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Stability Verification of Quantum non-i.i.d. sources


arXivPhysica Scripta

Abstract: We introduce the problem of stability verification of quantum sources which are non-i.i.d.. These sources are better described by a Markov chain. This problem is closely related to the problem of quantum verification first proposed by Pallister et. al. [1], however, it extends the notion of the original problem. We introduce a family of states that come from these non-i.i.d. sources which we call a Markov state. We prove in theorem 1 that these states are not well described with tensor products over a changing source. In theorem 2 we further provide a lower bound on the trace distance between two Markov states, which is the simplest way to solve the problem of stability verification of the quantum sources we introduce

Classicality from Quantum Stochastic Processes



Abstract: We develop a theory of classicality from quantum systems. This theory stems from the study of classical and quantum stationary stochastic processes. The stochastic processes are characterized by polyhedral (classical) and semidefinite representative (quantum) cones. Based on a previous result \cite{2209.06806v1} we expand the study of fixed points from quantum channels. We give a semidefinite program that characterizes a quantum channel separating into a core and a part that decays with many iterations. In general, the solution is non-separable in the space it is defined. We present a characterization of channels in terms of their fixed points for the separable case. A quantum simulation of a polyhedral cone can then be constructed.

Relativistic time dilation as a quantum mechanism



Abstract: We propose a mechanism for time dilation using quantum systems. We introduce a family of operators that are sensitive to the changes of quantum states from different frames of reference. The change between reference frames is done via a Galilean transformation, therefore, the source of the dilation in our case comes from the observable. These observables grow linearly in time and depending on the reference frame of the state the linear growth changes its slope, therefore it takes longer to grow to the same point. Such mechanism implies a different view from the usual understanding of spacetime.

Quantum Stabilizer Channel for Thermalization



Abstract: We study the problem of quantum thermalization from a very recent perspective: via discrete interactions with thermalized systems. We thus extend the previously introduced scattering thermalization program by studying not only a specific channel but allowing any possible one. We find a channel that solves a fixed point condition using the Choi matrix approach that is in general non-trace-preserving. We find the explicit channel that solves this problem which yields a condition for trace preservation. From a quantum computing perspective, the results thus obtained can be interpreted as a condition for quantum error correction that also reminds of quantum error avoiding.

Sequential Analysis of a finite number of Coherent states



Abstract: We investigate an advantage for information processing of ordering a set of states over making a global quantum processing with a fixed number of copies of coherent states. Suppose Alice has $N$ copies of one of two quantum states $\sigma_0$ or $\sigma_1$ and she gives these states to Bob. Using the optimal sequential test, the SPRT, we ask if processing the states in batches of size $l$ is advantageous to optimally distinguish the two hypotheses. We find that for the symmetric case $\{|\gamma\rangle,|−\gamma\rangle\}$ there is no advantage of taking any batch size $l$. We give an expression for the optimal batch size $l_\text{opt}$ in the assymetric case. We give bounds $l_\text{min}$ and $l_\text{max}$ for when $P_S\approx1$.

Online identification of symmetric pure states

Gael Sentís, EMV, Ramon Muñoz-Tapia

Quantum 6, 658 (2022)

Abstract: We consider online strategies for discriminating between symmetric pure states with zero error when n copies of the states are provided. Optimized online strategies involve local, possibly adaptive measurements on each copy and are optimal at each step, which makes them robust in front of particle losses or an abrupt termination of the discrimination process. We first review previous results on binary minimum and zero error discrimination with local measurements that achieve the maximum success probability set by optimizing over global measurements, highlighting their online features. We then extend these results to the case of zero error identification of three symmetric states with constant overlap. We provide optimal online schemes that attain global performance for any n if the state overlaps are positive, and for odd n if overlaps have a negative value. For arbitrary complex overlaps, we show compelling evidence that online schemes fail to reach optimal global performance. The online schemes that we describe only require to store the last outcome obtained in a classical memory, and adaptiveness of the measurements reduce to at most two changes, regardless of the value of n.

Quantum Sequential Hypothesis Testing

EMV, Christoph Hirche, Gael Sentís, Michalis Skotiniotis, Marta Carrizo, Ramon Muñoz-Tapia, and John Calsamiglia

Phys. Rev. Lett. 126, 180502 (2021)
arXiv PRL

Abstract: We introduce sequential analysis in quantum information processing, by focusing on the fundamental task of quantum hypothesis testing. In particular, our goal is to discriminate between two arbitrary quantum states with a prescribed error threshold ε when copies of the states can be required on demand. We obtain ultimate lower bounds on the average number of copies needed to accomplish the task. We give a block-sampling strategy that allows us to achieve the lower bound for some classes of states. The bound is optimal in both the symmetric as well as the asymmetric setting in the sense that it requires the least mean number of copies out of all other procedures, including the ones that fix the number of copies ahead of time. For qubit states we derive explicit expressions for the minimum average number of copies and show that a sequential strategy based on fixed local measurements outperforms the best collective measurement on a predetermined number of copies. Whereas for general states the number of copies increases as log(1/ε), for pure states sequential strategies require a finite average number of samples even in the case of perfect discrimination, i.e., ε=0.

Quantum measurement optimization by decomposition of measurements into extremals

EMV, Carlos Pineda, Pablo Barberis-Blostein

Scientific Reports volume 10, Article number: 9375 (2020)
Scientific Reports

Abstract: Using the convex structure of positive operator value measurements and several quantities used in quantum metrology, such as quantum Fisher information or the quantum Van Trees information, we present an efficient numerical method to find the best strategy allowed by quantum mechanics to estimate a parameter. This method explores extremal measurements thus providing a significant advantage over previously used methods. We exemplify the method for different cost functions in a qubit and in a harmonic oscillator and find a strong numerical advantage when the desired target error is sufficiently small.

Certified answers for ordered quantum discrimination problems

EMV and Ramon Muñoz-Tapia

Phys. Rev. A 100, 042331 (2019)

Abstract: We investigate the quantum state discrimination task for sets of linear independent pure states with an intrinsic ordering. These structured discrimination problems allow for a scheme that provides a certified level of error; that is, answers that never deviate from the true value by more than a specified distance and hence control the desired quality of the results. We obtain an efficient semidefinite program and also find a general lower bound valid for any error distance that only requires the knowledge of an optimal minimum-error scheme. We apply our results to the cases of quantum change point and quantum state anomaly detection.

Online optimal exact identification of a quantum change point

Gael Sentís, EMV, Ramon Muñoz-Tapia

Phys. Rev. A 98, 052305 (2018)

Abstract: We consider online detection strategies for identifying a change point in a stream of quantum particles allegedly prepared in identical states. We show that the identification of the change point can be done without error via sequential local measurements while attaining the optimal performance bound set by quantum mechanics. In this way, we establish the task of exactly identifying a quantum change point as an instance where local protocols are as powerful as global ones. The optimal online detection strategy requires only one bit of memory between subsequent measurements, and it is amenable to experimental realization with current technology.

Quantum estimation of unknown parameters

EMV, Carlos Pineda, François Leyvraz, and Pablo Barberis-Blostein

Phys. Rev. A 95, 012136 (2017)

Abstract: We discuss the problem of finding the best measurement strategy for estimating the value of a quantum system parameter. In general the optimum quantum measurement, in the sense that it maximizes the quantum Fisher information and hence allows one to minimize the estimation error, can only be determined if the value of the parameter is already known. A modification of the quantum Van Trees inequality, which gives a lower bound on the error in the estimation of a random parameter, is proposed. The suggested inequality allows us to assert if a particular quantum measurement, together with an appropriate estimator, is optimal. An adaptive strategy to estimate the value of a parameter, based on our modified inequality, is proposed.