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Physical Review A

Unbounded and lossless compression of multiparameter quantum information

Joe H. Jenne, David R. M. Arvidsson-Shukur

Published Oct 4, 2022• 18 citations
Publisher

Abstract

Several tasks in quantum-information processing involve quantum learning. For example, quantum sensing, quantum machine learning, and quantum-computer calibration involve learning and estimating unknown parameters $\mathbit{\ensuremath{\theta}}=({\ensuremath{\theta}}_{1},\ensuremath{\cdots},{\ensuremath{\theta}}_{M})$ from measurements of many copies of a quantum state ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}$. This type of metrological information is described by the quantum Fisher information matrix, which bounds the average amount of information learned about $\mathbit{\ensuremath{\theta}}$ per measurement of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}$. In several scenarios, it is advantageous to compress the multiparameter information encoded in ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\bigotimes}n}$ into ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}^{\text{ps}}\phantom{\rule{0.16em}{0ex}}{}^{\ensuremath{\bigotimes}m}$, where $m\ensuremath{\ll}n$. Here, we present a ``go-go'' theorem proving that $m/n$ can be made arbitrarily small, and that the information compression can happen without loss of information. We also demonstrate how to construct filters that perform this unbounded and lossless information compression. These filters can, for example, reduce arbitrarily the quantum-state intensity on experimental detectors, while retaining all initial information. Finally, we prove that the ability to compress quantum Fisher information is a nonclassical advantage that stems from the negativity of a particular quasiprobability distribution, a quantum extension of a probability distribution.

DOI
10.1103/physreva.106.042404

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Several tasks in quantum-information processing involve quantum learning. For example, quantum sensing, quantum machine learning, and quantum-computer calibration involve learning and estimating unknown parameters \mathbit\ensuremathθ=(\ensuremathθ1,\ensuremath⋯,\ensuremathθM)\mathbit{\ensuremath{\theta}}=({\ensuremath{\theta}}_{1},\ensuremath{\cdots},{\ensuremath{\theta}}_{M})\mathbit\ensuremathθ=(\ensuremathθ1​,\ensuremath⋯,\ensuremathθM​) from measurements of many copies of a quantum state \ensuremathρ\ifmmode^\elseˆ\fi\mathbit\ensuremathθ{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}\ensuremathρ\ifmmode^\elseˆ\fi​\mathbit\ensuremathθ​. This type of metrological information is described by the quantum Fisher information matrix, which bounds the average amount of information learned about \mathbit\ensuremathθ\mathbit{\ensuremath{\theta}}\mathbit\ensuremathθ per measurement of \ensuremathρ\ifmmode^\elseˆ\fi\mathbit\ensuremathθ{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}\ensuremathρ\ifmmode^\elseˆ\fi​\mathbit\ensuremathθ​. In several scenarios, it is advantageous to compress the multiparameter information encoded in \ensuremathρ\ifmmode^\elseˆ\fi\mathbit\ensuremathθ\ensuremath⨂n{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\bigotimes}n}\ensuremathρ\ifmmode^\elseˆ\fi​\mathbit\ensuremathθ​\ensuremath⨂n into \ensuremathρ\ifmmode^\elseˆ\fi\mathbit\ensuremathθps\ensuremath⨂m{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}^{\text{ps}}\phantom{\rule{0.16em}{0ex}}{}^{\ensuremath{\bigotimes}m}\ensuremathρ\ifmmode^\elseˆ\fi​\mathbit\ensuremathθps​\ensuremath⨂m, where m\ensuremath≪nm\ensuremath{\ll}nm\ensuremath≪n. Here, we present a ``go-go'' theorem proving that m/nm/nm/n can be made arbitrarily small, and that the information compression can happen without loss of information. We also demonstrate how to construct filters that perform this unbounded and lossless information compression. These filters can, for example, reduce arbitrarily the quantum-state intensity on experimental detectors, while retaining all initial information. Finally, we prove that the ability to compress quantum Fisher information is a nonclassical advantage that stems from the negativity of a particular quasiprobability distribution, a quantum extension of a probability distribution.
Quantum learning (in metrology and machine learning) involves estimating unknown parameters from measurements of quantum states. The quantum Fisher information matrix can bound the average amount of information learnt about the unknown parameters per experimental trial. In several scenarios, it is advantageous to concentrate information in as few states as possible. Here, we present two "go-go" theorems proving that negativity, a narrower nonclassicality concept than noncommutation, enables unbounded and lossless distillation of Fisher information about multiple parameters in quantum learning.
Suppose that a finite dimensional quantum source is known to have von Neumann entropy less than or equal to SSS but is otherwise completely unspecified. We describe a method of universal quantum data compression which will faithfully compress the quantum information of any such source to SSS qubits per signal (in the limit of large block lengths).
In every parameter-estimation experiment, the final measurement or the postprocessing incurs a cost. Postselection can improve the rate of Fisher information (the average information learned about an unknown parameter from a trial) to cost. We show that this improvement stems from the negativity of a particular quasiprobability distribution, a quantum extension of a probability distribution. In a classical theory, in which all observables commute, our quasiprobability distribution is real and nonnegative. In a quantum-mechanically noncommuting theory, nonclassicality manifests in negative or nonreal quasiprobabilities. Negative quasiprobabilities enable postselected experiments to outperform optimal postselection-free experiments: postselected quantum experiments can yield anomalously large information-cost rates. This advantage, we prove, is unrealizable in any classically commuting theory. Finally, we construct a preparation-and-postselection procedure that yields an arbitrarily large Fisher information. Our results establish the nonclassicality of a metrological advantage, leveraging our quasiprobability distribution as a mathematical tool.
Multiparameter quantum estimation faces two fundamental obstacles: sloppiness, i.e., anisotropy of the quantum Fisher information matrix (QFIM) that renders some parameter directions insensitive, and incompatibility, the non-commutativity of optimal measurements for different parameters. The trade-off bound CTC_TCT​ captures their joint impact on precision, but it has remained unclear how the distribution of incompatibility across parameter planes affects its overall cost. Here we separate the total amount of incompatibility from its location. We introduce a dimensionless quantity Gn(F)G_n^{(F)}Gn(F)​ that measures the alignment between the incompatibility distribution and the eigenvalues of the QFIM, and show how the Frobenius scale of the incompatibility contribution factorizes. We obtain a bound and prove the incompatibility cost lies between this bound and a rank-dependent multiple thereof. We also prove that at fixed sloppiness, or equivalently fixed Fisher volume, concentrating incompatibility into a single parameter plane reduces the optimized trade-off cost because the Fisher geometry can then be reshaped to allocate more Fisher area to that plane. A qutrit SU(2)SU(2)SU(2) encoding numerically confirms that states with larger incompatibility strength can nevertheless incur a smaller cost if the matching factor GGG is sufficiently small. Our results establish that the distribution of incompatibility relative to the Fisher eigenbasis is a central diagnostic for multiparameter estimation, beyond the total incompatibility strength.