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Optimal measurement strategies for quantum states and quantum channels estimation

Nadia Milazzo

Published Jul 14, 2022• 0 citations
Publisher

Abstract

The rapid advance of quantum information technology requires precise control and manipulation of quantum systems and of their properties; in particular, it is essential to certify that quantum processors truly work quantum mechanically, in order to validate experiments and their results. The problem of certification of quantum states and devices is a demanding one, thus many attempts have been put forward to find ways of efficiently test their basic functionalities, such as their entanglement properties. The main problems addressed in this thesis concern indeed finding optimal strategies for the estimation and characterization of quantum states and channels, with a special focus on entanglement correlations. In our first work we considered the certification of entanglement across a given partition of a multi-qubit system, when only partial information about the corresponding quantum state is available; we tackled the problem of finding the best measurement strategy introducing the statistics of lengths of measurement sequences, choosing multi-qubit Pauli-measurements as observables. Using a sample of random unknown states, we were able to identify the (on average) shortest, i.e. most efficient, measurement sequence to detect entanglement. The investigation was carried out with an algorithm based on the truncated moment sequences (tms) problem, which provides necessary and sufficient conditions for entanglement or separability of a quantum state of a finite dimensional system. This study results in a very efficient strategy especially for symmetric states, for which only a tiny fraction ($10^{-6}$) of randomly chosen entangled states are undetected for systems with 6 qubits or more. We continued in our second work with the problem of separability of quantum channels to answer the question whether a given quantum device is able to create entanglement or not. The Choi matrix representation, provided by the Choi–Jamiołkowski isomorphism, casts the problem in terms of systems and ancillas, giving different classes of separability depending on the cut considered between them. Once again, the tms framework turned out to be well suited for this study, giving a unifying approach for the different separability problems; a solution is found in terms of the tms associated to the coordinates of the Choi state, in a fixed basis, and semidefinite programming is used to get a separability certificate. We explored examples of families of 2-qubit and single-qutrit channels, for which our algorithm can give an answer in cases where other criteria remain inconclusive. In a further work we shifted our attention to a different direction, highlighting a connection between the fields of quantum information, Bose-Einstein condensation (BEC) and analogue gravity, thus showing the relevance of quantum information notions in a broader context. We studied the entanglement properties of a BEC analogue of a black hole, which is described by a three-mode Gaussian state; we investigated bipartite and tripartite entanglement measures based on the covariance matrix description, both at zero and finite temperature, providing the best experimental configuration for entanglement detection. In a final stage of this thesis, we started looking at different measurement strategies for testing quantum devices whose functionality can be described by a single parameter; we used a Bayesian approach to estimate this parameter, exploiting the information collected through measurements to update a conditional probability distribution, without the need of estimating expected values of observables. The problem of finding the optimal experimental design can be in this case translated in the optimization of a utility function, which usually guides the update process adaptively at each step; we tried to outperform this approach looking ahead not only at the next step, but at a few ones, in order to obtain an answer of correct or wrong functioning as soon as possible. Moreover, we chose the input state for the quantum channel considered minimizing the error probability given by the Chernoff distance between the outputs of an ideal and a faulty channel; finally, we considered two different final decision criteria and compared their efficiency.

DOI
10.15496/publikation-70830

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The rapid advance of quantum information technology requires precise control and manipulation of quantum systems and of their properties; in particular, it is essential to certify that quantum processors truly work quantum mechanically, in order to validate experiments and their results. The problem of certification of quantum states and devices is a demanding one, thus many attempts have been put forward to find ways of efficiently test their basic functionalities, such as their entanglement properties. The main problems addressed in this thesis concern indeed finding optimal strategies for the estimation and characterization of quantum states and channels, with a special focus on entanglement correlations. In our first work we considered the certification of entanglement across a given partition of a multi-qubit system, when only partial information about the corresponding quantum state is available; we tackled the problem of finding the best measurement strategy introducing the statistics of lengths of measurement sequences, choosing multi-qubit Pauli-measurements as observables. Using a sample of random unknown states, we were able to identify the (on average) shortest, i.e. most efficient, measurement sequence to detect entanglement. The investigation was carried out with an algorithm based on the truncated moment sequences (tms) problem, which provides necessary and sufficient conditions for entanglement or separability of a quantum state of a finite dimensional system. This study results in a very efficient strategy especially for symmetric states, for which only a tiny fraction (10−610^{-6}10−6) of randomly chosen entangled states are undetected for systems with 6 qubits or more. We continued in our second work with the problem of separability of quantum channels to answer the question whether a given quantum device is able to create entanglement or not. The Choi matrix representation, provided by the Choi–Jamiołkowski isomorphism, casts the problem in terms of systems and ancillas, giving different classes of separability depending on the cut considered between them. Once again, the tms framework turned out to be well suited for this study, giving a unifying approach for the different separability problems; a solution is found in terms of the tms associated to the coordinates of the Choi state, in a fixed basis, and semidefinite programming is used to get a separability certificate. We explored examples of families of 2-qubit and single-qutrit channels, for which our algorithm can give an answer in cases where other criteria remain inconclusive. In a further work we shifted our attention to a different direction, highlighting a connection between the fields of quantum information, Bose-Einstein condensation (BEC) and analogue gravity, thus showing the relevance of quantum information notions in a broader context. We studied the entanglement properties of a BEC analogue of a black hole, which is described by a three-mode Gaussian state; we investigated bipartite and tripartite entanglement measures based on the covariance matrix description, both at zero and finite temperature, providing the best experimental configuration for entanglement detection. In a final stage of this thesis, we started looking at different measurement strategies for testing quantum devices whose functionality can be described by a single parameter; we used a Bayesian approach to estimate this parameter, exploiting the information collected through measurements to update a conditional probability distribution, without the need of estimating expected values of observables. The problem of finding the optimal experimental design can be in this case translated in the optimization of a utility function, which usually guides the update process adaptively at each step; we tried to outperform this approach looking ahead not only at the next step, but at a few ones, in order to obtain an answer of correct or wrong functioning as soon as possible. Moreover, we chose the input state for the quantum channel considered minimizing the error probability given by the Chernoff distance between the outputs of an ideal and a faulty channel; finally, we considered two different final decision criteria and compared their efficiency.
With the advance of quantum information technology, the question of how to most efficiently test quantum circuits is becoming of increasing relevance. Here we introduce the statistics of lengths of measurement sequences that allows one to certify entanglement across a given bipartition of a multiqubit system over the possible sequence of measurements of random unknown states and identify the best measurement strategies in the sense of the (on average) shortest measurement sequence of (multiqubit) Pauli measurements. The approach is based on the algorithm of truncated moment sequences, which allows one to deal naturally with incomplete information, i.e., information that does not fully specify the quantum state. We find that the set of measurements corresponding to diagonal matrix elements of the moment matrix of the state are particularly efficient. For symmetric states their number increases only like the third power of the number NNN of qubits. Their efficiency increases rapidly with NNN, leaving already for N=4N=4N=4 less than a fraction ${10}^{\ensuremath{-}6}$ of randomly chosen entangled states undetected.
We consider the problem of separability of quantum channels via the Choi matrix representation given by the Choi-Jamio\l{}kowski isomorphism. We explore three classes of separability across different cuts between systems and ancillae, and we provide a solution based on the mapping of the coordinates of the Choi state (in a fixed basis) to a truncated moment sequence (tms) yyy. This results in an algorithm which gives a separability certificate using semidefinite programming. The computational complexity and the performance of it depend on the number of variables nnn in the tms and on the size of the moment matrix Mt(y){M}_{t}(y)Mt​(y) of order ttt. We exploit the algorithm to numerically investigate separability of families of two-qubit and single-qutrit channels; in the latter case we can provide an answer for examples explored earlier through the criterion based on the negativity NNN, a criterion which remains inconclusive for Choi matrices with N=0N=0N=0.
Stratégies de mesure optimales pour l'estimation des états quantiques et des canaux quantiques L'avancée rapide de la technologie de l'information quantique nécessite un contrôle et une manipulation précis des systèmes quantiques. En particulier, il est essentiel de certifier que les processeurs quantiques fonctionnent réellement de manière quantique, afin de valider les expériences et leurs résultats. Les principaux problèmes abordés dans cette thèse concernent la recherche de stratégies optimales pour l'estimation et la caractérisation des états et des canaux quantiques, avec un accent particulier sur les corrélations d'intrication. Nous avons considéré la certification de l'intrication à travers une partition donnée d'un système multi-qubit, lorsque seulement une information partielle sur l'état quantique correspondant est disponible ; nous avons abordé le problème avec un algorithme basé sur le problème des séquences de moments tronqués (tms), identifiant ainsi la séquence de mesure la plus efficace pour détecter l'intrication. Nous avons également étudié le problème de la séparabilité des canaux quantiques en termes de représentation de la matrice de Choi ; dans ce cas, le cadre tms fournit une condition nécessaire et suffisante unifiée pour les différentes classes de séparabilité des canaux quantiques. Quelques résultats préliminaires sur les "tests fonctionnels quantiques" sont discutés ; nous avons utilisé des stratégies adaptatives bayésiennes pour obtenir une réponse sur le fonctionnement correct ou incorrect d'un dispositif quantique décrit par un seul paramètre dès que possible. Enfin, nous avons également mis en évidence la pertinence des notions d'information quantique par une connexion entre les domaines de l'information quantique, de la condensation de Bose-Einstein (BEC) et de la gravité analogue, en étudiant les propriétés d'intrication d'un BEC analogue à un trou noir.
If only limited control over a multiparticle quantum system is available, a viable method to characterize correlations is to perform random measurements and consider the moments of the resulting probability distribution. We present systematic methods to analyze the different forms of entanglement with these moments in an optimized manner. First, we find the optimal criteria for different forms of multiparticle entanglement in three-qubit systems using the second moments of randomized measurements. Second, we present the optimal inequalities if entanglement in a bipartition of a multi-qubit system shall be analyzed in terms of these moments. Finally, for higher-dimensional two-particle systems and higher moments, we provide criteria that are able to characterize various examples of bound entangled states, showing that detection of such states is possible in this framework.
If only limited control over a multiparticle quantum system is available, a viable method to characterize correlations is to perform random measurements and consider the moments of the resulting probability distribution. We present systematic methods to analyze the different forms of entanglement with these moments in an optimized manner. First, we find the optimal criteria for different forms of multiparticle entanglement in three-qubit systems using the second moments of randomized measurements. Second, we present the optimal inequalities if entanglement in a bipartition of a multiqubit system shall be analyzed in terms of these moments. Finally, for higher-dimensional two-particle systems and higher moments, we provide criteria that are able to characterize various examples of bound entangled states, showing that detection of such states is possible in this framework.
Certifying high-dimensional quantum channels is essential for ensuring the reliability of quantum communication protocols. Existing certification schemes often rely on fully trusted internal devices, which is difficult to achieve in realistic scenarios. Here, we propose a semi-device-independent framework for certifying channel properties directly from observed statistics, assuming only that the system dimension is known. By explicitly incorporating the full set of structural constraints inherent to Choi states, our approach exploits the Choi-Jamiołkowski isomorphism for rigorous certification of quantum channels. The entanglement dimensionality of quantum channels is first certified by introducing a witness and numerically determining its Schmidt-number-dependent bounds. This certification method reproduces known analytical benchmarks and is applied to dephasing and depolarizing noise channels, thereby confirming its validity. To provide a more complete assessment of channel performance, the entanglement fidelity of quantum channels is also certified using a hierarchy of semidefinite programming relaxations based on localizing matrices. Lower bounds on the entanglement fidelity are obtained that are compatible with either the full set of observed statistics or a single witness value.