QuantumPhysics.dvi

The contribution of the second factor is being kept as a subdominant effect, which may be evaluated since it corresponds to the ...

16 Mixtures and Statistical Entropy In this section, the notion of statistically mixed ensembles, of pure and mixed states, of c ...

get, up to an immaterial overall phase, |n〉= cosθ eiψ|+〉+ sinθe−iψ|−〉 (16.2) so that Pn= ∣∣ ∣acosθe−iψ+bsinθ eiψ ∣∣ ∣ 2 (16.3) D ...

Since the superpositions involved here are incoherent, theweightswimay be thought of as classical probability weights assigned t ...

16.2.1 Ensemble averages of expectation values in mixtures In a pure normalized state|ψ〉, we defined theexpectation value of an ...

Polarized along theSzdirection with eigenvalue− ̄h/2, the state is|z−〉, and the correspond- ing density operator isρ=|z−〉〈z−|. ...

tr(ρ) = 1, at least one of the eigenvalues must be positive. The sign of the other eigenvalue is then determined by the sign of ...

16.5 Quantum Statistical Mechanics One particularly important special case of mixed states andensembles is provided by equilibri ...

introduce Lagrange multipliers for the constant energy condition, as well as for the condition of unit trace ofρ. Thus, we extre ...

16.6 Classical information and Shannon entropy Entropy may be given a meaning beyond traditional statistical mechanics. In devel ...

Thus, the logarithm of the number of micro-states Ω is an extensive quantity, proportional to the lengthNof the message. The sta ...

Additivityupon combination of two subsystems which arestatistically uncorrelated. Let the systems be described by Hilbert space ...

16.7.1 Density matrix for a subsystem LetHab=Ha⊗Hbbe the Hilbert space organized as the tensor product of two subspacesHa andHb. ...

The general density matrixρabof the combined system may be parametrized as follows, ρab=ρa⊗ρb+ ∑^3 i,j=1 Cijσia⊗σbj (16.55) wher ...

The first two terms on the right hand side simplify as follows, ∑ i,j (pi−p′j)|〈φi|φ′j〉|^2 = ∑ i pi− ∑ j′ pj′= 0 (16.62) in view ...

The combinations of projectors obeys tr ( PiPj′ ) ≥ 0 (16.68) As a result, every term in the double sum is either positive orzer ...

Next, we takeρ′=ρa⊗ρb, so that S(ρab) ≤ −kTrHaTrHb ( ρ(lnρa⊗Ib)⊕ρ(Ia⊗lnρb) ) ≤ −kTrHa ( ρalnρa ) −kTrHb ( ρblnρb ) (16.74) which ...

The formulas for the population fractions directly result from the normalization of the pure states |ψβ;a〉, and|ψα;b〉. Notice th ...

17 Entanglement, EPR, and Bell’s inequalities Quantum systems exhibit correlations (entanglement) thatappear counter-intuitive f ...

expressed here with respect to thez-basis. Basic observables inHaand inHbrespectively are the spinna·Saandnb·Sb. In addition, we ...