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arXiv:2104.02848 · Physics Letters A

Experimental test of the majorization uncertainty relation with mixed states

Shuang Wang, Fang-Xia Meng, Hui Wang, Cong‐Feng Qiao

Published Mar 31, 2022• 1 citation
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Abstract

The uncertainty relation lies at the heart of quantum theory and behaves as a non-classical constraint on the indeterminacies of incompatible observables in a system. In the literature, many experiments have been devoted to the test of the uncertainty relations which mainly focus on the pure states. In this work we test the novel majorization uncertainty relations of three incompatible observables using a series of mixed states with adjustable mixing degrees, and compare the compactness of various entropy uncertainty relations. The experimental results confirm that for general mixed quantum system, the majorization uncertainty relation tends to be the tightest constraint on uncertainty, and indicate that the entropy uncertainty relation obtained from the majorization uncertainty relation is the optimal one. Our experimental setup provides an easy means for preparing mixed states, and based on this simple optical elements can be utilized to realize the required quantum states.

DOI
10.1016/j.physleta.2022.128105

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The uncertainty relation lies at the heart of quantum theory and behaves as a non-classical constraint on the indeterminacies of incompatible observables in a system. In the literature, many experiments have been devoted to the test of the uncertainty relations which mainly focus on the pure states. In this work we test the novel majorization uncertainty relations of three incompatible observables using a series of mixed states with adjustable mixing degrees, and compare the compactness of various entropy uncertainty relations. The experimental results confirm that for general mixed quantum system, the majorization uncertainty relation tends to be the tightest constraint on uncertainty, and indicate that the entropy uncertainty relation obtained from the majorization uncertainty relation is the optimal one. Our experimental setup provides an easy means for preparing mixed states, and based on this simple optical elements can be utilized to realize the required quantum states.
The uncertainty principle, which gives the constraints on obtaining precise outcomes for incompatible measurements, provides a new vision of the real world that we are not able to realize from the classical knowledge. In recent years, numerous theoretical and experimental developments about the new forms of the uncertainty principle have been achieved. Among these efforts, one attractive goal is to find tighter bounds of the uncertainty relation. Here, using an all optical setup, we experimentally investigate a most recently proposed form of uncertainty principle-the fine-grained uncertainty relation assisted by a quantum memory. The experimental results on the case of two-qubit state with maximally mixed marginal demonstrate that the fine-graining method can help to get a tighter bound of the uncertainty relation. Our results might contribute to further understanding and utilizing of the uncertainty principle.
Abstract Majorization uncertainty relations are generalized for an arbitrary mixed quantum state ρ of a finite size N . In particular, a lower bound for the sum of two entropies characterizing the probability distributions corresponding to measurements with respect to two arbitrary orthogonal bases is derived in terms of the spectrum of ρ and the entries of a unitary matrix U relating both bases. The results obtained can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as an uncertainty relation for the sum of conditional entropies.
The uncertainty relation is a distinguishing feature of quantum theory, characterizing the incompatibility of noncommuting observables in the preparation of quantum states. Recently, many uncertainty relations were proposed with improved lower bounds and were deemed capable of incorporating multiple observables. Here we report an experimental verification of seven uncertainty relations of this type with single-photon measurements. The results, while confirming these uncertainty relations, show as well the relative stringency of various uncertainty lower bounds.
The indeterminacy inherent in quantum measurements is an outstanding character of quantum theory, which manifests itself typically in the uncertainty principle. In the last decade, several universally valid forms of error-disturbance uncertainty relations were derived for completely general quantum measurements for arbitrary states. Subsequently, Branciard established a form that is optimal for spin measurements for some pure states. However, the bound in his inequality is not stringent for mixed states. One of the present authors recently derived a new bound tight in the corresponding mixed state case. Here, a neutron-optical experiment is carried out to investigate this new relation: it is tested whether error and disturbance of quantum measurements disappear or persist in mixing up the measured ensemble. The attainability of the new bound is experimentally observed, falsifying the tightness of Branciard's bound for mixed spin states.
Uncertainty relations are one of the most important foundations of quantum physics. In the textbook literatures, uncertainty relations usually refer to the preparation uncertainty. Its original formulation based on variances of two observables limits on the ability to prepare an ensemble of quantum systems for which non-commuting observables will have arbitrary uncertainty. The preparation uncertainty relation has been widely investigated. On the other hand, a unitary operator is a fundamental tenet of quantum theory. Every evolution of a closed quantum system is governed by acting unitary operators on the state of the system and the evolution of an open system can be represented by acting unitary operators on an enlarged system consisting of the quantum system as a subsystem. Therefore, naturally, to understand and quantitatively capture the essence of uncertainty relations for unitary operators is important and timely. Here we report an experimental investigation of a set of uncertainty relations for two unitary operators, which are theoretically derived by using a sequence of fine-grained inequalities. We test these uncertainty relations with single photons and interferometric networks. The unitary uncertainty relation is saturated by any pure qubit state. For higher-dimensional states, it is stronger than the best known bound introduced in the previous literatures. The lower bounds of the unitary uncertainty relations can be even further strengthened by the symmetry of permutation. The experimental findings agree with the predictions of quantum theory and respect the new uncertainty relations.
The well-known Robertson-Schr\"odinger uncertainty relations miss an irreducible lower bound. This is widely attributed to the lower bound's state dependence. Therefore, Abbott et al. introduced a general approach to derive tight state-independent uncertainty relations for qubit measurements [Mathematics 4, 8 (2016)]. The relations are expressed in two measures of uncertainty, which are standard deviation and entropy, both functions of the expectation value. Here, we present a neutron polarimetric test of the tight state-independent preparation uncertainty relations for orthogonal, as well as nonorthogonal, Pauli spin observables. The final results, obtained with pure and mixed spin states, reproduce the theoretical predictions clearly for arbitrary initial states of variable degree of polarization.