By Robert Olkiewicz, Wojciech Cegla, Andrzej Frydryszak, Piotr Garbaczewski, Lech Jakobczyk

The relevant subject of this lecture assortment is quantum dynamics, looked more often than not because the dynamics of entanglement and that of decoherence phenomena. either those techniques seem to check with the habit of strangely fragile good points of quantum platforms presupposed to version quantum stories and to enforce quantum date processing exercises. This assortment may perhaps function a necessary source for these attracted to either theoretical description and sensible purposes of basics of quantum mechanics.

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**Extra info for Quantum Dynamics and Information: Proceedings of the 46th Karpacz Winter School of Theoretical Physics**

**Example text**

2 ) 3(1 + r∞ = ✁ ✁ (57) One can thus compare the asymptotic entanglement content of the two– qubit states ρ12 (α) with respect to the states ρ∞ (α) in (49) when they are not subjected to the protocol. We shall consider: (1) the concurrence C(ρ12 ∞ (α)) when C(ρ∞ (α)) = 0; (2) the entanglement gain (50) after application of the protocol, ∆12 (α) := C(ρ12 ∞ (α)) − C(ρ(α)) , when, without protocol, ∆(α) = C(ρ∞ (α)) − C(ρ(α)) ≤ 0; (3) the entanglement gain (50) after applying the protocol when, without protocol, ∆(α) = C(ρ∞ (α)) − C(ρ(α)) > 0.

There is a simple relation between block positive operators and positive maps:45 W is block positive if and only if W = (id ⊗ Λ)Pd+ , (7) for some positive map Λ in Md (C). In the above formulae Pd+ denotes a canonical maximally entangled state in Cd ⊗ Cd d Pd+ = 1 eij ⊗ eij , d i,j=1 (8) where {e1 , . . , ed } stands for an orthonormal basis in Cd ⊗ Cd , and eij = |ei ej | denotes an orthonormal basis in Md (C) (recall, that Md (C) is a 5th August 2010 40 12:12 WSPC - Proceedings Trim Size: 9in x 6in karp46proc D.

D, it is clear that A ≥ 0. The partial transposition of B reads as follows L B Γ = µ 1 Id ⊗ I d − α=1 Γ (|λ− α | + µ1 )Pα . (51) 5th August 2010 12:12 WSPC - Proceedings Trim Size: 9in x 6in karp46proc Constructing positive maps in matrix algebras 49 Let us recall that the spectrum of the partial transposition of rank-1 projector |ψ ψ| is well know: the nonvanishing eigenvalues of |ψ ψ|Γ are given by s2α (ψ) and ±sα (ψ)sβ (ψ), where s1 (ψ) ≥ . . ≥ sd (ψ) are Schmidt coefficients of ψ. Therefore, the smallest eigenvalue of B Γ (call it bmin ) satisfies L bmin ≥ µ1 − α=1 2 (|λ− α | + µ1 )||ψα ||1 , (52) and using the definition of µ1 (cf.