Ascertaining the uncertainty relations via quantum correlations

Abstract

We propose a new scheme to express the uncertainty principle in form of inequality of the bipartite correlation functions for a given multipartite state, which provides an experimentally feasible and model-independent way to verify various uncertainty and measurement disturbance relations. By virtue of this scheme the implementation of experimental measurement on the measurement disturbance relation to a variety of physical systems becomes practical. The inequality in turn also imposes a constraint on the strength of correlation, i.e. it determines the maximum value of the correlation function for two-body system and a monogamy relation of the bipartite correlation functions for multipartite system.The uncertainty principle lies at the heart of quantum mechanics and is one of the most fundamental features which distinguish it from the classical mechanics. The original form, where the anticommutator is defined as {A, B} ≡ AB + BA. * Corresponding author: qiaocf@ucas.ac.cn 1 Note that in the form involving standard deviations, the uncertainty relation represents the property of the ensemble of arbitrary quantum state in Hilbert space and does not concern with the specific measurements. Thus such uncertainty relation is not related to the precision of measurement on one observable and the disturbance to its conjugate.If we assume ǫ(A) to be the precision of the measurement on A and η(B) to be the disturbance of the same measurement on B, the Heisenberg-type relation with regard to measurement and disturbance would readIn recently, Ozawa found that the this form of measurement disturbance relation (MDR) (1) is not a universal one, and a new MDR was proposed [4], which are thought to be generallyEq.(2) is of fundamental importance, for example, it leads to a totally different accuracy limit ǫ(A) for non-disturbing measurements (η(B) = 0) comparing to the Heisenberg-type MDR. In quantum information science, the uncertainty principle in general is also crucial to the security of certain protocols in quantum cryptography [5], and additionally, it plays an important role in the quantum metrology [6].Despite the importance of the uncertainty principle, only the uncertainty relation in form of standard deviations has been well verified in various situations, e.g., see [7] Here σ = (σ x , σ y , σ z ) are Pauli matrices, n a = a/| a|, and a general commutative relation holds for such operatorswherep be the two eigenvectors of operator P = σ · n p with eigenvalues ±1, the following complete relations holdHere n p is a unit vector, |n For the Robertson-Schrödinger uncertainty relation we have the following theorem:Theorem 1 The Robertson-Schrödinger uncertainty relation imply the following inequality on the correlation functions of arbitrary bipartite quantum statewherethe corresponding partite, S is the parallelogram area formed by a, b.This theorem indicates that the correlation functions between one specific operator (P ) and two other operators (A, B) and their commutator (C) in bipartite states are constrained by ...

Reformulating the Quantum Uncertainty Relation

“…An arbitrary quantum state may be constructed as p = | p |(sin θ cos φ, sin θ sin φ, cos θ), where θ, φ are the polar and azimuthal angles in the 3-dimensional real space. For pure states of | p | = 1, substituting the values of cos θ pa , cos θ pb , and cos θ pc into equations (7), (8) and (9), one then has the following trade-off relation for ∆A 2 , ∆B 2 , and ∆C 2 ,…”

Connection between measurement disturbance relation and multipartite quantum correlation

“…Except for the fundamental physical implications for multipartite correlations, the MDR's unique constraint on the bipartite correlation function in the tripartite state makes the experimental test on MDR applicable in various physical systems, e.g., atoms, ions, and even higher energy particles, through measurement of correlation functions [16]. One schematic optical experimental setup for a qubit system is shown in Fig.…”

“…Note that in (1), only the properties of two observables in the ensemble of a quantum state are involved, and the relation is independent of any specific measurement. The MDR has been intensively studied both theoretically [12][13][14][15][16] and experimentally [17][18][19][20]. However, the implication of the MDR for quantum information and quantum measurement is still unclear [21][22][23].…”

“…U, the subscripts 1 and 2 signify that the operators are acting on states |ψ and |φ , respectively, U is a unitary interaction between |ψ and |φ , and I is the identity operator. The measurement operator M may be set to A if it has the same possible outcomes as operator A, i.e., A = U † (I 1 ⊗ A 2 )U [15,16]. Note that in addition to this operator formalism, which we will work with, there are also other types of definitions for the measurement precision and disturbance, e.g., the probability distribution formalism [21].…”

Mutually Exclusive Uncertainty Relations

“…The variance-based uncertainty relations possess clear physical meanings and have a variety of applications in quantum information processings such as quantum spin squeezing [11][12][13][14][15] , quantum metrology [16][17][18] , and quantum nonlocality. 19,20 While the early forms of variance-based uncertainty relations are vital to the foundation of quantum theory, there are two problems still need to be addressed: (i) Homogeneous product of variances may not fully capture the concept of incompatibility. In other words, a weighted relation may produce a better approximation (e.g., the uncertainty relation with Rényi entropy 21 and variance-based uncertainty relation for a weighted sum), for more details and examples, see Ref.…”