracle decision problems in continuously-parameterized quantum systems is developed.
In the infinite-dimensional case, we study continuous-variable quantum algorithms for the solution of the Deutsch–Jozsa oracle decision problem implemented within a single harmonic-oscillator. Orthogonal states are used as the computational bases, and we show that, contrary to a previous claim in the literature, this implementation of quantum information processing has limitations due to a position-momentum trade-off of the Fourier transform. We further demonstrate that orthogonal encoding bases are not unique, and using the coherent states of the harmonic oscillator as the computational bases, our formalism enables quantifying the relative performances of different choices of the encoding bases.
We extend our formalism to include quantum algorithms in the continuously parameterized yet finite-dimensional Hilbert space of a coherent spin system. We show that the highest-squeezed spin state possible can be approximated by a superposition of two states thus transcending the usual model of using a single basis state as algorithm input. As a particular example, we show that the close Hadamard oracle-decision problem, which is related to the Hadamard codewords of digital communications theory, can be solved quantitatively more efficiently using this computational model than by any known classical algorithm.