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By Wernerz R.F.

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A 54 (1996) 2614 [33] P. Shor: “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM J. Computing 26 (1997) 1484 (quant-ph/9508027). J. F. Werner: “Maximal violation of Bell’s inequality is generic in quantum field theory”, Commun. Math. Phys. 110 (1987) 247-259 [35] H. Halvorson and R. Phys. 41 (2000) 1711-1717 [36] W. Arveson: “Subalgebras of C*-algebras”, Acta math. 123 (1969) 141 - 224 [37] T. Beth, D. Jungnickel, H. Lenz: Design Theory, 2nd edition, Encyclopedia of Mathematics and Its Applications , Vols.

For dense coding case we get the same result, although along other routes. 40) follows easily by writing the teleportation equation as | Ω, (Ux∗ ⊗ 1I)Φx |2 = δxy . The problem is to show that Ω has to be maximally entangled. Using the reduced density operator ω1 of ω, this becomes tr(ω1 Ux∗ Uy ) = Ω, (Ux∗ Uy ⊗ 1I)Ω = Φx , Φy = δxy . 42) We claim that this equation, for a positive operator ω1 , and d2 unitaries Ux , implies that ω1 = d−1 1I. To see this, expand the operator A = |φ ek |ω1−1 in the basis Ux according to the formula A = x Ux tr(Ux∗ Aω1 ): ek , Ux∗φ Ux = |φ ek |ω1−1 .

38) for all φ, ψ, and coefficients which must satisfy x |λx|2 = 1. Note how in this equation a scalar product between the vectors in the first and third tensor factor is generated. This type of equation, which is clearly the core of the teleportation process may be solved in general: Lemma 9 Let H, K be finite dimensional Hilbert spaces, and let Ω1 ∈ K ⊗ H and Ω2 ∈ H ⊗ K be unit vectors such that, for all φ, ψ ∈ H, φ ⊗ Ω1 , Ω2 ⊗ ψ = λ φ, ψ . 39) Then |λ| ≤ 1/ dim H with equality iff Ω1 and Ω2 are maximally entangled and equal up to the exchange of the tensor factors H and K.

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