© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2303.04949 · International Journal of Quantum Information

Pretty good measurement for bosonic Gaussian ensembles

Hemant K. Mishra, Ludovico Lami, Prabha Mandayam, Mark M. Wilde

Published Apr 30, 2024• 5 citations

Abstract

The pretty good measurement is a fundamental analytical tool in quantum information theory, giving a method for inferring the classical label that identifies a quantum state chosen probabilistically from an ensemble. Identifying and constructing the pretty good measurement for the class of bosonic Gaussian states is of immediate practical relevance in quantum information processing tasks. Holevo recently showed that the pretty good measurement for a bosonic Gaussian ensemble is a bosonic Gaussian measurement that attains the accessible information of the ensemble [IEEE Trans. Inf. Theory 66(9) (2020) 5634]. In this paper, we provide an alternate proof of Gaussianity of the pretty good measurement for a Gaussian ensemble of multimode bosonic states, with a focus on establishing an explicit and efficiently computable Gaussian description of the measurement. We also compute an explicit form of the mean square error of the pretty good measurement, which is relevant when using it for parameter estimation. Generalizing the pretty good measurement is a quantum instrument, called the pretty good instrument. We prove that the post-measurement state of the pretty good instrument is a faithful Gaussian state if the input state is a faithful Gaussian state whose covariance matrix satisfies a certain condition. Combined with our previous finding for the pretty good measurement and provided that the same condition holds, it follows that the expected output state is a faithful Gaussian state as well. In this case, we compute an explicit Gaussian description of the post-measurement and expected output states. Our findings imply that the pretty good instrument for bosonic Gaussian ensembles is no longer merely an analytical tool, but that it can also be implemented experimentally in quantum optics laboratories.

DOI: 10.1142/s0219749924400100

Related papers

Pretty Good Measures in Quantum Information Theory

Raban Iten, Joseph M. Renes, David Sutter

Quantum generalizations of Rényi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Rényi divergence D̅αand the minimal quantum Rényi divergence D̃α. In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely, αD̅α(Q∥σ) ≤ D̃α(Q∥σ) for α ∈[0,1] and where Q and σ are density operators. This bound suggests defining a ”pretty good fidelity,” whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction. We also find a new necessary and sufficient condition for optimality of the pretty good measurement and singlet fraction.

arXiv:1608.08229•IEEE Transactions on Information Theory•Dec 14, 2016•25 citations

Accessible information of a general quantum Gaussian ensemble

Alexander Semenovich Holevo

Accessible information, which is a basic quantity in quantum information theory, is computed for a general quantum Gaussian ensemble under certain “threshold conditions.” It is shown that the maximizing measurement is Gaussian, constituting a far-reaching generalization of optical heterodyning. This substantially extends the previous result concerning the gauge-invariant case even for a single bosonic mode. A simple sufficient condition is provided that implies the threshold condition for a general Gaussian ensemble. The tightness of the sufficient condition is demonstrated in the single-mode case.

arXiv:2102.01981•Journal of Mathematical Physics•Sep 1, 2021•9 citations

Near-optimal performance of square-root measurement for general score functions and quantum ensembles

Hemant K. Mishra, Ludovico Lami, Mark M. Wilde

The Barnum-Knill theorem states that the optimal success probability in the multiple state discrimination task is not more than the square root of the success probability when the pretty good or square-root measurement is used for this task. An assumption of the theorem is that the underlying ensemble consists of finitely many quantum states over a finite-dimensional quantum system. Motivated in part by the fact that the success probability is not a relevant metric for continuous ensembles, in this paper we provide a generalization of the notion of pretty good measurement and the Barnum-Knill theorem for general quantum ensembles, including those described by a continuous parameter space and an infinite-dimensional Hilbert space. To achieve this, we also design a general metric of performance for quantum measurements that generalizes the success probability, namely, the expected gain of the measurement with respect to a positive score function. A notable consequence of the main result is that, in a Bayesian estimation task, the mean square error of the generalized pretty good measurement does not exceed twice the optimal mean square error.

arXiv:2505.20447•May 26, 2025•0 citations

Error rates of Belavkin weighted quantum measurements and a converse to Holevo’s asymptotic optimality theorem

Jon Tyson

We compare several instances of pure-state Belavkin weighted square-root measurements from the standpoint of minimum-error discrimination of quantum states. The quadratically weighted measurement is proven superior to the so-called ``pretty good measurement'' (PGM) in a number of respects: (1) Holevo's quadratic weighting unconditionally outperforms the PGM in the case of two-state ensembles, with equality only in trivial cases. (2) A converse of a theorem of Holevo is proven, showing that a weighted measurement is asymptotically optimal only if it is quadratically weighted. Counterexamples for three states are constructed. The cube-weighted measurement of Ballester, Wehner, and Winter is also considered. Sufficient optimality conditions for various weights are compared.

arXiv:0907.1884•Physical Review A•Mar 31, 2009•20 citations

Finding Gaussian Structure in Bosonic States

Alvan Arulandu, Sitan Chen, Ziyun Chen +2 more

We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary $n$-mode bosonic state $ρ$, the goal is to output a pure Gaussian state whose infidelity with $ρ$ is at most $\mathrm{opt} + ε$, where $\mathrm{opt}$ is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When $\mathrm{opt}$ is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in $n, 1/ε$ and $\log \log E$, where $E$ is the energy of the closest pure Gaussian state. For arbitrary $\mathrm{opt}$, our protocol uses $(n+1)^{\mathrm{poly}(1/ε)} \mathrm{poly}\left(1+\log\log(E)\right)$ copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether $\mathrm{opt} > c + ε$ or $\mathrm{opt} < c - ε$, for any threshold $c\in(0,1)$. We also prove $\mathrm{poly}(n,1/ε)$ runtime is impossible, unless $\mathrm{NP}\subseteq\mathrm{BQP}$. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.

quant-pharXiv:2610.06810•Oct 6, 2026•0 citations

Pretty good measures in quantum information theory

Raban Iten, Joseph M. Renes, David Sutter

Quantum generalizations of Rényi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Rényi divergence D̅αand the minimal quantum Rényi divergence D̅α. In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely that αD̅α(ρ∥σ) ≤ D̅α(ρ∥σ) for α ϵ [0, 1] and where ρ and σ are density operators. This bound suggests defining a “pretty good fidelity”, whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2017•1 citation

The pretty good measurement is a fundamental analytical tool in quantum information theory, giving a method for inferring the classical label that identifies a quantum state chosen probabilistically from an ensemble. Identifying and constructing the pretty good measurement for the class of bosonic Gaussian states is of immediate practical relevance in quantum information processing tasks. Holevo recently showed that the pretty good measurement for a bosonic Gaussian ensemble is a bosonic Gaussian measurement that attains the accessible information of the ensemble [IEEE Trans. Inf. Theory 66(9) (2020) 5634]. In this paper, we provide an alternate proof of Gaussianity of the pretty good measurement for a Gaussian ensemble of multimode bosonic states, with a focus on establishing an explicit and efficiently computable Gaussian description of the measurement. We also compute an explicit form of the mean square error of the pretty good measurement, which is relevant when using it for parameter estimation. Generalizing the pretty good measurement is a quantum instrument, called the pretty good instrument. We prove that the post-measurement state of the pretty good instrument is a faithful Gaussian state if the input state is a faithful Gaussian state whose covariance matrix satisfies a certain condition. Combined with our previous finding for the pretty good measurement and provided that the same condition holds, it follows that the expected output state is a faithful Gaussian state as well. In this case, we compute an explicit Gaussian description of the post-measurement and expected output states. Our findings imply that the pretty good instrument for bosonic Gaussian ensembles is no longer merely an analytical tool, but that it can also be implemented experimentally in quantum optics laboratories.

Quantum generalizations of Rényi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Rényi divergence D̅αand the minimal quantum Rényi divergence D̃α. In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely, αD̅α(Q∥σ) ≤ D̃α(Q∥σ) for α ∈[0,1] and where Q and σ are density operators. This bound suggests defining a ”pretty good fidelity,” whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction. We also find a new necessary and sufficient condition for optimality of the pretty good measurement and singlet fraction.

Accessible information, which is a basic quantity in quantum information theory, is computed for a general quantum Gaussian ensemble under certain “threshold conditions.” It is shown that the maximizing measurement is Gaussian, constituting a far-reaching generalization of optical heterodyning. This substantially extends the previous result concerning the gauge-invariant case even for a single bosonic mode. A simple sufficient condition is provided that implies the threshold condition for a general Gaussian ensemble. The tightness of the sufficient condition is demonstrated in the single-mode case.

The Barnum-Knill theorem states that the optimal success probability in the multiple state discrimination task is not more than the square root of the success probability when the pretty good or square-root measurement is used for this task. An assumption of the theorem is that the underlying ensemble consists of finitely many quantum states over a finite-dimensional quantum system. Motivated in part by the fact that the success probability is not a relevant metric for continuous ensembles, in this paper we provide a generalization of the notion of pretty good measurement and the Barnum-Knill theorem for general quantum ensembles, including those described by a continuous parameter space and an infinite-dimensional Hilbert space. To achieve this, we also design a general metric of performance for quantum measurements that generalizes the success probability, namely, the expected gain of the measurement with respect to a positive score function. A notable consequence of the main result is that, in a Bayesian estimation task, the mean square error of the generalized pretty good measurement does not exceed twice the optimal mean square error.

We compare several instances of pure-state Belavkin weighted square-root measurements from the standpoint of minimum-error discrimination of quantum states. The quadratically weighted measurement is proven superior to the so-called ``pretty good measurement'' (PGM) in a number of respects: (1) Holevo's quadratic weighting unconditionally outperforms the PGM in the case of two-state ensembles, with equality only in trivial cases. (2) A converse of a theorem of Holevo is proven, showing that a weighted measurement is asymptotically optimal only if it is quadratically weighted. Counterexamples for three states are constructed. The cube-weighted measurement of Ballester, Wehner, and Winter is also considered. Sufficient optimality conditions for various weights are compared.

We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary

n

-mode bosonic state

ρ

, the goal is to output a pure Gaussian state whose infidelity with

ρ

is at most

\mathrm{opt} + ε

, where

\mathrm{opt}

is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When

\mathrm{opt}

is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in

n, 1/ε

and

\log \log E

, where

E

is the energy of the closest pure Gaussian state. For arbitrary

\mathrm{opt}

, our protocol uses

(n+1)^{\mathrm{poly}(1/ε)} \mathrm{poly}\left(1+\log\log(E)\right)

copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether

\mathrm{opt} > c + ε

or

\mathrm{opt} < c - ε

, for any threshold

c\in(0,1)

. We also prove

\mathrm{poly}(n,1/ε)

runtime is impossible, unless

\mathrm{NP}\subseteq\mathrm{BQP}

. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.

Quantum generalizations of Rényi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Rényi divergence D̅αand the minimal quantum Rényi divergence D̅α. In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely that αD̅α(ρ∥σ) ≤ D̅α(ρ∥σ) for α ϵ [0, 1] and where ρ and σ are density operators. This bound suggests defining a “pretty good fidelity”, whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction.