Products of weak values: Uncertainty relations, complementarity, and incompatibility

Hall, Michael J. W.; Pati, Arun Kumar; Wu, Junde

doi:10.1103/physreva.93.052118

Cited by 39 publications

(26 citation statements)

References 66 publications

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“…However, it has been shown that the expectation value of a non-Hermitian operator can be inferred by measuring the weak value of the Hermitian operator into which the non-Hermitian operator can be polar decomposed [2]. Weak Measurements and weak values have not only found technological applications in ultra sensitive measurements [3] but also in exploring foundational issues in quantum mechanics [4,5]. In this manuscript, the novel idea regarding application of weak value to obtain expectation value of non-Hermitian operator given in Ref.…”

Section: Introductionmentioning

confidence: 99%

Measuring average of non-Hermitian operator with weak value in a Mach-Zehnder interferometer

et al. 2019

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Quantum theory allows direct measurement of the average of a non-Hermitian operator using the weak value of the positive semidefinite part of the non-Hermitian operator. Here, we experimentally demonstrate the measurement of weak value and average of non-Hermitian operators by a novel interferometric technique. Our scheme is unique as we can directly obtain the weak value from the interference visibility and the phase shift in a Mach Zehnder interferometer without using any weak measurement or post selection. Both the experiments discussed here were performed with laser sources, but the results would be the same with average statistics of single photon experiments. Thus, the present experiment opens up the novel possibility of measuring weak value and the average value of non-Hermitian operator without weak interaction and post-selection, which can have several technological applications.

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Section: Introductionmentioning

confidence: 99%

Measuring average of non-Hermitian operator with weak value in a Mach-Zehnder interferometer

et al. 2019

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“…The above relation can be generalized for mixed states also as given in Appendix A using an alternate derivation. Recently, uncertainty relation-1 in Eq(4) is derived using the product representation formula for the weak values [29].…”

Section: Uncertainty Relations For Two Arbitrary Unitary Operatorsmentioning

confidence: 99%

Uncertainty relations for general unitary operators

Bagchi

Pati

2016

Phys. Rev. A

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We derive several uncertainty relations for two arbitrary unitary operators acting on physical states of a Hilbert space. We show that our bounds are tighter in various cases than the ones existing in the current literature. Using the uncertainty relation for the unitary operators we obtain the tight state-independent lower bound for the uncertainty of two Pauli observables and anticommuting observables in higher dimensions. With regard to the minimum uncertainty states, we derive the minimum uncertainty state equation by analytic method, and relate this to the ground-state problem of the Harper Hamiltonian. Furthermore, the higher dimensional limit of the uncertainty relations and minimum uncertainty states are explored. From an operational point of view, we show that the uncertainty in the unitary operator is directly related to the visibility of quantum interference in an interferometer where one arm of the interferometer is affected by a unitary operator. This shows a principle of preparation uncertainty, i.e., for any quantum system, the amount of visibility for two general non-commuting unitary operators is non-trivially upper bounded.

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“…There exist two kinds of operators in quantum mechanics: Hermitian and non-Hermitian operators, but it should be paid particular attention that the previous uncertainty relations are contradictory with the non-Hermitian operators, i.e., lots of uncertainty relations will be violated when applied to non-Hermitian operators [68][69][70] 2 for all qubit systems, where the non-Hermitian operator σ + (σ − ) is the raising (lowering) operator of the single qubit system. That is to say, different from the Hermitian operators, the uncertainties of the non-Hermitian operators are not lower-bounded by the quantities related with the commutator and anti-commutator.…”

Section: The Applicability Of the Unified And Exact Framework To Non-mentioning

confidence: 99%

Unified and Exact Framework for Variance-Based Uncertainty Relations

Zheng

Fan

et al. 2020

Sci Rep

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We provide a unified and exact framework for the variance-based uncertainty relations. This unified framework not only recovers some well-known previous uncertainty relations, but also fixes the deficiencies of them. Utilizing the unified framework, we can construct the new uncertainty relations in both product and sum form for two and more incompatible observables with any tightness we require. Moreover, one can even construct uncertainty equalities to exactly express the uncertainty relation by the unified framework, and the framework is therefore exact in describing the uncertainty relation. Some applications have been provided to illustrate the importance of this unified and exact framework. Also, we show that the contradiction between uncertainty relation and non-Hermitian operator, i.e., most of uncertainty relations will be violated when applied to non-Hermitian operators, can be fixed by this unified and exact framework. Quantum uncertainty relations 1-3 , expressing the impossibility of the joint sharp preparation of the incompatible observables 4,5 , are the most fundamental differences between quantum and classical mechanics 6-9. The uncertainty relation has been widely used in the quantum information science 10,11 , such as quantum non-cloning theorem 12,13 , quantum cryptography 14-17 , entanglement detection 18-22 , quantum spins squeezing 23-26 , quantum metrology 27-29 , quantum synchronization 30,31 and mixedness detection 32,33. In general, the improvement in uncertainty relations will greatly promote the development of quantum information science 18,28,34-36. The variance-based uncertainty relations for two incompatible observables A and B can be divided into two forms: the product form ΔA 2 ΔB 2 ≥ LB p 2,3,5,37,38 and the sum form ΔA 2 + ΔB 2 ≥ LB s 39-42 , where LB p and LB s represent the lower bounds of the two forms uncertainty relations, and ΔQ 2 is the variance of Q (To make sure that the quantity measuring the uncertainty will be a real number, the variance is taken as 〈(Q − 〈Q〉) † (Q − 〈Q〉) for non-Hermitian operators. Here the 〈Q〉 represents the expected value of Q). The product form uncertainty relation cannot fully capture the concept of the incompatible observables, because it can be trivial; i.e., the lower bound LB p can be null even for incompatible observables 39,40,43,44. This deficiency is referred to as the triviality problem of the product form uncertainty relation. In order to fix the triviality problem, Maccone and Pati deduced a sum form uncertainty relation with a non-zero lower bound for incompatible observables 44 , firstly showing that the triviality problem can be addressed by the sum form uncertainty relation. Thus, the sum form uncertainty relations were considered to be stronger than the product form uncertainty relations, and since then, lots of effort has been made to investigate the uncertainty relation in the sum form 18,39,45-48. However, most of the sum form uncertainty relations depend on the orthogonal state to the state of the system, and thus are diffic...

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Products of weak values: Uncertainty relations, complementarity, and incompatibility

Cited by 39 publications

References 66 publications

Measuring average of non-Hermitian operator with weak value in a Mach-Zehnder interferometer

Measuring average of non-Hermitian operator with weak value in a Mach-Zehnder interferometer

Uncertainty relations for general unitary operators

Unified and Exact Framework for Variance-Based Uncertainty Relations

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