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arXiv:2301.10982 · Physical Review Letters

Inferring Physical Properties of Symmetric States from the Fewest Copies

Da-Jian Zhang, D. M. Tong

Published Jul 25, 2024• 14 citations
arXivHTMLPDF

Abstract

Learning physical properties of high-dimensional states is crucial for developing quantum technologies but usually consumes an exceedingly large number of samples which are difficult to afford in practice. In this Letter, we use the methodology of quantum metrology to tackle this difficulty, proposing a strategy built upon entangled measurements for dramatically reducing sample complexity. The strategy, whose characteristic feature is symmetrization of observables, is powered by the exploration of symmetric structures of states which are ubiquitous in physics. It is provably optimal under some natural assumption, efficiently implementable in a variety of contexts, and capable of being incorporated into existing methods as a basic building block. We apply the strategy to different scenarios motivated by experiments, demonstrating exponential reductions in sample complexity.

DOI
10.1103/physrevlett.133.040202

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Learning physical properties of high-dimensional states is crucial for developing quantum technologies but usually consumes an exceedingly large number of samples which are difficult to afford in practice. In this Letter, we use the methodology of quantum metrology to tackle this difficulty, proposing a strategy built upon entangled measurements for dramatically reducing sample complexity. The strategy, whose characteristic feature is symmetrization of observables, is powered by the exploration of symmetric structures of states which are ubiquitous in physics. It is provably optimal under some natural assumption, efficiently implementable in a variety of contexts, and capable of being incorporated into existing methods as a basic building block. We apply the strategy to different scenarios motivated by experiments, demonstrating exponential reductions in sample complexity.
Estimating the properties of quantum states is a fundamental task in both theoretical and experimental quantum physics. While numerous Pauli-based measurement strategies have been proposed, none explicitly exploit prior knowledge of structural features such as symmetry. Here, we introduce a compact measurement scheme that exploits symmetry to speed up quantum measurements. By incorporating symmetry into the measurement design, our approach can rigorously reduce sample complexity for symmetric states and suitable observables. We provide a theoretical analysis showing that, for specific observables, the variance of our estimator admits polynomial or even exponential improvements in the required number of state copies. To demonstrate practicality, we implement the scheme on photonic quantum platforms, preparing high-fidelity Greenberger-Horne-Zeilinger (GHZ) and W states. The experimental results confirm substantial gains in estimation accuracy for both linear and nonlinear properties. Our work suggests that symmetry and other structural properties of quantum states can be systematically exploited as algorithmic resources, opening new avenues for scalable characterization and verification of complex quantum systems on near-term and fault-tolerant devices.
Using quantum measurements to extract information from states is a matter of routine in quantum science and technologies. A recent work [Phys. Rev. Lett. 133, 040202 (2024)] reported the finding that the symmetric structures of a state can be harnessed to dramatically reduce the sample complexity in extracting information from the state. However, due to the presence of noise, the actual state at hand is often corrupted, making its symmetric structures distorted before the execution of quantum measurements. Here, using the methodology of quantum metrology, we identify the optimal measurement that can retrieve maximum information of a symmetric state from its corrupted copies. We show that this measurement can be found by solving a semidefinite program in generic cases and can be explicitly determined for a large class of noise models covariant under the symmetry group in question. The results of this study nicely complement the recent work by providing a method to optimally utilize the distorted symmetric structures of corrupted states for information retrieval.
We study how entangled measurements, reflected states, and input-state classicality determine the sample complexity of learning bosonic continuous-variable quantum states through their characteristic function. We first analyze a simple entanglement-free protocol and show that its sample complexity is exponential in the number of modes n for general inputs. We then prove that this exponential scaling is intrinsic through three complementary lower bounds: one excluding entangled measurements, one for reflection-symmetric inputs, and one excluding reflected states even when entangled measurements on finitely many copies are allowed. Next, we show that the classicality parameter s max quantitatively determines when this barrier can be softened: positive classicality can reduce the entanglement-free sample cost, and for fully classical states heterodyne learning matches the ϵ -scaling of a Bell-measurement (BM) scheme. Finally, combining reflected states with a two-copy entangled measurement yields sample complexity independent of n . These results identify when entanglement-free strategies already suffice and when, within the resource models analyzed here, efficient continuous-variable state learning requires the full BM resource combination of entangled measurements and reflected-state access.
Recently, a technique known as quantum symmetry test has gained increasing attention for detecting bipartite entanglement in pure quantum states. In this work we show that, beyond qualitative detection, a family of well-defined measures of bipartite and multipartite entanglement can be obtained with symmetry tests. We propose and benchmark several efficient methods to estimate these measures, and derive near-optimal sampling strategies for each. Despite the nonlinearity of the methods, we demonstrate that the sampling error scales no worse than O(Ntot−1/2)O(N_{\mathrm{tot}}^{-1/2})O(Ntot−1/2​) with the total number of copies NtotN_{\mathrm{tot}}Ntot​, which suggests experimental feasibility. By exploiting symmetries we compute our measures for large number of copies, and derive the asymptotic decay exponents for relevant states in many-body systems. Using these results we identify tradeoffs between estimation complexity and sensitivity of the presented entanglement measures, oriented to practical implementations.
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We describe a procedure that simultaneously reduces the dimensions of many, potentially distinct quantum states while preserving properties invariant under the tensor power action of an isometry. This serves as a black-box method to replace the dimension with the maximum rank in the sample complexity of learning symmetric properties, even those depending on multiple input states. Combined with full state tomography, random dimension reduction yields improved upper bounds for estimating distances, fidelities, and relative entropies between pairs of states. Expressing the action of this procedure through the Choi--Jamiołkowski isomorphism reveals an intimate connection with the recently introduced random purification channel by Tang, Wright, and Zhandry. With this new perspective, we obtain a simple description of the rank-dependent version of the random purification channel and give an end-to-end analysis of sample-optimal tomography without requiring a reference to the Schur transform or Schur polynomials. Finally, we prove that there does not exist a random purification channel that simultaneously purifies copies of multiple, potentially different input states. Hence, the procedure is related to, but distinct from, random purification.