Quantum Information Theory Research Articles

Development of a thermodynamics of human cognition and human culture.

Inspired by foundational studies in classical and quantum physics, and by information retrieval studies in quantum information theory, we prove that the notions of 'energy' and 'entropy' can be consistently introduced in human language and, more generally, in human culture. More explicitly, if energy is attributed to words according to their frequency of appearance in a text, then the ensuing energy levels are distributed non-classically, namely, they obey Bose-Einstein, rather than Maxwell-Boltzmann, statistics, as a consequence of the genuinely 'quantum indistinguishability' of the words that appear in the text. Secondly, the 'quantum entanglement' due to the way meaning is carried by a text reduces the (von Neumann) entropy of the words that appear in the text, a behaviour which cannot be explained within classical (thermodynamic or information) entropy. We claim here that this 'quantum-type behaviour is valid in general in human language', namely, any text is conceptually more concrete than the words composing it, which entails that the entropy of the overall text decreases. In addition, we provide examples taken from cognition, where quantization of energy appears in categorical perception, and from culture, where entities collaborate, thus 'entangle', to decrease overall entropy. We use these findings to propose the development of a new 'non-classical thermodynamic theory' for human cognition, which also covers broad parts of human culture and its artefacts and bridges concepts with quantum physics entities. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'.

Information, mereology and vagueness

Classical systems of mereology identify a maximuml set of jointly exhaustive and pairwise disjoint (RCC5) relations. The amount of information that is carried by each member of this set of (crisp) relations is determined by the number of bits of information that are required to distinguish all the members of the set. It is postulated in this paper, that vague mereological relations are limited in the amount of information they can carry. That is, if a crisp mereological relation can carry N bits of information, then a vague mereological relation can carry only 1 ⩽ n < N bits. The aim of this paper is it to explore these ideas in the context of various systems of vague mereological relations. The resulting formalism is non-classical in the sense of quantum information theory.

Traveling discontinuity at the quantum butterfly front

We formulate a kinetic theory of quantum information scrambling in the context of a paradigmatic model of interacting electrons in the vicinity of a superconducting phase transition. We carefully derive a set of coupled partial differential equations that effectively govern the dynamics of information spreading in generic dimensions. Their solutions show that scrambling propagates at the maximal speed set by the Fermi velocity. At early times, we find exponential growth at a rate set by the inelastic scattering. At late times, we find that scrambling is governed by shock-wave dynamics with traveling waves exhibiting a discontinuity at the boundary of the light cone. Notably, we find perfectly causal dynamics where the solutions do not spill outside of the light cone.

Ontological indistinguishability as a central tenet of quantum theory.

Quantum indistinguishability directly relates to the philosophical debate on the notions of identity and individuality. They are crucial for our understanding of multipartite quantum systems. Furthermore, the correct interpretation of this feature of quantum theory has implications that transcend fundamental science and philosophy, given that quantum indistinguishability is a resource in quantum information theory. Most of the conceptual analysis of quantum indistinguishability is restricted to studying the permutational invariance of quantum states, the concomitant quantum statistics and their entanglement. Here, we analyse the role of indistinguishability and non-individuality in other areas of quantum theory. We start by analysing how a very peculiar use of indistinguishability underlies Feynman's rules for summing amplitudes in interference phenomena. Next, we study how quantum indistinguishability is underestimated in several topics of debate in the quantum physics literature, such as the Einstein-Podolsky-Rosen argument, Bell's inequalities and the Bell-Kochen-Specker theorem. Finally, we argue that an ontology of truly indistinguishable entities can serve as a basis for a quantum ontology that can give interesting answers to the interpretational problems of quantum mechanics. We claim that, in addition to superposition, contextuality and entanglement, indistinguishability (understood in a robust ontological sense) is one of the central features of quantum physics. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.

Indistinguishable entangled fermions: basics and future challenges.

The study of entanglement in systems composed of identical particles raises interesting challenges with far-reaching implications in both, our fundamental understanding of the physics of composite quantum systems, and our capability of exploiting quantum indistinguishability as a resource in quantum information theory. Impressive theoretical and experimental advances have been made in the last decades that bring us closer to a deeper comprehension and to a better control of entanglement. Yet, when it involves composites of indistinguishable quantum systems, the very meaning of entanglement, and hence its characterization, still finds controversy and lacks a widely accepted definition. The aim of the present paper is to introduce, within an accessible and self-contained exposition, the basic ideas behind one of the approaches advanced towards the construction of a coherent definition of entanglement in systems of indistinguishable particles, with focus on fermionic systems. We also inquire whether the corresponding tools developed for studying entanglement in identical-fermion systems can be exploited when analysing correlations in distinguishable-party systems, in which the complete information of the individual parts is not available. Further, we open the discussion on the broader problem of constructing a suitable framework that accommodates entanglement in the presence of generalized statistics. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.

Graph states and the variety of principal minors

In Quantum Information theory, graph states are quantum states defined by graphs. In this work we exhibit a correspondence between orbits of graph states and orbits in the variety of binary symmetric principal minors, under the action of SL(2,{mathbb {F}}_2)^{times n}rtimes {mathfrak {S}}_n. First we study the orbits of maximal abelian subgroups of the n-fold Pauli group under the action of {mathcal {C}}_n^{textrm{loc}}rtimes {mathfrak {S}}_n, where {mathcal {C}}_n^{textrm{loc}} is the n-fold local Clifford group, and we show that this action corresponds to the natural action of SL(2,{mathbb {F}}_2)^{times n}rtimes {mathfrak {S}}_n on the variety {mathcal {Z}}_nsubset {mathbb {P}}({mathbb {F}}_2^{2^n}) of principal minors of binary symmetric ntimes n matrices: the crucial step is in translating the action of SL(2,{mathbb {F}}_2)^{times n} into an action of the local symplectic group Sp_{2n}^{textrm{loc}}({mathbb {F}}_2). We conclude by showing how the former action restricts onto stabilizer groups, stabilizer states and graph states.

Quantum Network Discrimination

Discrimination between objects, in particular quantum states, is one of the most fundamental tasks in (quantum) information theory. Recent years have seen significant progress towards extending the framework to point-to-point quantum channels. However, with technological progress the focus of the field is shifting to more complex structures: Quantum networks. In contrast to channels, networks allow for intermediate access points where information can be received, processed and reintroduced into the network. In this work we study the discrimination of quantum networks and its fundamental limitations. In particular when multiple uses of the network are at hand, the rooster of available strategies becomes increasingly complex. The simplest quantum network that capturers the structure of the problem is given by a quantum superchannel. We discuss the available classes of strategies when considering n copies of a superchannel and give fundamental bounds on the asymptotically achievable rates in an asymmetric discrimination setting. Furthermore, we discuss achievability, symmetric network discrimination, the strong converse exponent, generalization to arbitrary quantum networks and finally an application to an active version of the quantum illumination problem.

The Necessary and Sufficient Conditions When Global and Local Fidelities Are Equal.

In the field of quantum information theory, the concept of quantum fidelity is employed to quantify the similarity between two quantum states. It has been observed that the fidelity between two states describing a bipartite quantum system A⊗B is always less than or equal to the quantum fidelity between the states in subsystem A alone. While this fidelity inequality is well understood, determining the conditions under which the inequality becomes an equality remains an open question. In this paper, we present the necessary and sufficient conditions for the equality of fidelities between a bipartite system A⊗B and subsystem A, considering pure quantum states. Moreover, we provide explicit representations of quantum states that satisfy the fidelity equality, based on our derived results.

New Density Matrix Renormalization Group Approaches for Strongly Correlated Systems Coupled with Large Environments.

Thanks to the high compression of the matrix product state (MPS) form of the wave function and the efficient site-by-site iterative sweeping optimization algorithm, the density matrix normalization group (DMRG) and its time-dependent variant (TD-DMRG) have been established as powerful computational tools in accurately simulating the electronic structure and quantum dynamics of strongly correlated molecules with a large number (101-2) of quantum degrees of freedom (active orbitals or vibrational modes). However, the quantitative characterization of the quantum many-body behaviors of realistic strongly correlated systems requires a further consideration of the interaction between the embedded active subsystem and the remaining correlated environment, e.g., a larger number (102-3) of external orbitals in electronic structure or infinite condensed-phase phononic modes in nucleus dynamics. To this end, we introduced three new post-DMRG and TD-DMRG approaches, namely (1) DMRG2sCI-MRCI and DMRG2sCI-ENPT by the reconstruction of selected configuration interaction (sCI) type of compact reference function from DMRG coefficients and the use of externally contracted MRCI (multireference configuration interaction) and Epstein-Nesbet perturbation theory (ENPT), without recourse to the expensive high order n-electron reduced density matrices (n-RDMs). (2) DMRG combined with RR-MRCI (renormalized residue-based MRCI), which improves the computational accuracy and efficiency of internally contracted (ic) MRCI by renormalizing the contracted bases with small-sized buffer environment(s) of a few external orbitals as probes based on quantum information theory. (3) HM (hierarchical mapping)-TD-DMRG in which a large environment is reduced to a small number of renormalized environmental modes (which accounts for the most vital system-environment interactions) through stepwise mapping transformation. These advances extend the efficacy of highly accurate DMRG/TD-DMRG computations to the quantitative characterization of the electronic structure and quantum dynamics in realistic strongly correlated systems coupled with large environments and are reviewed in this paper.

Tetrahedron genuine entanglement measure of four-qubit systems

Quantifying genuine entanglement is a key task in quantum information theory. We study the quantification of genuine multipartite entanglement for four-qubit systems. Based on the concurrence of nine different classes of four-qubit states, with each class being closed under stochastic local operation and classical communication, we construct a concurrence tetrahedron. Proper genuine four-qubit entanglement measure is presented by using the volume of the concurrence tetrahedron. For non genuine entangled pure states, the four-qubit entanglement measure classifies the bi-separable entanglement. We show that the concurrence tetrahedron based measure of genuine four-qubit entanglement is not equivalent to the genuine four-partite entanglement concurrence. We illustrate the advantages of the concurrence tetrahedron by detailed examples.

Quantum-information theory of magnetic field influence on circular dots with different boundary conditions

Influence of the transverse uniform magnetic field B on position (subscript ρ) and momentum (γ) Shannon quantum-information entropies S ρ,γ Fisher informations I ρ,γ and informational energies O ρ,γ is studied theoretically for the 2D circular quantum dots (QDs) whose circumference supports homogeneous either Dirichlet or Neumann boundary condition (BC). Comparative analysis reveals similarities and differences of the influence on the properties of the structure of the surface interaction with the magnetic field. A conspicuous distinction between the two spectra are crossings at the increasing induction of the Neumann energies with the same radial quantum number n and adjacent non-positive angular indices m. At the growing B, either system undergoes Landau condensation when its characteristics turn into their uniform field counterparts. It is shown that for the Dirichlet system this transformation takes place at the smaller magnetic intensities; e.g. the Dirichlet sum on its approach from above to a fundamental limit is at any B smaller than the corresponding Neumann quantity what physically means that the former geometry provides more total information about the position and motion of the particle. It is pointed out that the widely accepted disequilibrium uncertainty relation , with d being a dimensionality of the system, is violated by the Neumann QD in the magnetic field. Comparison with electrostatic harmonic confinement is performed; in particular, contrary to the hard-wall QDs, for this configuration the sums S ρ + S γ and the products I ρ I γ and O ρ O γ do not depend on the field. Physical interpretation is based on the different roles of the two BCs and their interplay with the field: Dirichlet (Neumann) surface is a repulsive (attractive) interface.

Entanglement Distillation Optimization Using Fuzzy Relations for Quantum State Tomography

Practical entanglement distillation is a critical component in quantum information theory. Entanglement distillation is often utilized for designing quantum computer networks and quantum repeaters. The practical entanglement distillation problem is formulated as a bilevel optimization problem. A fuzzy formulation is introduced to estimate the quantum state (density matrix) from pseudo-likelihood functions (i.e., quantum state tomography). A scale-independent relationship between fuzzy relations in terms of the pseudo-likelihood functions is obtained. The entanglement distillation optimization problem is solved using the combined coupled map lattice and dual annealing approach. Comparative analysis of the results is then conducted against a standard dual annealing algorithmic implementation.

Fisher information matrix as a resource measure in the resource theory of asymmetry with general connected-Lie-group symmetry

In recent years, in quantum information theory, there has been a remarkable development in the general theoretical framework for studying symmetry in dynamics. This development, called resource theory of asymmetry, is expected to have a wide range of applications, from accurate time measurements to black-hole physics. Despite its importance, the foundation of resource theory of asymmetry still has room for expansion. An important problem is in quantifying the amount of resource. When the symmetry prescribed U(1) or $\mathbb{R}$, i.e., with a single conserved quantity, the quantum Fisher information is known as a resource measure that has suitable properties and a clear physical meaning related to quantum fluctuation of the conserved quantity. However, it is not clear what is the resource measure with such suitable properties when a general symmetry prevails for which there are multiple conserved quantities. The purpose of this paper is to fill this gap. Specifically, we show that the quantum Fisher information matrix is a resource measure whenever a connected linear Lie group describes the symmetry. We also consider the physical meaning of this matrix and see which properties that the quantum Fisher information has when the symmetry is described by U(1) or $\mathbb{R}$ can be inherited by the quantum Fisher information matrix.

Sequences of the Projection-valued Measures and Functional Calculi

Let a pair be functional calculus, where is a homomorphism from the space of the measurable functions on into the space of all linear bounded operators on a reflexive Banach space . We define a norm of functional calculus by , the convergence of the sequence of functional calculi is a convergence relative to this norm. We study the correspondence between sequences of spectral decompositions, well-bounded operators defined on the reflexive Banach space , and their correspondence with the theory of functional calculus for such operators. In this article, we establish that if a sequence of the projection-valued measures strongly converges to then the sequence of the functional calculi converges to the functional calculus Results of the article can be employed in the modern extensions of the quantum theory and theory of quantum information.

$C^*$-Algebras

Operator algebras form a very active area of mathematics which, since its inception in the 1940s, has always been driven by interactions with other fields of mathematics and physics. The scope of these interactions is very wide, ranging over dynamical systems, (non-commutative) geometry, functional analysis, (geometric) group theory, topology, random matrices, harmonic analysis and quantum information theory. The goals of this workshop were to stimulate new collaborations across these fields of mathematics, to disseminate recent progress by giving participants a global view on the subject and to specially focus on two important developments: the solution of the Connes embedding problem by methods from quantum information theory and the progress on noncommutative dynamical systems, especially in the topological C^* -algebra context.

Character Theory and Categorification

Over a hundred years after the work of Frobenius and Schur, the sheer enormity of what is not known about the character theory of symmetric and alternating groups continues to surprise and awe the uninitiated. How does one decompose the tensor product of a pair of complex characters? Or the restriction of a complex character to a Sylow or wreath product subgroup? Can we understand the vanishing sets of complex characters? What about the asymptotic behaviour of complex characters? What are the dimensions of the modular characters? These questions have been hailed as some of the definitive open problems in representation theory and algebraic combinatorics, they have deep connections with Lie theory, group theory, local-global conjectures in representation theory of finite groups, symplectic geometry, complexity theory, statistical mechanics and quantum information theory. The overarching theme of this proposal is the use of hidden, richer representation theoretic structures arising in modular, local-global, and categorical representation theory in order to prove and disprove conjectures concerning characters of symmetric and alternating groups.

Spectral bounds for the quantum chromatic number of quantum graphs

Quantum graphs are an operator space generalization of classical graphs that have emerged in different branches of mathematics including operator theory, non-commutative topology and quantum information theory. In this paper, we obtain lower bounds for the classical, quantum and quantum-commuting chromatic number of a quantum graph using eigenvalues of the quantum adjacency matrix. In particular, we prove a quantum generalization of Hoffman's bound and introduce quantum analogues for the edge number, Laplacian and signless Laplacian. We generalize all the spectral bounds of Elphick & Wocjan [14] to the quantum graph setting and demonstrate the tightness of these bounds in the case of complete quantum graphs. Our results are achieved using techniques from linear algebra and a combinatorial definition of quantum graph coloring, which is obtained from the winning strategies of a quantum-to-classical nonlocal graph coloring game [5].

Tree-level entanglement in quantum electrodynamics

We report on a systematic study on the entanglement between helicity degrees of freedom generated at tree-level in quantum electrodynamics two-particle scattering processes. We determine the necessary and sufficient dynamical conditions for outgoing particles to be entangled with one another, and expose the hitherto unknown generation of maximal or nearly maximal entanglement through Bhabha and Compton scattering. Our work is an early step in revisiting quantum field theory and high-energy physics in the light of quantum information theory.

Mutually Unbiased Measurements, Hadamard Matrices, and Superdense Coding

Mutually unbiased bases (MUBs) are highly symmetric bases on complex Hilbert spaces, and the corresponding rank-1 projective measurements are ubiquitous in quantum information theory. In this work, we study a recently introduced generalisation of MUBs called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mutually unbiased measurements</i> (MUMs). These measurements inherit the essential property of complementarity from MUBs, but the Hilbert space dimension is no longer required to match the number of outcomes. This operational complementarity property renders MUMs highly useful for device-independent quantum information processing. It has been shown that MUMs are strictly more general than MUBs. In this work we provide a complete proof of the characterisation of MUMs that are direct sums of MUBs. We then construct new examples of MUMs that are not direct sums of MUBs. A crucial technical tool for this construction is a correspondence with quaternionic Hadamard matrices, which allows us to map known examples of such matrices to MUMs that are not direct sums of MUBs. Furthermore, we show that–in stark contrast with MUBs–the number of MUMs for a fixed outcome number is unbounded. Next, we focus on the use of MUMs in quantum communication. We demonstrate how any pair of MUMs with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> outcomes defines a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -dimensional superdense coding protocol. Using MUMs that are not direct sums of MUBs, we disprove a recent conjecture due to Nayak and Yuen on the rigidity of superdense coding, for infinitely many dimensions. The superdense coding protocols arising in the refutation reveal how shared entanglement may be used in a manner heretofore unknown.

Open Systems, Quantum Probability, and Logic for Quantum-like Modeling in Biology, Cognition, and Decision-Making.

The aim of this review is to highlight the possibility of applying the mathematical formalism and methodology of quantum theory to model behavior of complex biosystems, from genomes and proteins to animals, humans, and ecological and social systems. Such models are known as quantum-like, and they should be distinguished from genuine quantum physical modeling of biological phenomena. One of the distinguishing features of quantum-like models is their applicability to macroscopic biosystems or, to be more precise, to information processing in them. Quantum-like modeling has its basis in quantum information theory, and it can be considered one of the fruits of the quantum information revolution. Since any isolated biosystem is dead, modeling of biological as well as mental processes should be based on the theory of open systems in its most general form-the theory of open quantum systems. In this review, we explain its applications to biology and cognition, especially theory of quantum instruments and the quantum master equation. We mention the possible interpretations of the basic entities of quantum-like models with special interest given to QBism, as it may be the most useful interpretation.