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A quantum measure of coherence and incompatibility

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

Abstract. The well-known two-slit interference is understood as a special relation between observable (localization at the slits) and state (being on both slits). Relation between an observable and a quantum state is investigated in the general case. It is assumed that the amount of coherence equals that of incompatibility between observable and state. On ground of this, an argument is presented that leads to a natural quantum measure of coherence, called ”coherence or incompatibility information”. Its properties are studied in detail making use of ’the mixing property of relative entropy ’ derived in this article. A precise relation between the measure of coherence of an observable and that of its coarsening is obtained and discussed from the intuitive point of view. Convexity of the measure is proved, and thus the fact that it is an information entity is established. A few more detailed properties of coherence information are derived with a view to investigate final-state entanglement in general repeatable measurement, and, more importantly, general bipartite entanglement in follow ups of this study.

Related papers

A quantum measure of coherence and incompatibility

Fedor Herbut

The well-known two-slit interference is understood as a special relation between an observable (localization at the slits) and a state (being on both slits). The relation between an observable and a quantum state is investigated in the general case. It is assumed that the amount of coherence equals that of the incompatibility between an observable and a state. On these grounds, an argument is presented that leads to a natural quantum measure of coherence, called 'coherence or incompatibility information'. Its properties are studied in detail, making use of 'the mixing property of relative entropy' derived in this paper. A precise relation between the measure of coherence of an observable and that of its coarsening is obtained and discussed from the intuitive point of view. Convexity of the measure is proved, and thus the fact that it is an information entity is established. A few more detailed properties of the coherence information are derived with a view to investigating the final-state entanglement in general repeatable measurement, and, more importantly, general bipartite entanglement in follow-ups of this study.

arXiv:quant-ph/0503077•Journal of Physics A Mathematical and General•Mar 15, 2005•34 citations

On the meaning of entanglement in quantum measurement

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Measurement interaction between a measured object and a measuring instrument, if both are initially in a pure state, produces a (final) bipartite entangled state vector. The quasi-classical part of the correlations in it is connected with transmission of information in the measurement. But, prior to "reading" the instrument, there is also purely quantum entanglement in the final state vector. It is shown that in repeatable measurement quantitatively the entanglement equals the amount of incompatibility between the measured observable and the final state. It also equals the amount of incompatibility of the observable and the initial state of the object.

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Quantum coherence measures based on the Tsallis relative entropies

Alexey E. Rastegin

The concept of coherence is one of cornerstones in physics. The development of quantum information science has lead to renewed interest in properly approaching the coherence at the quantum world. Various measures could be proposed for quantifying the coherence of a quantum state with respect to the prescribed orthonormal basis. To be a proper measure of coherence, each candidate should enjoy certain properties. It seems that the monotonicity property plays a crucial role here. Indeed, there is known an intuitive measure of coherence that does not share this condition. We study coherence measures induced by quantum divergences of the Tsallis type. It is proved that considered measures of coherence satisfy all the required properties. The only point is that monotonicity under incoherent selective measurements has to be reformulated. The proposed formulation can naturally be treated as a parametric extension of its standard form. This extension is reduced to the standard form when the standard relative entropy is dealt. Finally, two coherence measures quadratic in moduli of matrix elements are compared from the monotonicity viewpoint.

•Dec 21, 2015•0 citations
Abstract. The well-known two-slit interference is understood as a special relation between observable (localization at the slits) and state (being on both slits). Relation between an observable and a quantum state is investigated in the general case. It is assumed that the amount of coherence equals that of incompatibility between observable and state. On ground of this, an argument is presented that leads to a natural quantum measure of coherence, called ”coherence or incompatibility information”. Its properties are studied in detail making use of ’the mixing property of relative entropy ’ derived in this article. A precise relation between the measure of coherence of an observable and that of its coarsening is obtained and discussed from the intuitive point of view. Convexity of the measure is proved, and thus the fact that it is an information entity is established. A few more detailed properties of coherence information are derived with a view to investigate final-state entanglement in general repeatable measurement, and, more importantly, general bipartite entanglement in follow ups of this study.
The well-known two-slit interference is understood as a special relation between an observable (localization at the slits) and a state (being on both slits). The relation between an observable and a quantum state is investigated in the general case. It is assumed that the amount of coherence equals that of the incompatibility between an observable and a state. On these grounds, an argument is presented that leads to a natural quantum measure of coherence, called 'coherence or incompatibility information'. Its properties are studied in detail, making use of 'the mixing property of relative entropy' derived in this paper. A precise relation between the measure of coherence of an observable and that of its coarsening is obtained and discussed from the intuitive point of view. Convexity of the measure is proved, and thus the fact that it is an information entity is established. A few more detailed properties of the coherence information are derived with a view to investigating the final-state entanglement in general repeatable measurement, and, more importantly, general bipartite entanglement in follow-ups of this study.
Measurement interaction between a measured object and a measuring instrument, if both are initially in a pure state, produces a (final) bipartite entangled state vector. The quasi-classical part of the correlations in it is connected with transmission of information in the measurement. But, prior to "reading" the instrument, there is also purely quantum entanglement in the final state vector. It is shown that in repeatable measurement quantitatively the entanglement equals the amount of incompatibility between the measured observable and the final state. It also equals the amount of incompatibility of the observable and the initial state of the object.
Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in different reference bases? We show that the quantum coherence of a state as quantified by the relative entropy of coherence in two or more noncommuting reference bases respects uncertainty like relations for a given state of single and bipartite quantum systems. In the case of bipartite systems, we find that the presence of entanglement may tighten the above relation. Further, we find an upper bound on the sum of the relative entropies of coherence of bipartite quantum states in two noncommuting reference bases. Moreover, we provide an upper bound on the absolute value of the difference of the relative entropies of coherence calculated with respect to two incompatible bases.
The concept of physical twin observables (PTO) for bipartite quantum states, introduced and proved relevant for quantum information theory in recent work, is substantially simplified. The relation of observable and state is studied in detail from the point of view of coherence entropy. Properties of this quantity are further explored. It is shown that, besides for pure states, also for a class of mixed states, quantum discord (measure of entanglement) can be expressed through the coherence entropy of a PTO complete in relation to the state.
We propose a new measure of relative incompatibility for a quantum system with respect to two non-commuting observables, and call it quantumness of relative incompatibility. In case of a classical state, order of observation is inconsequential, hence probability distribution of outcomes of any observable remains undisturbed. We define relative entropy of the two marginal probability distributions as a measure of quantumness in the state, which is revealed only in presence of two non-commuting observables. Like all other measures, we show that the proposed measure satisfies some basic axioms. Also, we find that this measure depicts complementarity with quantum coherence. The relation is more vivid when we choose one of the observables in such a way that its eigen basis matches with the basis in which the coherence is measured. Our result indicates that the quantumness in a single system is still an interesting question to explore and there can be an inherent feature of the state which manifests beyond the idea of quantum coherence.
The concept of coherence is one of cornerstones in physics. The development of quantum information science has lead to renewed interest in properly approaching the coherence at the quantum world. Various measures could be proposed for quantifying the coherence of a quantum state with respect to the prescribed orthonormal basis. To be a proper measure of coherence, each candidate should enjoy certain properties. It seems that the monotonicity property plays a crucial role here. Indeed, there is known an intuitive measure of coherence that does not share this condition. We study coherence measures induced by quantum divergences of the Tsallis type. It is proved that considered measures of coherence satisfy all the required properties. The only point is that monotonicity under incoherent selective measurements has to be reformulated. The proposed formulation can naturally be treated as a parametric extension of its standard form. This extension is reduced to the standard form when the standard relative entropy is dealt. Finally, two coherence measures quadratic in moduli of matrix elements are compared from the monotonicity viewpoint.