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Experimental investigation of quantum uncertainty relations with classical shadows

Lu Liu, Ting Zhang, Xiao Yuan, He Lu

arXiv:2202.06441v1quant-ph

TL;DR

Experiments had not yet compared quantum uncertainty bounds across coherence measures beyond relative entropy, while tomography becomes costly as state dimension increases. This paper uses classical shadows to test bounds on optically prepared states, finding that their tightness depends on reference basis and purity.

  • Problem

    Experimental comparisons of quantum uncertainty relations across coherence measures beyond relative entropy were lacking, and tomography becomes costly as state dimension increases.

    Source: §I · p. 1, §III · p. 2

  • Method

    The experiment uses classical shadows to estimate purity and test coherence-based uncertainty bounds on optically prepared states.

    Source: §III · p. 2, §IV · p. 3

  • Results

    The tightness of relative-entropy coherence bounds depends on reference basis and state purity, while measured l1-norm and formation-coherence values are bounded by their measured lower bounds.

    Source: §VI · p. 4, §V · p. 4

  • Takeaways & Limitations

    The authors report that classical-shadow measurement cost is small and independent of state dimension, and that the method can be generalized to multipartite states while retaining efficiency.

    Source: §VI · pp. 4–5

Abstract

from arXiv · show

The quantum component in uncertainty relation can be naturally characterized by the quantum coherence of a quantum state, which is of paramount importance in quantum information science. Here, we experimentally investigate quantum uncertainty relations construed with relative entropy of coherence, $l_1$ norm of coherence and coherence of formation. In stead of quantum state tomographic technology, we employ the classical shadow algorithm for the detection of lower bounds in quantum uncertainty relations. With an all optical setup, we prepare a family of quantum states whose purity can be fully controlled. We experimentally explore the tightness of various lower bounds in different reference bases on the prepared states. Our results indicate the tightness of quantum coherence lower bounds dependents on the reference bases as well as the purity of quantum state.

I. INTRODUCTION

Quantum uncertainty relations describe limits on incompatible measurements, while coherence is used to characterize the quantum component of uncertainty. This paper experimentally tests coherence-based lower bounds using classical shadows and finds their tightness varies with reference basis and state purity.

I. INTRODUCTION

  • Entropic uncertainty relations bound the sum of measurement entropies using the maximal overlap between the two measurement bases.The introduction traces this relation to Deutsch, Kraus, and Maassen and Uiffink.Source: §I · p. 1
  • The same Z measurement entropy, H(Z) = 1, occurs for both |+⟩ and I/2, despite their different quantum states.The passage uses this example to distinguish measurement entropy from the quantum or classical origin of uncertainty.Source: §I · p. 1
  • Coherence-based uncertainty relations address a gap in experiments: comparisons across coherence measures beyond relative entropy and tests of theoretically constructed bounds were lacking.The passage also identifies quantum state tomography as costly as state dimension increases.Source: §I · p. 1
  • The experiment tests quantum uncertainty relations built from relative entropy, l1-norm, and formation coherence, using classical shadows to detect their lower bounds.The tested states are single-photon states, and the reported tightness depends on reference basis and purity.Source: §I · p. 1

II. QUANTUM UNCERTAINTY RELATIONS

The section introduces coherence measures and notes that different measures serve different roles in quantum information processing.

II. QUANTUM UNCERTAINTY RELATIONS

  • A functional qualifies as a coherence measure when it satisfies non-negativity, monotonicity, strong monotonicity, and convexity.The passage attributes these postulates to.Source: §II · p. 2
  • Relative entropy of coherence is linked to coherence distillation, coherence freezing, and the secret key rate in quantum key distribution.The passage cites,, and for these roles, respectively.Source: §II · p. 2
  • Coherence of formation represents coherence cost, while l1-norm coherence is linked to multi-slit interference and quantum algorithm performance.The passage describes coherence cost as the minimum rate of maximally coherent pure states consumed in preparation under incoherent and strictly incoherent operations.Source: §II · p. 2

A. QURs using relative entropy of coherence

This section describes relative-entropy coherence in a measurement basis and relates its lower-bound expressions to state purity.

A. QURs using relative entropy of coherence

  • In basis J, the passage identifies H(J) with the von Neumann entropy of the state’s diagonal part, ρd.It defines ρd as the diagonal part of ρ in measurement basis J.Source: A. QURs using relative entropy of… · p. 2
  • The passage presents the relative-entropy coherence lower bound and references entropic uncertainty relations by S´anches-Ruiz [39], Berta et al., and Korzekwa et al..It defines h(x) as binary entropy and P as state purity, but the supplied passage does not include the displayed bound itself.Source: A. QURs using relative entropy of… · p. 2
  • For a qubit with eigenvalues λ and 1 − λ, purity is P = 2λ^2 − 2λ + 1.The state’s von Neumann entropy is given as the binary entropy of λ.Source: A. QURs using relative entropy of… · p. 2

B. QUR of the l1 norm of coherence

This section introduces l1-norm coherence in a fixed measurement basis and the associated quantum uncertainty relation.

B. QUR of the l1 norm of coherence

  • The l1 norm of coherence is defined for a fixed measurement basis J.The supplied passage introduces the definition but does not include its formula.Source: B. QUR of the l1 norm of coherence · p. 2
  • The named quantity in this section is l1-norm coherence, with its definition tied to measurement basis J.This connects the measure introduced in the section to the fixed-basis qualification in its definition.Source: B. QUR of the l1 norm of coherence · p. 2
  • The section presents a quantum uncertainty relation using the l1 norm of coherence.The supplied passage does not include the lower-bound expression.Source: B. QUR of the l1 norm of coherence · p. 2

C. QUR using coherence of formation

Coherence of formation is defined for a fixed measurement basis through an optimization over state decompositions, and is used to construct a quantum uncertainty relation.

C. QUR using coherence of formation

III. CLASSICAL SHADOW

The section motivates estimating state purity without costly tomography and describes classical shadows as randomized single-qubit measurements whose estimators can be used to estimate purity.

III. CLASSICAL SHADOW

  • Purity is needed to test the quantum uncertainty relations, while reconstructing the state by tomography is costly as Hilbert-space dimension increases.A two-copy alternative uses P = tr[Πρ⊗ρ], with Π the local swap operator.Source: §III · p. 2
  • Classical shadows randomly measure a single-qubit state in one of the Pauli X, Y, or Z bases and construct an estimator from each outcome.The three bases are selected uniformly, and the estimator is 3|k⟩⟨k| − I.Source: §III · p. 2
  • Each shadow estimator requires one sample, and purity is estimated from pairs of independently constructed estimators.The passage specifies that two randomly selected estimators are used.Source: §III · p. 2

IV. EXPERIMENT REALIZATIONS

The experiment prepares a single-photon family ρ(τ) and uses optical elements to vary its purity. The setup then measures coherence-related uncertainty bounds and compares classical-shadow purity estimates with tomography.

IV. EXPERIMENT REALIZATIONS

  • The experimental apparatus uses optical components including a laser diode, single-photon detector, attenuator, filters, and polarization optics.Source: §IV · p. 3
  • The optical setup generates the state family ρ(τ) using a heralded photon prepared in |+⟩, a tunable path ratio, and a dephasing interferometer.The two beams are incoherently mixed to erase path information and produce ρ(τ).Source: §IV · p. 3, Figure 1 · p. 3
  • Figure 2 presents classical-shadow purity estimates across the 11 prepared states and compares them with tomography and ideal-state predictions.Source: Figure 2 · p. 3, §IV · p. 3
  • The parameter τ is varied from 0 to 1 in steps of 0.1, producing 11 states for the uncertainty-relation measurements.The setup measures lower bounds using classical-shadow purity and coherence measurements in selected bases.Source: §IV · p. 3

V. EXPERIMENTAL RESULTS

The experiments compare coherence-based uncertainty bounds across state purities and reference bases. Their relative tightness changes with both, while the measured coherence is bounded for the l1-norm and coherence-of-formation relations.

V. EXPERIMENTAL RESULTS

  • With 2,000 samples, |PQST − PCS| = 0.0036, and the classical-shadow purity estimates agree well with theoretical predictions across the 11 prepared states.The reported absolute difference is below 0.1 when Ns ≥ 600.Source: §V · pp. 3–4
  • For lower-purity states, Eq. 3 and Eq. 4 bounds become negative and are set to zero at sufficiently large c; at c = 0.5, Eq. 4 and Eq. 5 tie and outperform the other bounds for the pure state.For the pure state, Eq. 2 and Eq. 3 become stricter as c increases.Source: §V · p. 4, Figure 3 · p. 4
  • The relative-entropy bounds were tested with A fixed at J(0°) and B at J(90°), J(66.42°), or J(36.86°), corresponding to c = 0.5, 0.7, and 0.9.The relative entropy of coherence is calculated as H(J(θ)) − SVN(ρ(τ)), with SVN estimated from PCS.Source: §V · p. 4
  • For c = 0.5, RE(ρ) is lower bounded by 1 − SVN(ρ) according to Eqs. 4 and 5; for larger c, Eq. 2 gives a stricter lower bound than the others.Source: §V · p. 4
  • For the l1-norm and coherence-of-formation relations at c = 0.5, the measured coherence is well bounded by the measured lower bounds.The observables are Z and X, respectively corresponding to A and B.Source: §V · p. 4

VI. CONCLUSION

The experiment uses classical shadows to detect lower bounds for quantum uncertainty relations built from three coherence measures. Bound tightness depends on the reference basis and state purity, and the method is proposed for efficient generalization to multipartite states.

VI. CONCLUSION

  • The tightness of relative-entropy coherence lower bounds depends on both the reference basis and the quantum-state purity.The study also tests relations using l1-norm coherence and coherence of formation.Source: §VI · p. 4
  • Classical shadows detect the quantum-uncertainty lower bounds with measurement cost described as small and independent of state dimension.The experiment evaluates relative entropy, l1 norm, and coherence of formation.Source: §VI · p. 4
  • The authors state that their method can be generalized to multipartite states while retaining efficiency, and that the results could inform bound choice under experimental imperfections..Source: §VI · pp. 4–5
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