We derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus. It is also proven that this bound can be achieved in any statistical model in the qubit system. Moreover, we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models. That is, in such a statistical model, the quantum correlation can improve the accuracy of the estimation in an asymptotic setting.We derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus. It is also proven that this bound can be achieved in any statistical model in the qubit system. Moreover, we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models. That is, in such a statistical model, the quantum correlation can improve the accuracy of the estimation in an asymptotic setting.

Journal of Mathematical PhysicsOct 1, 2008101 citations

A Theory of Quantum (Statistical) Measurement

Walter Felipe Wreszinski

arXiv:2205.12906Journal of Statistical PhysicsFeb 9, 20231 citation

Statistical Inference and Quantum Measurement

Masanao Ozawa

Annals of the Japan Association for Philosophy of ScienceJan 1, 19891 citation

On the Role of Entanglement and Statistics in Learning

Srinivasan Arunachalam, Vojtech Havlicek, Louis Schatzki

In this work we make progress in understanding the relationship between learning models with access to entangled, separable and statistical measurements in the quantum statistical query (QSQ) model. To this end, we show the following results. $\textbf{Entangled versus separable measurements.}$ The goal here is to learn an unknown $f$ from the concept class $C\subseteq \{f:\{0,1\}^n\rightarrow [k]\}$ given copies of $\frac{1}{\sqrt{2^n}}\sum_x \vert x,f(x)\rangle$ . We show that, if $T$ copies suffice to learn $f$ using entangled measurements, then $O(nT^2)$ copies suffice to learn $f$ using just separable measurements. $\textbf{Entangled versus statistical measurements}$ The goal here is to learn a function $f \in C$ given access to separable measurements and statistical measurements. We exhibit a class $C$ that gives an exponential separation between QSQ learning and quantum learning with entangled measurements (even in the presence of noise). This proves the "quantum analogue" of the seminal result of Blum et al. [BKW'03]. that separates classical SQ and PAC learning with classification noise. $\textbf{QSQ lower bounds for learning states.}$ We introduce a quantum statistical query dimension (QSD), which we use to give lower bounds on the QSQ learning. With this we prove superpolynomial QSQ lower bounds for testing purity, shadow tomography, Abelian hidden subgroup problem, degree- $2$ functions, planted bi-clique states and output states of Clifford circuits of depth $\textsf{polylog}(n)$ . $\textbf{Further applications.}$ We give and $\textit{unconditional}$ separation between weak and strong error mitigation and prove lower bounds for learning distributions in the QSQ model. Prior works by Quek et al. [QFK+'22], Hinsche et al. [HIN+'22], and Nietner et al. [NIS+'23] proved the analogous results $\textit{assuming}$ diagonal measurements and our work removes this assumption.In this work we make progress in understanding the relationship between learning models with access to entangled, separable and statistical measurements in the quantum statistical query (QSQ) model. To this end, we show the following results. $\textbf{Entangled versus separable measurements.}$ The goal here is to learn an unknown $f$ from the concept class $C\subseteq \{f:\{0,1\}^n\rightarrow [k]\}$ given copies of $\frac{1}{\sqrt{2^n}}\sum_x \vert x,f(x)\rangle$. We show that, if $T$ copies suffice to learn $f$ using entangled measurements, then $O(nT^2)$ copies suffice to learn $f$ using just separable measurements. $\textbf{Entangled versus statistical measurements}$ The goal here is to learn a function $f \in C$ given access to separable measurements and statistical measurements. We exhibit a class $C$ that gives an exponential separation between QSQ learning and quantum learning with entangled measurements (even in the presence of noise). This proves the "quantum analogue" of the seminal result of Blum et al. [BKW'03]. that separates classical SQ and PAC learning with classification noise. $\textbf{QSQ lower bounds for learning states.}$ We introduce a quantum statistical query dimension (QSD), which we use to give lower bounds on the QSQ learning. With this we prove superpolynomial QSQ lower bounds for testing purity, shadow tomography, Abelian hidden subgroup problem, degree-$2$ functions, planted bi-clique states and output states of Clifford circuits of depth $\textsf{polylog}(n)$. $\textbf{Further applications.}$ We give and $\textit{unconditional}$ separation between weak and strong error mitigation and prove lower bounds for learning distributions in the QSQ model. Prior works by Quek et al. [QFK+'22], Hinsche et al. [HIN+'22], and Nietner et al. [NIS+'23] proved the analogous results $\textit{assuming}$ diagonal measurements and our work removes this assumption.

arXiv:2306.03161neural information processing systemsJan 1, 20235 citations

Quantum statistical complexity measure

Andre Tanus Cesario de Souza

Statistical complexity measures are distinctive tools that can detect significant physical phenomena. Many of these phenomena occur in such a way that they leave signatures that can be interpreted as changes in the state of the system, which can be read as statistical complexity through these measures, measured on a scale of order and disorder. We investigate the classical statistical complexity measure (CSCM) defined by López-Ruiz et al. and apply it to a simple model of discrete time evolution, characterized by finite Markov chains in dimension 2, for stochastic and bistochastic evolutions. Additionally, we introduce a quantum version for measuring statistical complexity: the quantum statistical complexity measure (QSCM), in the context of quantum information theory, and use it as a signaling function for quantum correlation transitions, measured on a scale of order and disorder. We consider the possibility that such transitions characterize intriguing physical phenomena, such as quantum phase transitions or sudden variations in correlation distributions. Our measure is applied to two analytically solvable Hamiltonian models: the one-dimensional quantum Ising model in the reduced single-particle state, and the Heisenberg chain XXZ of spin-1/2 in the state reduced by two particles. We analyze the measurement behavior in quantum phase transitions for finite system sizes and in the thermodynamic limit using the Bethe ansatz technique. We conclude that the quantum statistical complexity measure can serve as a signaling function of correlation transitions and distinguish quantum phase transitions, due to the abruptness that physical quantities experience at these transition points and the way in which the reduced states of these critical systems change within a scale of order and disorder.Statistical complexity measures are distinctive tools that can detect significant physical phenomena. Many of these phenomena occur in such a way that they leave signatures that can be interpreted as changes in the state of the system, which can be read as statistical complexity through these measures, measured on a scale of order and disorder. We investigate the classical statistical complexity measure (CSCM) defined by López-Ruiz et al. and apply it to a simple model of discrete time evolution, characterized by finite Markov chains in dimension 2, for stochastic and bistochastic evolutions. Additionally, we introduce a quantum version for measuring statistical complexity: the quantum statistical complexity measure (QSCM), in the context of quantum information theory, and use it as a signaling function for quantum correlation transitions, measured on a scale of order and disorder. We consider the possibility that such transitions characterize intriguing physical phenomena, such as quantum phase transitions or sudden variations in correlation distributions. Our measure is applied to two analytically solvable Hamiltonian models: the one-dimensional quantum Ising model in the reduced single-particle state, and the Heisenberg chain XXZ of spin-1/2 in the state reduced by two particles. We analyze the measurement behavior in quantum phase transitions for finite system sizes and in the thermodynamic limit using the Bethe ansatz technique. We conclude that the quantum statistical complexity measure can serve as a signaling function of correlation transitions and distinguish quantum phase transitions, due to the abruptness that physical quantities experience at these transition points and the way in which the reduced states of these critical systems change within a scale of order and disorder.

Dec 9, 20220 citations

On the Quantum Statistical Theory of the Measuring Process

Wolfgang Weidlich, Fritz Haake

Jan 1, 19690 citations

\textbf{Entangled versus separable measurements.}

The goal here is to learn an unknown

f

from the concept class

C\subseteq \{f:\{0,1\}^n\rightarrow [k]\}

given copies of

\frac{1}{\sqrt{2^n}}\sum_x \vert x,f(x)\rangle

. We show that, if

T

copies suffice to learn

f

using entangled measurements, then

O(nT^2)

copies suffice to learn

f

using just separable measurements.

\textbf{Entangled versus statistical measurements}

The goal here is to learn a function

f \in C

given access to separable measurements and statistical measurements. We exhibit a class

C

that gives an exponential separation between QSQ learning and quantum learning with entangled measurements (even in the presence of noise). This proves the "quantum analogue" of the seminal result of Blum et al. [BKW'03]. that separates classical SQ and PAC learning with classification noise.

\textbf{QSQ lower bounds for learning states.}

We introduce a quantum statistical query dimension (QSD), which we use to give lower bounds on the QSQ learning. With this we prove superpolynomial QSQ lower bounds for testing purity, shadow tomography, Abelian hidden subgroup problem, degree-

2

functions, planted bi-clique states and output states of Clifford circuits of depth

\textsf{polylog}(n)

\textbf{Further applications.}

We give and

\textit{unconditional}

separation between weak and strong error mitigation and prove lower bounds for learning distributions in the QSQ model. Prior works by Quek et al. [QFK+'22], Hinsche et al. [HIN+'22], and Nietner et al. [NIS+'23] proved the analogous results

\textit{assuming}

diagonal measurements and our work removes this assumption.