Quantum and non-causal stochastic calculus

Lindsay, J. Martin

doi:10.1007/bf01199312

Cited by 37 publications

(40 citation statements)

References 26 publications

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“…exponential product systems, cf. [36]. Since we obtain a similar ∑ -lemma for quasistationary quasifactorizing measures µ (at least if µ * µ ∼ µ) below, we expect a calculus on the product systems E M too.…”

Section: Example 44 Followingmentioning

confidence: 99%

“…Below, there are indications that there should exist a quantum stochastic calculus on type II product systems too. E.g., we can derive a variant of the ∑ lemma of Boson stochastic calculus [36] for stationary factorizing measure types of random closed sets, see Corollary 4.12. In the context of dilation theory, we should mention that the recent notion of tensor product systems of Hilbert modules of BHAT and SKEIDE [12] is deeply related to this topic too.…”

Section: ∈ P)mentioning

confidence: 99%

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Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces

Liebscher¹

2009

Memoirs of the AMS

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In a series of papers [66,65,62,63,64] TSIRELSON constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by ARVESON [4] for classifying E 0 -semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So we connect each continuous tensor product systems of Hilbert spaces with measure types of distributions of random (closed) sets in [0, 1] or R + . These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the TSIRELSON examples, that the classification scheme for product systems into types I n , II n and III is not complete. Moreover, based on a detailed study of this kind of measure types, we construct for each stationary factorizing measure type a continuous tensor product systems of Hilbert spaces such that this measure type arises as the before mentioned invariant.These results are a further step in the classification of all (separable) continuous tensor product systems of Hilbert spaces of type II in completion to the classification of type I done by [4] and combine well with other invariants like the lattice of product subsystems of a given product system. Although these invariants relate to type II product systems mainly, they are of general importance. Namely, the measure types of the above described kind are connected with representations of the corresponding L ∞ -spaces. This leads to direct integral representations of the elements of a given product system which combine well under tensor products. Using this structure in a constructive way, we can relate to any (type III) product system a product system of type II 0 preserving isomorphy classes. Thus, the classification of type III product systems reduces to that of type II (and even type II 0 ) ones.In this circle of ideas it proves useful that we reduce the problem of finding a compatible measurable structure for product systems to prove continuity of one periodic unitary group on a single Hilbert space. As a consequence, all admissible measurable structures (if there are any) on an algebraic continuous tensor product systems of Hilbert spaces yield isomorphic product systems. Thus the measurable structure of a continuous tensor product systems of Hilbert spaces is essentially determined by its algebraic one.

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Section: Example 44 Followingmentioning

confidence: 99%

Section: ∈ P)mentioning

confidence: 99%

Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces

Liebscher¹

2009

Memoirs of the AMS

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“…The three basic quantum stochastic processes become LððEÞ; ðEÞ Ã Þ-valued continuous (in fact, infinitely many times differentiable) maps defined on R. Moreover, so is the quantum white noise fa t ; a where the right-hand sides are in terms of the classical stochastic gradient and the classical Hitsuda-Skorohod integral. Thus, our definitions of the Hitsuda-Skorohod quantum stochastic integrals coincide with the ones introduced by Belavkin [3] and Lindsay [15] for a common integrand. In fact, their definition starts with the right-hand sides of (1:3) for suitably chosen Ä and .…”

Section: Introductionmentioning

confidence: 76%

“…Beyond the Itô type, non-adapted (or non-causal) quantum stochastic integrals have been discussed by Belavkin [3] and Lindsay [15]. In this paper, we propose a functional analytic method (kernel theorem and duality, see e.g., Treves [22]) for further investigation, in particular, towards regularity problems of quantum stochastic integrals.…”

Section: Introductionmentioning

confidence: 99%

Quantum Stochastic Gradients

Obata

2009

IIS

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“…We will introduce the Fock space in a way adapted to the language of counting measures. For details we refer to [6,7,8,2,9] and other papers cited in [8].…”

Section: Basic Notions and Notationsmentioning

confidence: 99%

Quantum Teleportation with Entangled States¶Given by Beam Splittings

Fichtner¹,

Ohya

2001

Communications in Mathematical Physics

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Quantum teleportation is rigorously discussed with coherent entangled states given by beam splittings. The mathematical scheme of beam splitting has been used to study quantum communication [2] and quantum stochastic [8]. We discuss the teleportation process by means of coherent states in this scheme for the following two cases: (1) Delete the vacuum part from coherent states, whose compensation provides us a perfect teleportation from Alice to Bob. (2) Use fully realistic (physical) coherent states, which gives a non-perfect teleportation but shows that it is exact when the average energy (density) of the coherent vectors goes to infinity.

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Quantum and non-causal stochastic calculus

Cited by 37 publications

References 26 publications

Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces

Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces

Quantum Stochastic Gradients

Quantum Teleportation with Entangled States¶Given by Beam Splittings

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