Experimental investigation of conditional majorization uncertainty relations in the presence of quantum memory

Letter

Experimental investigation of conditional majorization uncertainty relations in the presence of quantum memory

Gaoyan Zhu, Aoxiang Liu, Lei Xiao, Kunkun Wang, Dengke Qu, Junli Li, Congfeng Qiao, and Peng Xue

Phys. Rev. A 108, L050202 – Published 7 November 2023

Abstract

We report an experimental investigation of conditional majorization uncertainty relations (CMURs) in the presence of quantum memory. We find that the CMUR bounds are always physically nontrivial even if the particle of interest is strongly entangled with a quantum memory, whereas the previous conditional entropic uncertainty relation bounds may be trivial and physically unreachable. We deploy vectorized measures of uncertainty relations and quantum correlations, and the result reveals the sophisticated structures of them. In addition, we demonstrate an application of the CMURs, to witness steerability of bipartite states. Such a method applies to an arbitrary number of measurement settings and can be efficiently implemented. Aside from the CMURs' fundamental significance, our result also shows its impact on the development of future quantum technologies.

Received 14 February 2022
Revised 11 May 2023
Accepted 19 October 2023

DOI:https://doi.org/10.1103/PhysRevA.108.L050202

Physics Subject Headings (PhySH)

Nonlocality Quantum correlations in quantum information Quantum entanglement Quantum measurements Quantum memories Quantum optics

General PhysicsQuantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Gaoyan Zhu ¹, Aoxiang Liu ², Lei Xiao^3,1, Kunkun Wang⁴, Dengke Qu¹, Junli Li^2,*, Congfeng Qiao^2,†, and Peng Xue ^3,1,‡

¹Beijing Computational Science Research Center, Beijing 100084, China
²School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
³School of Physics, Southeast University, Nanjing 211189, China
⁴School of Physics and Optoelectronics Engineering, Anhui University, Hefei 230601, China

^*Deceased.
^†qiaocf@ucas.ac.cn
^‡gnep.eux@gmail.com

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Issue

Vol. 108, Iss. 5 — November 2023

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Images

Figure 1
Experimental setup. Photon pairs shared by $A$ and $B$ are generated in the state $| ψ_{ξ} 〉$ via a type-I spontaneous parametric down-conversion process by pumping two adjacent nonlinear crystals ( $β$ -barium borate, BBO) with a $405 − nm$ laser diode. The parameter $ξ$ is set by the half-wave plate (HWP) in front of the BBO crystals. Two $α$ -BBO crystals are inserted to compensate for the walk-off effect. Two interference filters restrict the photon bandwidth to $3 nm$ . Mixed states are prepared by letting one of the photons pass through the unbalanced Mach-Zehnder interferometer (UMZI). In the UMZI, two beam splitters first split the photon into three paths and then recombine them into one. Coherence in two of the arms is completely destroyed by inserting a quartz. The parameter $p$ is controlled by manipulating the adjustable shutters. Local measurements on $A$ and $B$ are carried out via a sequence of a quarter-wave plate (QWP), an HWP, and a polarizing beam splitter (PBS) on each side, respectively. Coincidence measurements are then performed by avalanche photodiodes (APDs).
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Figure 2
Experimental results of the CMUR in the presence of quantum memory. (a) Upper bounds ${\vec{s}}^{(\oplus)}$ of the CMURs for observables ${X = σ (θ, 0), Y = σ (θ, π)}$ and the state $| ψ_{ξ} 〉$ with $θ = π / 4$ and different $ξ$ illustrated by Lorenz curves. Blue color scheme curves denote the theoretical predictions of the bounds ${\vec{s}}^{(\oplus)}$ and symbols for the corresponding experimental results, which violate the limit $\vec{s}$ for a single-particle system, represented by the red curve. (b) Lower bounds for CMUR and conditional entropic uncertainty relation (3) with quantum memory for different states. The solid curves indicate the theoretical predictions of the reduced lower bounds of $H ({\vec{s}}^{(\otimes)})$ and the symbols are for the experimental results. The dotted lines represent the CEUR bounds $[{log}_{2} \frac{1}{c} + S (A | B)]$ . Error bars indicate the statistical uncertainty which is obtained via the Monte Carlo simulation method by assuming Poissonian photon-counting statistics.
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Figure 3
Illustration of the measurement-setting choices. We take the Bloch-space directions through antipodal pairs of vertices of Platonic polyhedra as axes for measurement. (a) Icosahedron for $M = 6$ . (b) Dodecahedron for $M = 10$ .
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Figure 4
Experimental results of the steerability of the states $ρ_{ξ}$ . (a) Distribution of experimental states. The blue and red curves represent the theoretical predictions of states $ρ_{ξ}$ when ${\vec{s}}^{(+)} = \vec{ε}$ for the case of six and ten measurement settings, respectively. The symbols under the blue (red) curve (in the light blue area) indicate that $A$ cannot steer $B$ . The dots above the blue (red) curve indicate that $A$ can steer $B$ . Dot textures are colored according to the degree of the violation. The blue and red dotted lines represent the theoretical predictions for the case of two and three settings, respectively. For the states in the green area, steerability can only be observed for six settings (of our method) but failed for both two and three settings (former methods). (b) A larger version of the blue dotted box in (a) to show more details.
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