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arXiv:2311.16886 · Quantum

Maximal Elements of Quantum Communication

Teiko Heinosaari, Oskari Kerppo

Published Nov 7, 2024• 3 citations
arXivHTMLPDF

Abstract

A prepare-and-measure scenario is naturally described by a communication matrix that collects all conditional outcome probabilities of the scenario into a row-stochastic matrix. The set of all possible communication matrices is partially ordered via the possibility to transform one matrix to another by pre- and post-processings. By considering maximal elements in this preorder for a subset of matrices implementable in a given theory, it becomes possible to identify communication matrices of maximum utility, i.e., matrices that are not majorized by any other matrices in the theory. The identity matrix of an appropriate size is the greatest element in classical theories, while the maximal elements in quantum theory have remained unknown. We completely characterize the maximal elements in quantum theory, thereby revealing the essential structure of the set of quantum communication matrices. In particular, we show that the identity matrix is the only maximal element in quantum theory but, as opposed to a classical theory, it is not the greatest element. Quantum theory can hence be seen to be distinct from classical theory by the existence of incompatible communication matrices.

DOI
10.22331/q-2024-11-07-1515

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A prepare-and-measure scenario is naturally described by a communication matrix that collects all conditional outcome probabilities of the scenario into a row-stochastic matrix. The set of all possible communication matrices is partially ordered via the possibility to transform one matrix to another by pre- and post-processings. By considering maximal elements in this preorder for a subset of matrices implementable in a given theory, it becomes possible to identify communication matrices of maximum utility, i.e., matrices that are not majorized by any other matrices in the theory. The identity matrix of an appropriate size is the greatest element in classical theories, while the maximal elements in quantum theory have remained unknown. We completely characterize the maximal elements in quantum theory, thereby revealing the essential structure of the set of quantum communication matrices. In particular, we show that the identity matrix is the only maximal element in quantum theory but, as opposed to a classical theory, it is not the greatest element. Quantum theory can hence be seen to be distinct from classical theory by the existence of incompatible communication matrices.
We look into the task of differentiating between any two quantum channels and reconstructing them from the obtained measurement statistics with possibly limited information about the experimental set-up. We employ the communication matrix formalism where the measurement statistics of a prepare-and-measure scenario is represented as a stochastic communication matrix. In order to differentiate between any two quantum channels, the informational completeness of the set-up is both necessary and sufficient. On the other hand, if we want to uniquely characterize any quantum channel, in addition we also need to have a complete description of the set-up. We show that in many important cases we can deduce this information directly from the communication matrix of the set-up before applying the channel. Given that we trust the dimension of the system, we show that we can deduce the information completeness of the set-up directly from the rank of the communication matrix. Furthermore, we show that another quantity of the communication matrix, called the information storability, can be used to self-test the set-up (up to unitary or antiunitary freedom) for an important class of states and measurements. This provides us a semi-device-independent way to identify quantum channels from the prepare-and-measure statistics. Lastly, we consider scenarios where we might have some additional information about the channels or additional resources at our disposal which could help us relax some of the assumptions of the proposed scenario.
The notion of quantum group replaces ordinary matrices with matrices with non-commutative elements. In TGD framework I have proposed that the notion should relate to the inclusions of von Neumann algebras allowing one to describe mathematically the notion of finite measurement resolution. In this article I will consider the notion of quantum matrix inspired by the recent view about quantum TGD relying on the notion of finite measurement resolution. Complex matrix elements are replaced with operators expressible as products of non-negative Hermitian operators and unitary operators analogous to the products of modulus and phase as a representation for complex numbers. The condition that determinant and sub-determinants exist is crucial for the well-defined eigenvalue problem in the generalized sense.  Strong/weak permutation symmetry of determinant requires its invariance under permutations of rows and/or columns. Weak permutation symmetry means development of determinant with respect to a fixed row or column and does not pose additional conditions. For weak permutation symmetry the permutation of rows/columns would however has a natural interpretation as braiding for the Hermitian operators defined by the moduli of operator valued matrix elements and here quantum group structure emerges. The commutativity of all sub-determinants is essential for the replacement of eigenvalues with eigenvalue spectra of Hermitian operators and sub-determinants define mutually commuting set of operators. Quantum matrices define a more general structure than quantum group but provide a concrete representation and interpretation for quantum group  in terms of finite measurement resolution, in particular when q is a root of unity. One can also understand the fractal structure of inclusion sequences of hyper-finite factors resulting by replacing operators appearing as matrix elements with quantum matrices.
We consider the communication systems sending the classical (Shannon) information through the quantum channel. The classical information source X is represented with symbols and their a priori probabilities At a transmitter, each symbol is modulated to a quantum state and detected at a receiver by the quantum mechanically generalized measurement, so-called the probability operator-valued measure Here we call a detection operator since it represents a detection process deciding a received signal to be ‘j’. The detection operator is no-negative, and resolution of identity, where is the identity operator on the signal Hilbert space spanned by signal quantum states. This detection process can be regarded as a quantum channel characterized by the conditional probability defined as follows:
Abstract We investigate the question which communication tasks can be accomplished within a given operational theory. The concrete task is to find out which communication matrices have a prepare-and-measure implementation with multiple states and a single measurement from a given theory, without using shared randomness. To set a general framework for this question we develop the ultraweak matrix majorization in the set of communication matrices. This preorder gives us means to determine when one communication task is more difficult than another. Furthermore, we introduce several monotones which can be used to compare and characterize the communication matrices. We observe that not only do the monotones allow us to compare communication matrices, but also their maximal values in a given theory are seen to relate to some physical properties of the theory. The maximal values can then be thought as ‘dimensions’, allowing us to compare different theories to each other. We analyse the introduced monotones one by one and demonstrate how the set of implementable communication matrices is different in several theories with the focus being mainly on the difference between classical and quantum theories of a given dimension.
Stochastic matrices and positive maps in matrix algebras have proved to be very important tools for analysing classical and quantum systems. In particular they represent a natural set of transformations for classical and quantum states, respectively. Here we introduce the notion of pseudo-stochastic matrices and consider their semigroup property. Unlike stochastic matrices, pseudo-stochastic matrices are permitted to have matrix elements which are negative while respecting the requirement that the sum of the elements of each column is one. They also allow for convex combinations, and carry a Lie group structure which permits the introduction of Lie algebra generators. The quantum analog of a pseudo-stochastic matrix exists and is called a pseudo-positive map. They have the property of transforming a subset of quantum states (characterized by maximal purity or minimal von Neumann entropy requirements) into quantum states. Examples of qubit dynamics connected with ‘diamond’ sets of stochastic matrices and pseudo-positive maps are dealt with.
The usual interpretation of quantum theory is clouded by the following points. (1) Invalid classical concepts are ascribed fundamental status. (2) The process of measurement is not describable within the framework of the theory. (3) The subject-object distinction is blurred. (4) The observed system is required to be isolated in order to be defined, yet interacting in order to be observed. These problems are resolved in a straightforward way by the unequivocal acceptance of a thoroughly pragmatic SSS-matrix viewpoint. This interpretation is described, and it is explained how and why the full physical content of quantum theory resides in the SSS matrix.