Adaptiveness is a key principle in information processing including statistics and machine learning. We investigate the usefulness adaptive methods in the framework of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel and observing outputs. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and nonadaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. The following results are obtained: (1) We prove that for classical-quantum channels, adaptive and nonadaptive strategies lead to the same error exponents both in the symmetric (Chernoff) and asymmetric (Hoeffding, Stein) settings. (2) The first separation between adaptive and nonadaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for nonadaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. (3) We prove, in some sense generalizing the previous statement, that for general channels adaptive strategies restricted to classical feed-forward and product state channel inputs are not superior in the asymptotic limit to nonadaptive product state strategies. (4) As an application of our findings, we address the discrimination power of an arbitrary quantum channel and show that adaptive strategies with classical feedback and no quantum memory at the input do not increase the discrimination power of the channel beyond nonadaptive tensor product input strategies.
Adaptiveness is a key principle in information processing including statistics and machine learning. We investigate the usefulness adaptive methods in the framework of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel and observing outputs. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and nonadaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. The following results are obtained: (1) We prove that for classical-quantum channels, adaptive and nonadaptive strategies lead to the same error exponents both in the symmetric (Chernoff) and asymmetric (Hoeffding, Stein) settings. (2) The first separation between adaptive and nonadaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for nonadaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. (3) We prove, in some sense generalizing the previous statement, that for general channels adaptive strategies restricted to classical feed-forward and product state channel inputs are not superior in the asymptotic limit to nonadaptive product state strategies. (4) As an application of our findings, we address the discrimination power of an arbitrary quantum channel and show that adaptive strategies with classical feedback and no quantum memory at the input do not increase the discrimination power of the channel beyond nonadaptive tensor product input strategies.
We present a broad investigation of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel and observing its output. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and non-adaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. The following results are obtained: (1) The first separation between adaptive and non-adaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for non-adaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. (2) We prove that for classical-quantum channels, adaptive and non-adaptive strategies lead to the same error exponents both in the symmetric (Chernoff) and asymmetric (Hoeffding, Stein) settings. (3) We prove, in some sense generalizing the previous statement, that for general channels adaptive strategies restricted to classical feed-forward and product state channel inputs are not superior in the asymptotic limit to non-adaptive product state strategies. (4) As an application of our findings, we address the discrimination power of quantum channels and show that adaptive strategies with classical feedback and no quantum memory at the input do not increase the discrimination power of entanglement-breaking channel beyond non-adaptive tensor product input strategies.
We present a broad investigation of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel multiple times and observing its output at the end. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and non-adaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. Our main result is the first separation between adaptive and non-adaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for non-adaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. Full details in [1].
The present thesis has three major thrusts: the first thrust presents a broad investigation of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and non-adaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. The following results are obtained: (1) The first separation between adaptive and non-adaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for non-adaptive strategies. (2) We prove that for classical-quantum channels, adaptive and non-adaptive strategies lead to the same error exponents both in the symmetric (Chernoff) and asymmetric (Hoeffding) settings. (3) We prove that, in some sense generalizing the previous statement, for general channels adaptive strategies restricted to classical feed-forward and product state channel inputs are not superior to non-adaptive product state strategies. (4) As application of our findings, we address the discrimination power of quantum channels and show that neither adaptive strategies nor input quantum memory can increase the discrimination power of an entanglement-breaking channel. In the second thrust, we construct new protocols for the tasks of converting noisy multipartite quantum correlations into noiseless classical and quantum ones using local operations and classical communications (LOCC). For the former, known as common randomness (CR) distillation, two new lower bounds are obtained. Our proof relies on a generalization of communication for omniscience (CO). Our contribution here is a novel simultaneous decoder for the compression of correlated classical sources by random binning with quantum side information at the decoder. For the latter, we derive two new lower bounds on the rate at which Greenberger-Horne-Zeilinger (GHZ) states can be asymptotically distilled from any given pure state under LOCC. Our approach consists in “making coherent” the proposed CR distillation protocols and recycling of resources. The final thrust studies communication over a single-serving two-receiver quantum broadcast channel with legitimate receiver and eavesdropper. We find inner and outer boundary regions for the tradeoff between common, individualized, confidential messages as well as the rate of the dummy randomness used for obfuscation. As applications, we find one-shot capacity bounds on the simultaneous transmission of classical and quantum information and re-derive a number of asymptotic results in the literature. Esta tesis está estructurada en tres ejes: En la primera parte se presenta una investigación extensa de los test de hipótesis binarios asintóticos cuando cada hipótesis representa diferentes instancias independientes de un canal cuántico. A diferencia del caso habitual ya conocido en el que los estados cuánticos se modelan como hipótesis, en este trabajo se distingue entre estrategias adaptativas y no adaptativas con respecto al uso del canal, y se presentan una serie de variantes adicionales de las tareas de discriminación imponiendo diferentes restricciones a la estrategia de test. Se obtienen los siguientes resultados: (1) Se obtiene por primera vez una separación entre los exponentes de test de hipótesis simétricos adaptativos y no adaptativos, derivándose por primera vez una cota inferior de la probabilidad de error para estrategias no adaptativas. (2) Se demuestra que para canales cuánticos clásicos, las estrategias adaptativas y no adaptativas conducen a los mismos exponentes de error tanto en el caso simétrico (Chernoff) como en configuraciones asimétricas (Hoeffding); (3) Se demuestra, generalizando el resultado anterior, que en general estrategias adaptativas restringidas al feedforward clásico y entradas de tipo producto no son superiores a estrategias de tipo producto no adaptativas; (4) Como aplicación de los resultados anteriores se aborda la discriminación de potencia en canales cuánticos. Se demuestra que, ni estrategias adaptativas, ni la utilización de entradas con memoria, permiten mejorar la discriminación de potencia de canales del tipo entanglement-breaking. En la segunda parte de la tesis se construyen nuevos protocolos para convertir correlaciones cuánticas ruidosas en correlaciones clásicas, o bien en correlaciones cuánticas, ambas libres de ruido, mediante la utilización de operaciones locales y comunicaciones clásicas (LOCC). Para la primera tarea, conocida como destilación de aleatoriedad común (CR), se obtienen dos nuevos límites inferiores de la aleatoriedad común destilable. Este trabajo supone una generalización de la comunicación para la omnisciencia. En la segunda tarea, se obtienen dos nuevos límites inferiores de la tasa a la que los estados Greenberger-Horne-Zeilinger (GHZ) pueden destilarse asintóticamente, desde cualquier estado puro, utilizando LOCC. El enfoque consiste en hacer coherente el protocolo de destilación CR propuesto, así como en la reutilización de recursos. La última parte de la tesis final estudia la comunicación mediante un solo uso del canal cuántico en presencia de un receptor legítimo y de un observador no autorizado. Se obtienen regiones interiores y exteriores asociadas al compromiso entre la tasa de transmisión común, confidencial, individualizada y de la fuente aleatoria. Aquesta tesi està estructurada en tres eixos: A la primera part es presenta una investigació extensa dels test d’hipòtesis binaris asimptòtics quan cada hipòtesi representa diferents instàncies independents d’un canal quàntic. A diferència del cas habitual ja conegut en el qual els estats quàntics es modelen com a hipòtesi, en aquest treball es distingeix entre estratègies adaptatives i no adaptatives pel que fa a l’ús del canal, i es presenten una sèrie de variants addicionals de les tasques de discriminació imposant diferents restriccions a la estratègies de test. S’obtenen els següents resultats: (1) S’obté per primera vegada una separació entre els exponents de test d’- hipòtesis simètrics adaptatius i no adaptatius, derivant per primera vegada una fita inferior de la probabilitat d’error per estratègies no adaptatives. (2) Es demostra que per a canals quàntics clàssics, les estratègies adaptatives i no adaptatives condueixen als mateixos exponents d’error tant en el cas simètric (Chernoff) com en configuracions asimètriques (Hoeffding); (3) Es demostra, generalitzant el resultat anterior, que en general estratègies adaptatives restringides a feed-forward clàssic i entrades de tipus producte no són superiors a estratègies de tipus producte no adaptatives; (4) Com a aplicació dels resultats anteriors s’aborda la discriminació de potència en canals quàntics. Es demostra que, ni estratègies adaptatives, ni la utilització d’entrades amb memòria, permeten millorar la discriminació de potència de canals de tipus entanglement-breaking. A la segona part de la tesi es construeixen nous protocols per convertir correlacions quàntiques sorolloses en correlacions clàssiques, o bé en correlacions quàntiques, ambdues lliures de soroll, mitjançant la utilització d’operacions locals i comunicacions clàssiques (LOCC). Per a la primera tasca, coneguda com destil·lació d’aleatorietat comú (CR), s’obtenen dos nous límits inferiors de l’aleatorietat comuna destil·lable. Aquest treball suposa una generalització de la comunicació per a l’omnisciència. En la segona tasca, s’obtenen dos nous límits inferiors de la taxa a la qual els estats Greenberger- Horne-Zeilinger (GHZ) poden destil·lar asimptòticament, des de qualsevol estat pur, utilitzant LOCC. L’enfocament consisteix a fer coherent el protocol de destil·lació CR proposat així com en la reutilització de recursos. L’última part de la tesi final estudia la comunicació mitjançant un sol ús de canal quàntic en presència d’un receptor legítim i d’un observador no autoritzat. S’obtenen regions interiors i exteriors associades a el compromís entre la taxa de transmissió comú, confidencial, individualitzada i de la font aleatòria.
Abstract In this paper, we study the problem of sampling complexity for channel discrimination with respect to two different strategies: product strategy and adaptive strategy. We first formally introduce the definitions of the sampling complexity of the channels under the framework of hypothesis testing , wherein the goal is to determine the minimum number of samples needed to reach a desired error probability. We then establish the lower and upper bounds on the sampling complexity of the symmetric, asymmetric, and error exponent hypothesis testing settings. We show that, by imposing product strategy on testing, the bounds are always characterized by the generalized channel divergence, while with adaptive strategy, the bounds are characterized by the amortized channel divergence. Finally, we analyze two concrete examples, and obtain that the adaptive strategy can not lead to an advantage to the problem of determining the sampling complexity for classical-quantum channels, which can bring advantages for generalized amplitude damping channels.
Quantum hypothesis testing is a central task in the entire field of quantum information theory. Understanding its ultimate limits will give insight into a wide range of quantum protocols and applications, from sensing to communication. Although the limits of hypothesis testing between quantum states have been completely clarified by the pioneering works of Helstrom in the 1970s, the more difficult problem of hypothesis testing with quantum channels, i.e., channel discrimination, is less understood. This is mainly due to the complications coming from the use of input entanglement and the possibility of employing adaptive strategies. In this Letter, we establish a lower limit for the ultimate error probability affecting the discrimination of an arbitrary number of quantum channels. We also show that this lower bound is achievable when the channels have certain symmetries. As an example, we apply our results to the problem of channel position finding, where the goal is to identify the location of a target channel among multiple background channels. In this general setting, we find that the use of entanglement offers a great advantage over strategies without entanglement, with nontrivial implications for data readout, target detection, and quantum spectroscopy.
We study the problem of binary composite channel discrimination in the asymmetric setting, where the hypotheses are given by fairly arbitrary sets of channels, and samples do not have to be identically distributed. In the case of quantum channels we prove: (i) a characterization of the Stein exponent for parallel channel discrimination strategies and (ii) an upper bound on the Stein exponent for adaptive channel discrimination strategies. We further show that already for classical channels this upper bound can sometimes be achieved and be strictly larger than what is possible with parallel strategies. Hence, there can be an advantage of adaptive channel discrimination strategies with composite hypotheses for classical channels, unlike in the case of simple hypotheses. Moreover, we show that classically this advantage can only exist if the sets of channels corresponding to the hypotheses are non-convex. As a consequence of our more general treatment, which is not limited to the composite i.i.d. setting, we also obtain a generalization of previous composite state discrimination results.