arXiv subjects

Martin Plávala

Publications and source records attributed to Martin Plávala.

At least 37 records · Page 2Linked to original sources

arXiv2024

The polarization hierarchy for polynomial optimization over convex bodies, with applications to nonnegative matrix rank

We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has previously been introduced for the study of polynomial optimization problems over state spaces of $C^*$-algebras, to convex cones in finite dimensions. If the convex bodies can be characterized by linear or semidefinite programs, then the same is true for our hierarchy. Convergence is proven by relating the problem to a certain de Finetti theorem for general probabilistic theories, which are studied as possible generalizations of quantum mechanics. We apply the method to the problem of nonnegative matrix factorization, and in particular to the nested rectangles problem. A numerical implementation of the third level of the hierarchy is shown to give rise to a very tight approximation for this problem.

math.OC↗

arXiv2024

No-broadcasting characterizes operational contextuality

Operational contextuality forms a rapidly developing subfield of quantum information theory. However, the characterization of the quantum mechanical entities that fuel the phenomenon has remained unknown with many partial results existing. Here, we present a resolution to this problem by connecting operational contextuality one-to-one with the no-broadcasting theorem. The connection works both on the level of full quantum theory and subtheories thereof. We demonstrate the connection in various relevant cases, showing especially that for quantum states the possibility of demonstrating contextuality is exactly characterized by non-commutativity, and for measurements this is done by a norm-1 property closely related to repeatability. Moreover, we show how techniques from broadcasting can be used to simplify known foundational results in contextuality.

quant-ph↗

arXiv2024

Simple Information Processing Tasks with Unbounded Quantum Advantage

Communication scenarios between two parties can be implemented by first encoding messages into some states of a physical system which acts as the physical medium of the communication and then decoding the messages by measuring the state of the system. We show that already in the simplest possible scenarios it is possible to detect a definite, unbounded advantage of quantum systems over classical systems. We do this by constructing a family of operationally meaningful communication tasks each of which on one hand can be implemented by using just a single qubit but which on the other hand require unboundedly larger classical system for classical implementation. Furthemore, we show that even though with the additional resource of shared randomness the proposed communication tasks can be implemented by both quantum and classical systems of the same size, the number of coordinated actions needed for the classical implementation also grows unboundedly. In particular, no finite storage can be used to store all the coordinated actions needed to implement all the possible quantum communication tasks with classical systems. As a consequence, shared randomness cannot be viewed as a free resource.

quant-ph↗

arXiv2023

Generalized dynamical theories in phase space and the hydrogen atom

We show that the phase-space formulation of general probabilistic theories can be extended to include a generalized time-evolution and that it can describe a nonquantum hydrogen-like system which is stable, has discrete energy levels, and includes the Zeeman effect. This allows us to study dynamical effects such as excitations of the hydrogen-like system by a resonant laser and Rutherford scattering. Our construction demonstrates that classical theory and quantum theory can be seen as specific choices of general probabilistic theory in phase space and that other probabilistic theories also lead to measurable predictions.

quant-ph↗

arXiv2023

Encoding and decoding of information in general probabilistic theories

Encoding and decoding are the two key steps in information processing. In this work we study the encoding and decoding capabilities of operational theories in the context of information-storability game, where the task is to freely choose a set of states from which one state is chosen at random and by measuring the state it must be identified; a correct guess results in as many utiles as the number of states in the chosen set and an incorrect guess means a penalty of a fixed number of utiles. We connect the optimal winning strategy of the game to the amount of information that can be stored in a given theory, called the information storability of the theory, and show that one must use so-called nondegradable sets of states and nondegradable measurements whose encoding and decoding properties cannot be reduced. We demonstrate that there are theories where the perfect discrimination strategy is not the optimal one so that the introduced game can be used as an operational test for super information storability. We further develop the concept of information storability by giving new useful conditions for calculating it in specific theories.

quant-ph↗

arXiv2023

Dynamic and Stochastic Rational Behavior

The (static) utility maximization model of Afriat (1967), which is the standard in analysing choice behavior, is under scrutiny. We propose the Dynamic Random Utility Model (DRUM) that is more flexible than the framework of Afriat (1967) and more informative than the static Random Utility Model (RUM) framework of McFadden and Richter (1990). Under DRUM, each decision-maker randomly draws a utility function in each period and maximizes it subject to a menu. DRUM allows for unrestricted time correlation and cross-section heterogeneity in preferences. We characterize DRUM for situations when panel data on choices and menus are available. DRUM is linked to a finite mixture of deterministic behaviors that can be represented as a product of static rationalizable behaviors. This link allows us to convert the characterizations of the static RUM to its dynamic form. In an application, we find that although the static utility maximization model fails to explain population behavior, DRUM can explain it.

econ.TH↗

arXiv2023

Semi-device-independent certification of number of measurements

We develop a method for semi-device-independent certification of number of measurements. We achieve this by testing whether Bob's steering equivalent observables (SEO) can be simulated by k measurements, which we do by testing whether they are k-compatible with separable joint observable. This test can be performed with the aid of hierarchy of semidefinite programs, and whenever it fails one can conclude that Alice must have access to at least k + 1 incompatible measurements.

quant-ph↗

arXiv2023

Dispensing of quantum information beyond no-broadcasting theorem -- is it possible to broadcast anything genuinely quantum?

No-broadcasting theorem is one of the most fundamental results in quantum information theory; it guarantees that the simplest attacks on any quantum protocol, based on eavesdropping and copying of quantum information, are impossible. Due to the fundamental importance of the no-broadcasting theorem, it is essential to understand the exact boundaries of this limitation. We generalize the standard definition of broadcasting by restricting the set of states which we want to broadcast and restricting the sets of measurements which we use to test the broadcasting. We show that in some of the investigated cases broadcasting is equivalent to commutativity, while in other cases commutativity is not necessary.

quant-ph↗

arXiv2022

Complete hierarchy for high-dimensional steering certification

High-dimensional quantum steering can be seen as a test for the dimensionality of entanglement, where the devices at one side are not characterized. As such, it is an important component in quantum informational protocols that make use of high-dimensional entanglement. Although it has been recently observed experimentally, the phenomenon of high-dimensional steering is lacking a general certification procedure. We provide necessary and sufficient conditions to certify the entanglement dimension in a steering scenario. These conditions are stated in terms of a hierarchy of semidefinite programs, which can also be used to quantify the phenomenon using the steering dimension robustness. To demonstrate the practical viability of our method, we characterize the dimensionality of entanglement in steering scenarios prepared with maximally entangled states measured in mutually unbiased bases. Our methods give significantly stronger bounds on the noise robustness necessary to experimentally certify high-dimensional entanglement.

quant-ph↗

arXiv2022

Operational Theories in Phase Space: Toy Model for the Harmonic Oscillator

We show how to construct general probabilistic theories that contain an energy observable dependent on position and momentum. The construction is in accordance with classical and quantum theory and allows for physical predictions, such as the probability distribution for position, momentum and energy. We demonstrate the construction by formulating a toy model for the harmonic oscillator that is neither classical nor quantum. The model features a discrete energy spectrum, a ground state with sharp position and momentum, an eigenstate with non-positive Wigner function as well as a state that has tunneling properties. The toy model demonstrates that operational theories can be a viable alternative approach for formulating physical theories.

quant-ph↗

arXiv2021

Incompatibility in restricted operational theories: connecting contextuality and steering

We investigate the connection between steering and contextuality in general probabilistic theories. We show that for a class of bipartite states the steerability of the state by given set of measurements is equivalent to non-existence of preparation noncontextual hidden variable model for certain restricted theory constructed from the given state and measurements. The connection between steering and contextuality is provided by the concept of incompatibility in restricted theories, which we also investigate.

quant-ph↗

arXiv2021

Entanglement and superposition are equivalent concepts in any physical theory

We prove that any two general probabilistic theories (GPTs) are entangleable, in the sense that their composite exhibits either entangled states or entangled measurements, if and only if they are both non-classical, meaning that neither of the state spaces is a simplex. This establishes the universal equivalence of the (local) superposition principle and the existence of global entanglement, valid in a fully theory-independent way. As an application of our techniques, we show that all non-classical GPTs exhibit a strong form of incompatibility of states and measurements, and use this to construct a version of the BB84 protocol that works in any non-classical GPT.

quant-ph↗

arXiv2021

General probabilistic theories: An introduction

We introduce the framework of general probabilistic theories (GPTs for short). GPTs are a class of operational theories that generalize both finite-dimensional classical and quantum theory, but they also include other, more exotic theories, such as the boxworld theory containing Popescu-Rohrlich boxes. We provide in-depth explanations of the basic concepts and elements of the framework of GPTs, and we also prove several well-known results. The review is self-contained and it is meant to provide the reader with consistent introduction to GPTs. Our tools mainly include convex geometry, but we also introduce diagrammatic notation and we often express equations via diagrams.

quant-ph↗

arXiv2020

Jordan products of quantum channels and their compatibility

Given two quantum channels, we examine the task of determining whether they are compatible - meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). We show several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolaring channels are compatible.

quant-ph↗

arXiv2020

Structure of quantum and classical implementations of Popescu-Rohrlich box

We construct implementations of the PR-box using quantum and classical channels as state spaces. In both cases our constructions are very similar and they share the same idea taken from general probabilistic theories and the square state space model. We construct all quantum qubit channels that maximally violate a given CHSH inequality, we show that they all are entanglement-breaking channels, that they have certain block-diagonal structure and we present some examples of such channels.

quant-ph↗

arXiv2020

Popescu-Rohrlich box implementation in general probabilistic theory of processes

It is shown that Popescu-Rohrlich nonlocal boxes (beating the Tsirelson bound for Bell inequality) do exist in the existing structures of both quantum and classical theory. In particular, we design an explicit example of measure-and-prepare nonlocal (but no-signaling) channel being the realization of nonlocal and no-signaling Popescu-Rohrlich box within the generalized probabilistic theory of processes. Further we present a post-selection-based spatially non-local implementation and show it does not require truly quantum resources, hence, improving the previously known results. Interpretation and potential (spatially non-local) simulation of this form of process nonlocality and the protocol is discussed.

quant-ph↗

arXiv2019

No-free-information principle in general probabilistic theories

In quantum theory, the no-information-without-disturbance and no-free-information theorems express that those observables that do not disturb the measurement of another observable and those that can be measured jointly with any other observable must be trivial, i.e., coin tossing observables. We show that in the framework of general probabilistic theories these statements do not hold in general and continue to completely specify these two classes of observables. In this way, we obtain characterizations of the probabilistic theories where these statements hold. As a particular class of state spaces we consider the polygon state spaces, in which we demonstrate our results and show that while the no-information-without-disturbance principle always holds, the validity of the no-free-information principle depends on the parity of the number of vertices of the polygons.

quant-ph↗

arXiv2019

On the properties of spectral effect algebras

The aim of this paper is to show that there can be either only one or uncountably many contexts in any spectral effect algebra, answering a question posed in [S. Gudder, Convex and Sequential Effect Algebras, (2018), arXiv:1802.01265]. We also provide some results on the structure of spectral effect algebras and their state spaces and investigate the direct products and direct convex sums of spectral effect algebras. In the case of spectral effect algebras with sharply determining state space, stronger properties can be proved: the spectral decompositions are essentially unique, the algebra is sharply dominating and the set of its sharp elements is an orthomodular lattice. The article also contains a list of open questions that might provide interesting future research directions.

quant-ph↗