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arXiv:2406.15179 · Journal of the Physical Society of Japan

Upper Bounds on Probabilities in Channel Measurements on Qubit Channels and Their Applications

Taihei Kimoto, Takayuki Miyadera

Published Jul 17, 2025• 0 citations
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Abstract

One of the fundamental tasks in quantum information processing is to measure the quantum channels. Similar to measurements of quantum states, measurements of quantum channels are inherently stochastic, that is, quantum theory provides a formula to calculate the probability of obtaining an outcome. The upper bound on each probability associated with the measurement outcome of the quantum channels is a fundamental and important quantity. In this study, we derived the upper bounds of the probability in a channel measurement for specific classes of quantum channels. We also present two applications for the upper bounds. The first is the notion of convertibility considered by Alberti and Uhlmann and the second is the detection problem of a quantum channel. These applications demonstrate the significance of the obtained upper bounds.

DOI
10.7566/jpsj.94.084004

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One of the fundamental tasks in quantum information processing is to measure the quantum channels. Similar to measurements of quantum states, measurements of quantum channels are inherently stochastic, that is, quantum theory provides a formula to calculate the probability of obtaining an outcome. The upper bound on each probability associated with the measurement outcome of the quantum channels is a fundamental and important quantity. In this study, we derived the upper bounds of the probability in a channel measurement for specific classes of quantum channels. We also present two applications for the upper bounds. The first is the notion of convertibility considered by Alberti and Uhlmann and the second is the detection problem of a quantum channel. These applications demonstrate the significance of the obtained upper bounds.
Informationally complete (IC) measurements are a useful class of measurements, as their outcome statistics uniquely determine an unknown quantum state. Hence, they are important for certain tasks such as quantum state tomography, quantum process tomography, etc. In this work, we study the quantification of informational completeness for arbitrary quantum measurements by introducing and characterizing a faithful measure for it. We explicitly evaluate the informational completeness of qubit symmetric informationally complete (SIC) measurements and show that it is an upper bound for all qubit minimal informationally complete measurements. Furthermore, by introducing a faithful measure, we try to quantify and characterize the ability of an arbitrary quantum channel to preserve the informational completeness of any IC measurement when the channel acts on it in the Heisenberg picture. We call this measure informational completeness-preservability (IC preservability) of quantum channels. After studying its properties, we establish its relation to another quantity, namely, the absolute output coherence of a quantum channel, which quantifies the minimum amount of coherence (w.r.t. an arbitrary incoherent basis) that can always be obtained from the output of that channel. Finally, we illustrate our results with an example from light-matter interaction scenarios. Thus, in this work, not only do we try to provide a quantitative framework for studying both the informational completeness of quantum measurements and the ability of quantum channels to preserve it, but we also try to offer key insight into the conceptual relation between informational completeness and quantum coherence.
A quantum channel is a mapping which sends density matrices to density matrices. The estimation of quantum channels is of great importance to the field of quantum information. In this thesis two topics related to estimation of quantum channels are investigated. The first of these is the upper bound of Sarovar and Milburn (2006) on the Fisher information obtainable by measuring the output of a channel. Two questions raised by Sarovar and Milburn about their bound are answered. A Riemannian metric on the space of quantum states is introduced, related to the construction of the Sarovar and Milburn bound. Its properties are characterized. The second topic investigated is the estimation of unitary channels. The situation is considered in which an experimenter has several non-identical unitary channels that have the same parameter. It is shown that it is possible to improve estimation using the channels together, analogous to the case of identical unitary channels. Also, a new method of phase estimation is given based on a method sketched by Kitaev (1996). Unlike other phase estimation procedures which perform similarly, this procedure requires only very basic experimental resources.
We apply algorithmic information theory to quantum mechanics in order to shed light on an algorithmic structure which is inherent in quantum mechanics.There are two equivalent ways to define the (classical) Kolmogorov ∞mplexity\infty \mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{t}\mathrm{y}∞mplexity K(s)K(s)K(s) of agiven classical finite binary string sss .In the standard way, K(s)K(s)K(s) is defined as the length of the shortest input string for the universal self-delimiting Turing machine to output sss .In the other way, we first introduce the so-called universal probability mmm , and then define K(s)K(s)K(s) as −log⁡2m(s)-\log_{2}m(s)−log2​m(s) without using the concept of program-size.We generalize the universal probability to amatrix-valued function, and identify this function with apositive operator-valued measure (POVM), which describes the statistics of outcomes in aquantum measurement in the general set- ting.Based on this identification, we study acomputable POVM measurement with countable measurement outcomes performed upon afinite dimensional quantum system.We show that, up to amultiplicative constant, 2−K(s)2^{-K(s)}2−K(s) is the upper bound for the probability of each measurement outcome sss in such aquantum measurement.In what follows, the upper bound 2−K(s)2^{-K(s)}2−K(s) is shown to be optimal in acertain sense.
We show that the sample complexity required in quantum learning theory within a general parametric framework is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error, with high success probability under maximum-likelihood estimation. Notably, both the upper and lower bounds are determined by the supremum of the maximum diagonal entry of the inverse Fisher information matrix. We then apply the general bounds to Pauli channel learning and Pauli expectation value learning, which serve as representative tasks in quantum channel and state learning, respectively, in the asymptotic small-error regime. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value learning without quantum memory by comparing the quantum Fisher information matrix and the classical Fisher information matrix. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators, deriving the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli channel learning with entangled probes. We highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, the learning sample complexity is governed by the inverse Fisher information matrix, which is the central quantity in quantum metrology that determines the ultimate achievable mean squared error.