Clapeyron's theorem (elasticity): Difference between revisions
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In [[quantum mechanics]], and especially [[quantum information| quantum information theory]], the '''linear entropy''' or '''impurity''' of a [[quantum state|state]] is a [[scalar (physics)|scalar]] defined as | |||
:<math>S_L \, \dot= \, 1 - \mbox{Tr}(\rho^2) \,</math> | |||
where ''ρ'' is the [[density matrix]] of the state. | |||
The linear entropy can range between zero, corresponding to a completely pure state, and (1 − 1/''d''), corresponding to a completely mixed state. (Here, ''d'' is the [[dimension]] of the density matrix.) | |||
The linear entropy is trivially related to the [[purity (quantum mechanics)|purity]] <math>\gamma \,</math> of a state by | |||
:<math>S_L \, = \, 1 - \gamma \, .</math> | |||
==Motivation== | |||
The linear entropy is a lower approximation to the (quantum) [[von Neumann entropy]] ''S'', which is defined as | |||
:<math>S \, \dot= \, -\mbox{Tr}(\rho \ln \rho) = -\langle \ln \rho \rangle \, .</math> | |||
The linear entropy then is obtained by expanding ln ''ρ'' = ln (1−(1−''ρ'')), around a pure state, ''ρ''<sup>2</sup>=''ρ''; that is, expanding in terms of the non-negative matrix 1−''ρ'' in the formal [[Mercator series]] for the logarithm, | |||
:<math> - \langle \ln \rho \rangle = \langle 1- \rho \rangle + \langle (1- \rho )^2 \rangle/2 + \langle (1- \rho)^3 \rangle /3 + ... ~,</math> | |||
and retaining just the leading term. | |||
The linear entropy and von Neumann entropy are similar measures of the degree of mixing of a state, although the linear entropy is easier to calculate, as it does not require [[Diagonalizable matrix|diagonalization]] of the density matrix. | |||
==Alternate definition== | |||
Some authors<ref>{{cite journal | author=Nicholas A. Peters, Tzu-Chieh Wei, Paul G. Kwiat|title = Mixed state sensitivity of several quantum information benchmarks| year=2004 |journal = Physical Review A|volume=70|pages=052309 | doi=10.1103/PhysRevA.70.052309|arxiv = quant-ph/0407172 |bibcode = 2004PhRvA..70e2309P | issue=5}}</ref> define linear entropy with a different normalization | |||
:<math>S_L \, \dot= \, \tfrac{d}{d-1} (1 - \mbox{Tr}(\rho^2) ) \, ,</math> | |||
which ensures that the quantity ranges from zero to unity. | |||
==References== | |||
<references/> | |||
[[Category:Quantum mechanics]] | |||
[[Category:Quantum mechanical entropy]] | |||
{{quantum-stub}} | |||
Revision as of 18:47, 2 September 2013
In quantum mechanics, and especially quantum information theory, the linear entropy or impurity of a state is a scalar defined as
where ρ is the density matrix of the state.
The linear entropy can range between zero, corresponding to a completely pure state, and (1 − 1/d), corresponding to a completely mixed state. (Here, d is the dimension of the density matrix.)
The linear entropy is trivially related to the purity of a state by
Motivation
The linear entropy is a lower approximation to the (quantum) von Neumann entropy S, which is defined as
The linear entropy then is obtained by expanding ln ρ = ln (1−(1−ρ)), around a pure state, ρ2=ρ; that is, expanding in terms of the non-negative matrix 1−ρ in the formal Mercator series for the logarithm,
and retaining just the leading term.
The linear entropy and von Neumann entropy are similar measures of the degree of mixing of a state, although the linear entropy is easier to calculate, as it does not require diagonalization of the density matrix.
Alternate definition
Some authors[1] define linear entropy with a different normalization
which ensures that the quantity ranges from zero to unity.
References
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