Anti- (conjugate) linearity

Uhlmann, Armin

doi:10.1007/s11433-015-5777-1

Cited by 38 publications

(29 citation statements)

References 71 publications

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“…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].…”

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confidence: 99%

Uncertainty relations based on skew information with quantum memory

Ma¹,

Chen²,

Fei³

2016

Sci. China Phys. Mech. Astron.

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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...

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confidence: 99%

Uncertainty relations based on skew information with quantum memory

Ma¹,

Chen²,

Fei³

2016

Sci. China Phys. Mech. Astron.

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“…In the following subsections, we analyze this structure for each interaction, particularly discussing the independence of blocks in terms of the free parameters, making a distinction between the effective parameters (those appearing in the final expression of (28)) and the physical parameters (those appearing as coefficients h I in the Hamiltonian). They are not the same because many physical parameters appear clustered in the same way in (28), because the entries of S U depend only on the parameters h 11 ± h 22 , h 12 . As a result, by grouping finally in the U(1) × SU(2) blocks, there will be only two or eight different blocks S U in U.…”

Section: Structure Of Diagonal-off Entries Belonging To a Specific Blockmentioning

confidence: 99%

“…As was stated previously, they are imaginary only if local interactions are in the direction j = 2. In this case, we separate the factor ±i for j = 2 cases in the diagonal-off entries, and the remaining coefficients in the opposite corners in each block are equal as expected from (28). Then, there is generally one term with the same sign through all diagonal-off entries (when k = 2, or otherwise when j s = 2 in the first four rows in the Table 3), leaving only two possibilities for the remaining term.…”

Section: Block Entries Of H Imentioning

confidence: 99%

“…Anti-linear operators are particularly useful to depict time-reversal operations or the action of some Einstein-Podolsky-Rosen channels. If these kinds of operations are being considered in the processing, an extension of the Hamiltonian (1) should be considered by the inclusion of anti-linear operations [28]. In this work, we have restricted our development to linear operators, as was settled in Sections 2 and 3.…”

Section: Su(2) Decomposition In the Context Of N−qubit Controlled Gatesmentioning

confidence: 99%

“…where the superscript i in h i mn points out the consecutive number of blocks and S H I,I fulfills the syntactic notation followed in (28) and (29). We will exploit this notation in the following section for simplicity, where the matrix notation will become hardly extensive.…”

Section: Appendix A2 Group Theory Basics In the Context Of The Su(2mentioning

confidence: 99%

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SU(2) Decomposition for the Quantum Information Dynamics in 2d-Partite Two-Level Quantum Systems

Delgado

2018

Entropy

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Abstract:The gate array version of quantum computation uses logical gates adopting convenient forms for computational algorithms based on the algorithms classical computation. Two-level quantum systems are the basic elements connecting the binary nature of classical computation with the settlement of quantum processing. Despite this, their design depends on specific quantum systems and the physical interactions involved, thus complicating the dynamics analysis. Predictable and controllable manipulation should be addressed in order to control the quantum states in terms of the physical control parameters. Resources are restricted to limitations imposed by the physical settlement. This work presents a formalism to decompose the quantum information dynamics in SU(2 2d ) for 2d-partite two-level systems into 2 2d−1 SU(2) quantum subsystems. It generates an easier and more direct physical implementation of quantum processing developments for qubits. Easy and traditional operations proposed by quantum computation are recovered for larger and more complex systems. Alternating the parameters of local and non-local interactions, the procedure states a universal exchange semantics on the basis of generalized Bell states. Although the main procedure could still be settled on other interaction architectures by the proper selection of the basis as natural grammar, the procedure can be understood as a momentary splitting of the 2d information channels into 2 2d−1 pairs of 2 level quantum information subsystems. Additionally, it is a settlement of the quantum information manipulation that is free of the restrictions imposed by the underlying physical system. Thus, the motivation of decomposition is to set control procedures easily in order to generate large entangled states and to design specialized dedicated quantum gates. They are potential applications that properly bypass the general induced superposition generated by physical dynamics.

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Positivity bounds on electromagnetic properties of media

Creminelli,

Janssen,

Salehian

et al. 2024

J. High Energ. Phys.

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We study the constraints imposed on the electromagnetic response of general media by microcausality (commutators of local fields vanish outside the light cone) and positivity of the imaginary parts (the medium can only absorb energy from the external field). The equations of motion for the average electromagnetic field in a medium — the macroscopic Maxwell equations — can be derived from the in-in effective action and the effect of the medium is encoded in the electric and magnetic permeabilities ε(ω, |k|) and μ(ω, |k|). Microcausality implies analyticity of the retarded Green’s functions when the imaginary part of the 4-vector (ω, k) lies in forward light cone. With appropriate assumptions about the behavior of the medium at high frequencies one derives dispersion relations, originally studied by Leontovich. In the case of dielectrics these relations, combined with the positivity of the imaginary parts, imply bounds on the low-energy values of the response, ε(0, 0) and μ(0, 0). In particular the quantities ε(0, 0) – 1 and ε(0, 0) – 1/μ(0, 0) are constrained to be positive and equal to integrals over the imaginary parts of the response. We discuss various improvements of these bounds in the case of non-relativistic media and with additional assumptions about the UV behavior.

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Anti- (conjugate) linearity

Cited by 38 publications

References 71 publications

Uncertainty relations based on skew information with quantum memory

Uncertainty relations based on skew information with quantum memory

SU(2) Decomposition for the Quantum Information Dynamics in 2d-Partite Two-Level Quantum Systems

Positivity bounds on electromagnetic properties of media

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