The problem of estimating quantity of information, which can be obtained in the process of binary consecutive measuring a qubit state, is considered in the paper. The studied quantity is expressed as Shannon mutual information between the initial qubit state and the outcomes of a consecutive measurement. It is demonstrated that the maximum of information is reached by projective measurements. The maximum quantity of accessible information is calculated. It is proved that in the case of arbitrary binary test the maximum is reached asymptotically for consecutive measurements.
The problem of estimating quantity of information, which can be obtained in the process of binary consecutive measuring a qubit state, is considered in the paper. The studied quantity is expressed as Shannon mutual information between the initial qubit state and the outcomes of a consecutive measurement. It is demonstrated that the maximum of information is reached by projective measurements. The maximum quantity of accessible information is calculated. It is proved that in the case of arbitrary binary test the maximum is reached asymptotically for consecutive measurements.
The accessible information of any given quantum ensemble quantifies the maximum amount of Shannon information that can be extracted from the ensemble by any quantum measurement. Almost three decades ago, Shor conjectured that the accessible information of any quantum dichotomy, that is, an ensemble of two states, is attained by a projective measurement. Recently, a proof of this conjecture restricted to the qubit case was published by Keil. Here, we conclusively settle this longstanding open problem. First, we show that Shor's conjecture follows, in arbitrary dimension, from a recent result by Fang, Fawzi and Fawzi, and we provide a self-contained, elementary proof. Second, for any given dichotomy and measurement, we provide the explicit construction of a projective measurement that outperforms such a measurement in extracting information from the given dichotomy. Third, while the computation of the accessible information is known to be non-convex in general, we frame the computation of the accessible information of dihotomies as a convex problem.
Quantum and classical informations are considered in quantum measurement processes described by pure operations. The quantum information is given by coherent information in the state change of the measured physical system, and the classical information represented by the Shannon mutual information is obtained from the measurement outcomes. It is shown that if the maximum classical information is obtained, the quantum information becomes zero and if any classical information is not obtained, all the quantum information that the physical system has can be transmitted.
Around the turn of the century, Shor formulated his well-known and still-open conjecture stating that the accessible information of any quantum dichotomy, that is the maximum amount of classical information that can be decoded from a binary quantum encoding, is attained by a von Neumann measurement. A quarter of a century later, new developments on the Lorenz curves of quantum dichotomies in the field of quantum majorization and statistical comparison may provide the key to unlock such a longstanding open problem. Here, we first investigate the tradeoff relations between accessible information and guessing probability in the binary case, thus disproving the claimed monotonicity of the former quantity in the latter that, if true, would have settled Shor's problem in the qubit case. Our second result is to provide a state-dependent generalization of extremality for quantum measurements, to characterize state-dependent extremality for qubit dichotomies, and to apply such results to tighten previous results on the accessible information of qubit dichotomies.
The accessible information acc(E) of an ensemble E is the maximum mutual information between a random variable encoded into quantum states, and the probabilistic outcome of a quantum measurement of the encoding. Accessible information is extremely difficult to characterize analytically; even bounds on it are hard to place. The celebrated Holevo bound states that accessible information cannot exceed chi(E), the quantum mutual information between the random variable and its encoding. However, for general ensembles, the gap between the acc(E) and chi(E) may be arbitrarily large. We consider the special case of a binary random variable, which often serves as a stepping stone towards other results in information theory and communication complexity. We give explicit lower bounds on the the accessible information acc(E) of an ensemble E = {(p, rho_0), (1-p, rho_1)}, with 0 <= p <= 1, as functions of p and chi(E). The bounds are incomparable in the sense that they surpass each other in different parameter regimes. Our bounds arise by measuring the ensemble according to a complete orthogonal measurement that preserves the fidelity of the states rho_0,rho_1. As an intermediate step, therefore, we give new relations between the two quantities acc(E), chi(E) and the fidelity B(rho_0,rho_1).
Quantum information processing and computation requires high accuracy qubit configuration readout. In many practical schemes, the initial qubit configuration has to be inferred from readout that is a time-dependent weak measurement record. However, a combination of the measurement scheme and intrinsic dynamics can end up scrambling the initial state and lose information irretrievably. Here, we study the information physics of quantum trajectories based on weak measurements in order to address the optimal achievable performance in qubit configuration readout for two realistic models of single qubit readout: (i) Model I is informationally complete, but without intrinsic dynamics; (ii) Model II is informationally incomplete weak measurements with intrinsic dynamics. We first use mutual information to characterize how much intrinsic information about the initial state is encoded in the measurement record. Using a fixed discrete time-step formulation, we compute the mutual information while varying the measurement strength, duration of measurement record, and the relative strength of intrinsic dynamics in our measurement schemes. We also exploit the emergence of continuum scaling and the Stochastic Master Equation in the weak measurement limit. We develop an asymptotic expansion in the measurement efficiency parameter to calculate mutual information, which captures qualitative and quantitative features of the numerical data. The bounds on information extraction are manifested as plateaux in mutual information, our analysis obtains these bounds and also optimal duration of measurement required to saturate them. Our results should be useful both for quantum device operation and optimization and also, possibly, for improving the performance of recent machine learning approaches for qubit and multiqubit configuration readout in current Noisy Intermediate-Scale Quantum (NISQ) experiment regimes.