A von Neumann algebra approach to quantum metrics. Quantum - download pdf or read online

By Greg Kuperberg

ISBN-10: 0821853414

ISBN-13: 9780821853412

Quantity 215, quantity 1010 (first of five numbers).

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For example, a single photon has two basis polarization states, so a binary string of length n could be encoded as the polarization of an array of n photons. The basic difference with classical information is the physical possibility of superposing base states. Thus C2 models not a single bit of information, but rather a single “quantum bit” or “qubit”. In quantum error correction one is concerned with the possibility that information encoded in this way could be corrupted. The quantum Hamming metric measures the number of errors that might be introduced.

Let V be a quantum metric on a von Neumann algebra M ⊆ B(H) and let P be a projection in M. Suppose δ(P ) > 0. Then δ(P ) = sup{t ≥ 0 : V˜t = V˜0 } = inf{ρ(P1 , P2 ) : P1 , P2 ≤ P ⊗ I and ρ(P1 , P2 ) > 0} ˜ is the induced pseudometric on P MP and P1 and P2 range over projecwhere V tions in M⊗B(l2 ). Proof. Let K = ran(P ). Observe first that V0 = M implies P V0 P = P M , and that P M = (P MP ) where the commutant on the right is taken in B(K). ) Now ˜ t = P V0 P if 0 ≤ t < δ(P ) W B(K) if t ≥ δ(P ) ˜ t for all t < δ(P ), is a quantum pseudometric on P MP with P Vt P = P V0 P ⊆ W ˜ ˜ ˜ which shows that Vt ⊆ Wt for all t and hence that Vt = P V0 P = V˜0 for all t < δ(P ).

Equivalently, for any > 0 and any vectors v1 , . . , vn , w1 , . . , wn ∈ H the sets [Vs ]1 and [Vsλ ]1 are, respectively, eventually within the -neighborhoods of the sets [Vtλ ]1 and [Vt ]1 for the seminorm |||A||| = | Awi , vi |. Indeed, it would be enough to show this with the vi and wi ranging over a spanning subset of H. The next result is an easy consequence of this characterization. 18. Let {dλ } be a net of pseudometrics on a set X. 5) locally converge to the quantum pseudometric V corresponding to a pseudometric d on X if and only if dλ (x, y) → d(x, y) for all x, y ∈ X.

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A von Neumann algebra approach to quantum metrics. Quantum relations by Greg Kuperberg

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