By Turchin
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A = |eα eβ |), we may compare coefficients, and obtain ψµ , ψν = λµ δµν . Hence eµ = λ−1/2 ψµ is the desired orthonormal system. (2) The existence of the purification is evident if one defines Φ as above, with the orthonormal system eµ chosen in an arbitrary way. Then ρB = µ λµ |eµ eµ |, and the above computation shows that choosing the basis eµ is the only freedom in this construction. But any two bases are linked by a unitary transformation. 1. A state ρ on A ⊗ B is called separable (or “classically correlated”) if it can be written as B λµ ρA µ ⊗ ρµ , ρ= µ B with states ρA µ and ρµ on A and B, respectively, and weights λµ > 0.
E. a version of Bell’s Telephone. Since the time dimension is not represented in this diagram, let us consider the steps in the proper order. The first step is that Alice and Bob each receive one half of an entangled system. The source can be a third party or can be Bob’s lab. The last choice is maybe best for illustrative purposes, because it makes clear that no information is flowing from Alice to Bob at this stage. Alice is next given the quantum system whose state (which is unknown to her) she is to teleport.
Again this leads to a family of maps Tx : B → A such that T : B → C(X) ⊗ A, with T (B) = x ex ⊗ Tx (B). The conditions on {Tx} are Tx : B → A completely positive, and Tx (1I) = 1I . Note that this looks very similar to the conditions for an instrument, but the normalization is different. An interesting special case is a “preparator”, for which A = C is trivial. This prepares B states that depend in an arbitrary way on the classical input x. • Kraus Form. Consider quantum systems with Hilbert spaces HA and HB , and let K : HA → HB be a bounded operator.