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arXiv:1705.06343 · Journal of Mathematical Physics

Operational framework for quantum measurement simulability

Leonardo Guerini, Jessica Bavaresco, Marcelo Terra Cunha, Antonio Acín

Published Sep 1, 2017• 65 citations
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Abstract

We introduce a framework for simulating quantum measurements based on classical processing of a set of accessible measurements. Well-known concepts such as joint measurability and projective simulability naturally emerge as particular cases of our framework, but our study also leads to novel results and questions. First, a generalisation of joint measurability is derived, which yields a hierarchy for the incompatibility of sets of measurements. A similar hierarchy is defined based on the number of outcomes necessary to perform a simulation of a given measurement. This general approach also allows us to identify connections between different kinds of simulability and, in particular, we characterise the qubit measurements that are projective-simulable in terms of joint measurability. Finally, we discuss how our framework can be interpreted in the context of resource theories.

DOI
10.1063/1.4994303

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We introduce a framework for simulating quantum measurements based on classical processing of a set of accessible measurements. Well-known concepts such as joint measurability and projective simulability naturally emerge as particular cases of our framework, but our study also leads to novel results and questions. First, a generalisation of joint measurability is derived, which yields a hierarchy for the incompatibility of sets of measurements. A similar hierarchy is defined based on the number of outcomes necessary to perform a simulation of a given measurement. This general approach also allows us to identify connections between different kinds of simulability and, in particular, we characterise the qubit measurements that are projective-simulable in terms of joint measurability. Finally, we discuss how our framework can be interpreted in the context of resource theories.
We investigate the compression of quantum information with respect to a given set M of high-dimensional measurements. This leads to a notion of simulability, where we demand that the statistics obtained from M and an arbitrary quantum state ρ are recovered exactly by first compressing ρ into a lower-dimensional space, followed by some quantum measurements. A full quantum compression is possible, i.e., leaving only classical information, if and only if the set M is jointly measurable. Our notion of simulability can thus be seen as a quantification of measurement incompatibility in terms of dimension. After defining these concepts, we provide an illustrative example involving mutually unbiased bases, and develop a method based on semidefinite programming for constructing simulation models. In turn we analytically construct optimal simulation models for all projective measurements subjected to white noise or losses. Finally, we discuss how our approach connects with other concepts introduced in the context of quantum channels and quantum correlations.
Strict hierarchy between nnn-wise measurement simulability, compatibility structures, and multi-copy compatibility
The incompatibility of quantum measurements, i.e. the fact that certain observable quantities cannot be measured jointly is widely regarded as a distinctive quantum feature with important implications for the foundations and the applications of quantum information theory. While the standard incompatibility of multiple measurements has been the focus of attention since the inception of quantum theory, its generalizations, such as measurement simulability, nnn-wise incompatibility, and mulit-copy incompatibility have only been proposed recently. Here, we point out that all these generalizations are differing notions of the question of how many measurements are genuinely contained in a measurement device. We then show, that all notions do differ not only in their operational meaning but also mathematically in the set of measurement assemblages they describe. We then fully resolve the relations between these different generalizations, by showing a strict hierarchy between these notions. Hence, we provide a general framework for generalized measurement incompatibility. Finally, we consider the implications our results have for recent works using these different notions.
Measurement incompatibility describes two or more quantum measurements whose expected joint outcome on a given system cannot be defined. This purely nonclassical phenomenon provides a necessary ingredient in many quantum information tasks such as violating a Bell inequality or nonlocally steering part of an entangled state. In this Letter, we characterize incompatibility in terms of programmable measurement devices and the general notion of quantum programmability. This refers to the temporal freedom a user has in issuing programs to a quantum device. For devices with a classical control and classical output, measurement incompatibility emerges as the essential quantum resource embodied in their functioning. Based on the processing of programmable measurement devices, we construct a quantum resource theory of incompatibility. A complete set of convertibility conditions for programmable devices is derived based on quantum state discrimination with postmeasurement information.
This PhD thesis is focused on the quantum measurement simulability problem, that is, deciding whether a given measurement can be simulated when only a restricted subset of measurements is accessible. We provide an operational framework for this problem based on classical manipulations over the set of simulators. Particular cases of interest are further investigated, in which the simulators are taken to be projective measurements, measurements of a fixed number of outcomes, and arbitrary sets of fixed cardinality. In each of these situations we derive either necessary or sufficient conditions for simulability, and full characterisations in terms of semidefinite programming for some specific cases. Since joint measurability is a particular case of simulability, we also present a natural generalisation for it. Besides deciding whether a given measurement is simulable by some set of simulators, we also pose the question of what are the most robust measurements against simulability. We provide a strategy for approximating the set of quantum measurements based on relaxing the positivity constraint. This allows us to identify the most robust qubit measurement in terms of projective simulability, as well as the most incompatible sets of N measurements, for N = 1, . . . , 5, which notably are found to be always projective. By applying our simulability results in the context of Einstein-Podolsky-Rosen steering and Bell nonlocality we are able to construct improved and more general local models. Starting from models for a finite number of measurements we obtain the first general method for constructing local models for arbitrary families of quantum states. Similarly, our study on projective simulability yields a strategy for extending models for projective measurements to arbitrary ones, culminating in the most efficient local model for two-qubit Werner states and general measurements Esta tese de doutorado é centrada no problema de simulação de medições quânticas, ou seja, em decidir se uma dada medição pode ser simulada quando temos acesso a apenas um subconjunto restrito de medições. Apresentamos um framework operacional para esse problema, baseado em manipulações clássicas sobre o conjunto de simuladores. Casos particulares de interesse são estudados em detalhe, nos quais o conjunto de simuladores é dado por medições projetivas, medições de um número fixo de outcomes, e conjuntos arbitrários de cardinalidade fixada. Em cada uma dessas situações, derivamos condições necessárias ou suficientes para simulabilidade, e uma caracterização completa em termos de programação semidefinida em alguns casos específicos. Como comensurabilidade é um caso particular de simulabilidade, apresentamos também uma generalização natural para esse conceito. Além de decidir se uma dada medição é simulável ou não, também exploramos a questão de quais são as medições mais robustas contra simulabilidade. Apresentamos então uma estratégia para aproximar o conjunto das medições quânticas baseada em uma relaxação da condição de positividade. Isso nos permite identificar a medição mais robusta contra simulabilidade projetiva em dimensão 2, assim como os conjuntos de N medições mais incompatíveis, para N = 1, . . . , 5, que notavelmente se revelam ser projetivas em todos esses casos. Aplicando nossos resultados de simulabilidade no contexto de Einstein-Podolsky-Rosen steering e não-localidade de Bell, somos capazes de construir modelos locais melhores e mais gerais. Partindo de modelos para um número finito de medições, obtemos o primeiro método geral para construção de modelos locais para famílias arbitrárias de estados quânticos. De forma similar, nosso estudo de simulabilidade projetiva fornece uma estratégia para estender modelos locais para medições projetivas a medições arbitrárias, culminando no mais eficiente modelo local para estados de Werner de dois qubits e medições quaisquer Aquesta tesi doctoral se centra en el problema de la simulació de mesures quàntiques, és a dir, en decidir si es pot simular una determinada mesura quan només tenim accés a un subconjunt restringit de mesures diferents. Presentem un marc operacional per a aquest problema, basat en manipulacions clàssiques sobre el conjunt de simuladors. Casos particulars d’interès son estudiat en detall, on el conjunt de simuladors està donat per mesures projectius, mesures d’un nombre fix de resultats i conjunts arbitraris de cardinalitat fixa. En cadascuna d’aquestes situacions, derivem condicions necessaris o suficients per a la simulació, i una caracterització completa en termes de programació semi-definida en alguns casos específics. Com la mensurabilitat conjunta és un cas particular de simulació, presentem també una generalització natural per a aquest concepte. A més de decidir si un mesura és simulable o no, també exploram la qüestió de quines son las mesures més robustes contra la simulabilitat. A continuació, presentem una estratègia per aproximar el conjunt de mesures quàntiques basat en una relaxació de la condició de positivitat. Això permet la identificació de la mesura més robusta envers la simulació projectiva en dimensió 2, així com els conjunts més incompatibles de N mesures, per N = 1, . . . , 5, que notablement resulten ser projectivas en tots aquests casos. Aplicant els nostres resultats de simulació en el context d’Einstein-Podolsky-Rosen steering i no-localitat de Bell, som capaços de construir models locals millors i més generals. A partir de models per a un nombre finit de mesures, obtenim el primer mètode general per a la construcció de models locals per a famílies arbitràries d’estats quàntics. De la mateixa manera, el nostre estudi de la simulació projectiva proporciona una estratégia per ampliar models locals per a mesures projectivas a mesures arbitraris, culminant en el model local més eficient per als estats de Werner de dos qubits i mesures generals
In this thesis, we consider the properties of measurements in quantum theory and other operational theories. After having introduced the framework of operational theories, we consider a communication scheme based on an experimental prepare-and-measure scenario and demonstrate this with different communication tasks. This gives us context for how the different communication tasks can be implemented in different theories, in doing so establishing quantum theory intuitively as an operational theory among other theories. The main property of measurements we focus on in this work is the simulation of measurements, which consists of manipulating the inputs and outputs of the measurement devices. We study how using this process on existing measurement devices can be used to operationally imitate new devices, and what kind of structure the simulation process induces on measurements. We also consider applications of simulability. Firstly, we consider operational restrictions imposed upon measurements. We argue that the restricted set of physical measurements must be closed with respect to the simulation process since the simulation of physical devices must lead to other physically feasible devices. We demonstrate different types of restrictions by classifying them. As a second application we see how the simulation of measurements relates to compatibility of measurements and how it can be viewed as a generalisation of it. This allows us to present an operational principle previously known to quantum theory, the no-free-information principle, according to which any measurement that is compatible with all other measurement must not provide any useful, and therefore free, information about the system. Whilst this principle holds in quantum theory, there are non-classical theories for which it is violated, and so enforcing this principle may be considered a way to exclude some unphysical theories.
Quantum instruments describe both the classical outcome and the updated quantum state in a measurement process. To do this in a non-trivial way, instruments must have the capability to interact coherently with the state that they measure. Here, we develop a resource theory for instruments. We consider a relevant quantifier of the separation between interactive and non-interactive instruments and show that it admits three distinct operational interpretations in terms of quantum information tasks. These concern (i) the preservation of maximally entangled states after a local measurement, (ii) the average ability to preserve random states after measurement, and (iii) the ability to recover the classical information generated from measuring half of a maximally entangled state. We also introduce a natural set of allowed operations and show that the third task fully characterises the resource content of instruments. Our general framework reproduces as special cases established resource theories for channels and measurements.