Distinguishability of generic quantum states

We address the problem of extracting information from a single quantum system whose state is known to be in one of several possible states. In the generic case, it is notoriously difficult to find the optimal measurement, that is the measurement that provides the most possible information about the system's state. We consider a simple general prescription for a measurement that is typically not optimal but appears to be quite good. It seems to be particularly good when the states to be distinguished are equally likely and almost orthogonal.We address the problem of extracting information from a single quantum system whose state is known to be in one of several possible states. In the generic case, it is notoriously difficult to find the optimal measurement, that is the measurement that provides the most possible information about the system's state. We consider a simple general prescription for a measurement that is typically not optimal but appears to be quite good. It seems to be particularly good when the states to be distinguished are equally likely and almost orthogonal.

Journal of Modern OpticsDec 1, 1994324 citations

A Measure of the Distinguishability of Quantum States

William K. Wootters

Springer eBooksJan 1, 19833 citations

A unified formulation for entanglement of distinguishable and indistinguishable particles

Swapnamay Mondal

Entanglement is well understood concept for distinguishable particles. However fundamental particles are inherently indistinguishable and when they occupy orthogonal modes, i.e. have orthogonal wave functions, they are effectively distinguishable. Thus there should exist a formulation for entanglement which works in the generic case, i.e. even when wave functions overlap. To achieve this, we consider R\`{e}nyi entropies which are measurable quantities. Outcomes of such measurements for a generic case are conjectured in terms of R\`{e}nyi entropies of spatial regions in a field theory. To leading order naive quantum mechanical answers are reproduced. Corrections depend on overlap of wave functions and exhibit novel features. In a generic case the Von Neumann entropy computed from these R\`{e}nyi entropies is argued to be a measure of entanglement of pure states and a formula for this is conjectured in terms of Von Neumann entropies of spatial regions in field theory. It seems, due to the corrections it is possible for the entanglement to exceed maximal quantum mechanical value, e.g. $log 2$ for a single qubit.Entanglement is well understood concept for distinguishable particles. However fundamental particles are inherently indistinguishable and when they occupy orthogonal modes, i.e. have orthogonal wave functions, they are effectively distinguishable. Thus there should exist a formulation for entanglement which works in the generic case, i.e. even when wave functions overlap. To achieve this, we consider R\`{e}nyi entropies which are measurable quantities. Outcomes of such measurements for a generic case are conjectured in terms of R\`{e}nyi entropies of spatial regions in a field theory. To leading order naive quantum mechanical answers are reproduced. Corrections depend on overlap of wave functions and exhibit novel features. In a generic case the Von Neumann entropy computed from these R\`{e}nyi entropies is argued to be a measure of entanglement of pure states and a formula for this is conjectured in terms of Von Neumann entropies of spatial regions in field theory. It seems, due to the corrections it is possible for the entanglement to exceed maximal quantum mechanical value, e.g. $log 2$ for a single qubit.

Jan 8, 20150 citations

Distinguishability Transitions from Global Quantum Snapshots

Catherine McCarthy, Sarang Gopalakrishnan, Romain Vasseur

Quantum states generated by generic time evolution are locally featureless: local measurements return random outcomes that are identical for all states. However, global snapshots in the computational basis are surprisingly effective at distinguishing between quantum states. We explore the ability of Bayesian classifiers to discriminate between many random quantum states using a small number of measurement outcomes. We identify a phase transition in the ability of the classifier to distinguish between $k$ candidate states with $m$ shots controlled by the variable $x = m / \ln k$ , and derive the threshold value $x_c$ . We extend our results to low-depth circuits near the onset of anti-concentration, as well as to noisy circuits and phase-random states.Quantum states generated by generic time evolution are locally featureless: local measurements return random outcomes that are identical for all states. However, global snapshots in the computational basis are surprisingly effective at distinguishing between quantum states. We explore the ability of Bayesian classifiers to discriminate between many random quantum states using a small number of measurement outcomes. We identify a phase transition in the ability of the classifier to distinguish between $k$ candidate states with $m$ shots controlled by the variable $x = m / \ln k$, and derive the threshold value $x_c$. We extend our results to low-depth circuits near the onset of anti-concentration, as well as to noisy circuits and phase-random states.

cond-mat.stat-mechCross-listedarXiv:2609.09296Sep 10, 20260 citations

Distinguishability of generic quantum states

Zbigniew Puchała, Łukasz Pawela, Karol Życzkowski

Properties of random mixed states of dimension $N$ distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large $N$ , due to the concentration of measure, the trace distance between two random states tends to a fixed number $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}=1/4+1/\ensuremath{\pi}$, which yields the Helstrom bound on their distinguishability. To arrive at this result, we apply free random calculus and derive the symmetrized Marchenko-Pastur distribution, which is shown to describe numerical data for the model of coupled quantum kicked tops. Asymptotic value for the root fidelity between two random states, $\sqrt{F}=\frac{3}{4}$ , can serve as a universal reference value for further theoretical and experimental studies. Analogous results for quantum relative entropy and Chernoff quantity provide other bounds on the distinguishablity of both states in a multiple measurement setup due to the quantum Sanov theorem. We study also mean entropy of coherence of random pure and mixed states and entanglement of a generic mixed state of a bipartite system.Properties of random mixed states of dimension $N$ distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large $N$, due to the concentration of measure, the trace distance between two random states tends to a fixed number $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}=1/4+1/\ensuremath{\pi}$, which yields the Helstrom bound on their distinguishability. To arrive at this result, we apply free random calculus and derive the symmetrized Marchenko-Pastur distribution, which is shown to describe numerical data for the model of coupled quantum kicked tops. Asymptotic value for the root fidelity between two random states, $\sqrt{F}=\frac{3}{4}$, can serve as a universal reference value for further theoretical and experimental studies. Analogous results for quantum relative entropy and Chernoff quantity provide other bounds on the distinguishablity of both states in a multiple measurement setup due to the quantum Sanov theorem. We study also mean entropy of coherence of random pure and mixed states and entanglement of a generic mixed state of a bipartite system.

arXiv:1507.05123Physical Review AJun 9, 201648 citations

log 2

for a single qubit.

k

candidate states with

m

shots controlled by the variable

x = m / \ln k

, and derive the threshold value

x_c

. We extend our results to low-depth circuits near the onset of anti-concentration, as well as to noisy circuits and phase-random states.

Properties of random mixed states of dimension

N

distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large

N

, due to the concentration of measure, the trace distance between two random states tends to a fixed number $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}=1/4+1/\ensuremath{\pi}$, which yields the Helstrom bound on their distinguishability. To arrive at this result, we apply free random calculus and derive the symmetrized Marchenko-Pastur distribution, which is shown to describe numerical data for the model of coupled quantum kicked tops. Asymptotic value for the root fidelity between two random states,

\sqrt{F}=\frac{3}{4}

, can serve as a universal reference value for further theoretical and experimental studies. Analogous results for quantum relative entropy and Chernoff quantity provide other bounds on the distinguishablity of both states in a multiple measurement setup due to the quantum Sanov theorem. We study also mean entropy of coherence of random pure and mixed states and entanglement of a generic mixed state of a bipartite system.