## Generalization of the Hughston-Jozsa-Wootters Theorem to Hyperfinite von Neumann Algebras (2003)

Citations: | 5 - 1 self |

### BibTeX

@MISC{Halvorson03generalizationof,

author = {Hans Halvorson},

title = {Generalization of the Hughston-Jozsa-Wootters Theorem to Hyperfinite von Neumann Algebras },

year = {2003}

}

### OpenURL

### Abstract

The Hughston-Jozsa-Wootters (HJW) theorem entails that any finite ensemble compatible with a given density operator can be prepared from a fixed initial state by operations on a spacelike separated system. In this paper, we generalize the HJW theorem to the case of arbitrary measures on the state space of a von Neumann algebra with hyperfinite commutant. In doing so, we also show that every POV measure with range in a hyperfinite von Neumann algebra induces a local, completely positive instrument.

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Citation Context ...not change her commitment, and such that B cannot determine A’s commitment until given some sort of key by A. An initial protocol using pairs of polarized photons was proposed by Bennett and Brassard =-=[3]-=-; however, Bennett and Brassard showed that the protocol is insecure due to the existence of a cheating strategy that exploits the nonlocal correlations of the EPR-Bohm state. A number of other bit co... |

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Citation Context ... Recall that for each measure µ on K, there is an xµ ∈ K such that xµ = ∫ K ωdµ(ω); xµ is said to be the barycenter of µ. Furthermore, two measures µ,ν on K are said to be equivalent if xµ = xν. (See =-=[1, 4]-=- for more details.) In each of the following examples, we construct a pair of equivalent continuous measures on K. These measures could be used to encode bits using a recipe similar to that used by Be... |

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Citation Context ... Recall that for each measure µ on K, there is an xµ ∈ K such that xµ = ∫ K ωdµ(ω); xµ is said to be the barycenter of µ. Furthermore, two measures µ,ν on K are said to be equivalent if xµ = xν. (See =-=[1, 4]-=- for more details.) In each of the following examples, we construct a pair of equivalent continuous measures on K. These measures could be used to encode bits using a recipe similar to that used by Be... |

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Citation Context ...ed that there can be no unconditionally secure bit commitment protocol. His proof depends on the following fact, which was first proved rigorously and generally by Hughston, Jozsa, and Wootters (HJW) =-=[10]-=-: HJW Theorem. Let H1, H2 be Hilbert spaces, let D be a density operator on H1, and let ψ be a unit vector in H1 ⊗ H2 such that TrH2 (|ψ〉〈ψ|) = D. Then, for any convex decomposition D = ∑ n i=1 λiDi, ... |

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Citation Context ...t show that the POV measure Aµ gives rise to a measurement operation that can be performed locally to R ′ . For this, we need to recall the definition of a completely positive (CP) instrument on B(H) =-=[6, 15]-=-. Definition. If (X,Σ) is a Borel space, a CP instrument on B(H) with value space (X,Σ) is a map E : Σ × B(H) → B(H) such that 1. for fixed B ∈ B(H), E( · ,B) is σ-additive in the ultraweak topology; ... |

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Citation Context ...the context of the quantum theory of the free Bose field, the algebra R = R(W) associated with a Rindler wedge is the hyperfinite type III1 factor, and the vacuum ρ is a faithful, mixed state of R(W) =-=[20]-=-. Since KR is not a Choquet simplex, there are distinct extremal measures on KR with barycenter ρ. Thus, a fully general version of the HJW theorem would show that any of of these “mixtures” (i.e., me... |

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17 | Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields on curved spacetime - Verch - 1997 |

16 | Generic Bell correlation between arbitrary local algebras in quantum field theory
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(Show Context)
Citation Context ... hyperfinite type II1 factor [20]) where all normal states of B(H) are nonseparable across (R, R ′ ). (A state is nonseparable if it is not contained in the weak* closed convex hull of product states =-=[8]-=-.) Therefore, in these cases, non-normal state transformations are needed to transform non-separable states to separable states. Each instrument E determines a POV measure A via the formula A(S) ≡ E(S... |

12 | Causal nets of operator algebras, Akademie - Baumgärtel, Wollenberg - 1992 |

12 |
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(Show Context)
Citation Context ...stablish this result for the case where R has property P. A von Neumann algebra R has property P just in case, for each B ∈ B(H), the weakly closed convex hull of {UBU ∗ : U ∈ R, U unitary} meets R ′ =-=[17]-=-. In what follows, we write coR(B) − for the weakly closed convex hull of {UBU ∗ : U ∈ R, U unitary}. Thus, R has property P if for each B ∈ B(H), coR(B) − ∩ R ′ is nonempty. Proposition 2. Let R be a... |

6 |
Infinitely entangled states
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Citation Context ...ausal spacetime, if the net satisfies the split property, then each R(O) is hyperfinite [2, Prop. 17.2.1]. Finally, for an exotic case with direct relevance to quantum information theory, Keyl et al. =-=[11]-=- have shown that “infinitely entangled” states are naturally represented as states across a pair (R, R ′ ) of hyperfinite type II1 factors. Acknowledgments This work was motivated by discussions with ... |

5 | On the decompositions of a quantum state
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(Show Context)
Citation Context ...〉〈ψ| by a measurement operation on H2. Of course, if D is not pure, then D also has infinite convex decompositions. For the case of countably infinite decompositions, the results of Cassinelli et al. =-=[5]-=- provide the appropriate generalization of the HJW theorem. However, the most general case — namely, integral decompositions of states — is not covered by Cassinelli et al.’s results. Furthermore, the... |

5 |
An Invitation to the Algebra of Canonical Commutation Relations (Leuven
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(Show Context)
Citation Context ...n δeK, and if xµ is normal, then µ(S) = 0 for every countable S ⊆ δeK. Example (Equivalent measures on the state space of a type II1 factor). Let A[R 2 ] be the Weyl algebra for one degree of freedom =-=[16, 18]-=-. That is, A[R 2 ] is the C ∗ -algebra generated by unitary operators W(a), a ∈ R 2 , satisfying the Weyl relations W(a)W(b) = exp{iσ(a,b)}W(a + b), (2) with σ((a,b),(a ′ ,b ′ )) = 1 2 (ab′ − ba ′ ), ... |

4 | The Einstein-Podolsky-Rosen State Maximally Violates
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(Show Context)
Citation Context ...ba ′ ), a,b ∈ R. (3) Let ω be the unique tracial state of A[R2 ], and let (πω, Hω,Ω) be the GNS representation of A[R2 ] induced by ω. Then R ≡ πω(A[R2 ])′′ and R ′ = πω(A[R2 ])′ are type II1 factors =-=[9, 18]-=-. Let A be the abelian von Neumann subalgebra of R generated by {π(W(a,0)) : a ∈ R}, and let B be the subalgebra generated by {π(W(0,b)) : b ∈ R}. Let D(A) ⊆ KR be the set of states that are dispersio... |

4 |
Collapse postulate for observables with continuous spectra
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(Show Context)
Citation Context ...weak topology; 2. for fixed S ∈ Σ, E(S, · ) is a CP map such that E(S,I) ≤ I. If in addition the CP maps E(S, · ), (S ∈ Σ), are ultraweakly continuous, then E is said to be normal. Following Srinivas =-=[19]-=-, we do not generally impose the normality condition, since this condition rules out a priori the possibility of generalizing Lüders’ rule to continuous spectrum observables. Furthermore, if the CP ma... |

4 |
Harmonic analysis on locally compact groups
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(Show Context)
Citation Context .... Finally, if R is a set of operators in B(H), we let [Rψ] denote the closed linear span in H of the set {Aψ : A ∈ R}. The following result is essentially a special case of a theorem proved by Tomita =-=[21]-=- in 1956 (compare with [4, Lemma 4.1.21, Prop. 4.1.22]). Proposition 1. Let R be a von Neumann algebra acting on H. Let ψ be a unit vector in H, let ρ = ωψ|R, and let µ be a measure on KR with barycen... |

3 |
Semicausal operations are semilocalizable,” Europhys
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(Show Context)
Citation Context ...same type (I,II, or III) as R [12, Thm. 9.1.3, Chap.11]); and therefore E ⊗ ι does not necessarily define an instrument on B(H). We are led then to the following more general definition (compare with =-=[7]-=-). Definition. Let R be a von Neumann algebra acting on a Hilbert space H. We say that an instrument E on B(H) is local to R just in case E(S,B) = E(S,I)B, for all B ∈ R ′ . When A is atomic, there is... |