Uncertainty relations based on skew information with quantum memory

Abstract

PACS number(s): 03.67. 75.10.Pq, 03.67.Mn Dear Editors, Uncertainty principle is one of the most fascinating features of the quantum world. It asserts a fundamental limit on the precision with which certain pairs of physical properties of a particle, such as position and momentum, can not be simultaneously known. The uncertainty principle has attracted considerable attention since the innovation of quantum mechanics and has been investigated in terms of various types of uncertainty inequalities: as informational recourses in entropic terms, by means of majorization technique and based on sum of variances and standard deviations.For a pair of observables R and S , the well-known Heisenberg-Robertson uncertainty relation [1] says that,, where [R, S ] = RS − S R is the commutator and V ρ (R) is the standard deviation of R. The entropies serve as appropriate measures of the information content. It is also used to quantify the quantum uncertainty: the sum of the Shannon entropy of the probability distribution of the outcomes is no less than log 2 1 c [2] when R and S are measured. The term 1/c quantifies the complementarity of the two observables R and S . It has been proved that the entropy uncertainty relations do imply the Heisenberg's uncertainty relation.Concerning bipartite systems, the authors in [3] provided a bound on the uncertainties which depends on the amount of entanglement between the measured particle A and the quantum memory B. The result of [3] was further improved to depend on the quantum discord between particles A and B in [4]. Recently the authors in [5] obtained entropic uncertainty relations for multiple measurements with quantum memory.The quantum uncertainty relation can be also described in terms of skew information. I (ρ, H) = − 1 2 Tr √ ρ, H 2 is introduced to quantify the degree of non-commutativity of a state ρ and an observable H, which is reduced to the variance V ρ (H) when ρ is a pure state [6]. It can be interpreted as quantum uncertainty of H in ρ. Luo introduced another quantity, where {X, Y} is the anticommutator, H 0 = H − Tr(ρH)I with I the identity operator. The following inequality holds [7],I (ρ, R) J ρ (R) can be regarded as a kind of measure for quantum uncertainty.Hence we define UN(ρ, R) := I (ρ, R) J ρ (R). Then we define the uncertainty of ρ associated to the projective measurement {φ k } as:, where φ k = |φ k φ k | and ψ k = |ψ k ψ k | are the rank one spectral projectors of two non-degenerate observables R and S with eigenvectors |φ k and |ψ k , respectively. Now we consider the case of bipartite state ρ AB in tensor space H A ⊗ H B [8]. Recall that quantum discord is a kind of quantum correlation that is different from the entanglement and has found many novel applications [9]. A bipartite state ρ AB is of zero discord if and only if it is a classical-quantum correlated state (CQ state). Besides the definition of the original discord, there are some other discord-like measures sharing the same properties such that their values are zero iff the state is a CQ sta...

Uncertainty Relation Based on Wigner–Yanase–Dyson Skew Information with Quantum Memory

“…We note that Equation ( 7 ) reduces to an inequality that agrees with the result of [ 32 ] when . Theorem 2 is a generalization of the theorem in [ 32 ].…”

“…( a ) Blue (red) solid line for the value of the left (right)-hand side of Equation ( 5 ) with ; black dotted (red dot-dashed) line represents the value of the left (right)-hand side of Equation ( 5 ) with . ( b ) Red solid (black dotted) line represents the value of the left (right)-hand side of Equation ( 7 ) with ; blue solid (green dotted) line represents the value of the left (right)-hand side of Equation ( 7 ) with , which corresponds to Figure 1 in [ 32 ].…”

“…Inspired by the works [ 32 , 35 ], in this paper, we study the uncertainty relations based on Wigner–Yanase–Dyson skew information in the presence of quantum memory, which generalize the results in [ 32 ] to the case of Wigner–Yanase–Dyson skew information, and the results in [ 35 ], which generalize to the case with the presence of quantum memory. We present uncertainty inequalities both in product and summation forms, and show that the lower bounds contain two terms: one concerns the compatibility of two measurement observables, and the other concerns the quantum correlations between the measured system and the quantum memory.…”

“…In [ 32 ], an uncertainty relation based on Wigner–Yanase skew information has been obtained with quantum memory, where quantifies the degree of non-commutativity between a quantum state and an observable H , which is reduced to the variance when is a pure state. In fact, the Wigner–Yanase skew information is generalized to Wigner–Yanase–Dyson skew information , (see [ 33 ]): …”

“…In [32], an uncertainty relation based on Wigner-Yanase skew information I(ρ, H) has been obtained with quantum memory, where…”