1,721,324 research outputs found
2-Point Markov Evolutions
Full text linkWe study the Markov evolutions associated to the expected Markov processes
Quasi-invariant states
Full text linkIn this paper, we develop the theory of quasi-invariant (respectively, strongly quasi-invariant) states under the action of a group G of normal & lowast;-automorphisms of a & lowast;-algebra (or von Neumann algebra) A. We prove that these states are naturally associated to left-G-1-cocycles. If G is compact, the structure of strongly G-quasi-invariant states is determined. For any G-strongly quasi-invariant state phi, we construct a unitary representation associated to the triple (A,G,phi). We prove, under some conditions, that any quantum Markov chain with commuting, invertible and Hermitian conditional density amplitudes on a countable tensor product of type I factors is strongly quasi-invariant with respect to the natural action of the group S-infinity of local permutations and we give the explicit form of the associated cocycle. This provides a family of nontrivial examples of strongly quasi-invariant states for locally compact groups obtained as inductive limit of an increasing sequence of compact groups
The First 40 Years of GKSL Generators and Some Proposal for the Future
No full textThis paper consists of two parts. Section 1 contains the main part of the talk given by the first named author at the 48th ToruÅ Symposium. Starting from Sect. 2 we illustrate the general program formulated at the end of Sect. 1 with an example of the simplest low density type interaction
The q q-bit (I): Central limits with left q-Jordan-Wigner embeddings, monotone interacting Fock space, Azema random variable, probabilistic meaning of q
No full textThe qq-bit is the q-deformation of the q-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the q-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of q-deformation (generalizing the Hudson-Parthasarathy bosonization technique and different from the usual one) and the Azema martingale was established by Parthasarathy. Inspired by this result, Schürmann first introduced left and right q-JW-embeddings of M2 (2 × 2 complex matrices) into the infinite tensor product ⊗n in; NM2, proved central limit theorems (CLT) based on these embeddings in the context of∗-bi-algebras and constructed a general theory of q-Levy processes on∗-bi-algebras. For q =-1, left q-JW-embeddings define the Jordan-Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For q = 1, they reduce to the usual tensor embeddings that were at the basis of the first quantum CLT due to von Waldenfels. The present paper is the first of a series of four in which we study these theorems in the tensor product context. We prove convergence of the CLT for all q in; C. The moments of the limit random variable coincide with those found by Parthasarathy in the case q [-1, +1). We prove that the space where the limit random variable is represented is not the Boson Fock space, as in Parthasarathy, but the monotone Fock space in the case q = 0 and a non-trivial deformation of it for q ≠-1, 0, +1. The main analytical tool in the proof is a non-trivial extension of a recently proved multi-dimensional, higher order Cesaro-type theorem. The present paper deals with the standard CLT, i.e. the limit is a single random variable. Paperdeals with the functional extension of this CLT, leading to a process. In paperthe left q-JW-embeddings are replaced by symmetric q-embeddings. The radical differences between the results of the present paper and those ofraise the problem to characterize those CLT for which the limit space provides the canonical decomposition of all the underlying classical random variables (see the Introduction, Lemma 4.5 and Sec. 5 of the present paper for the origin of this problem). This problem is solved in the paperfor CLT associated to states satisfying a generalized Fock property. The states considered in this series have this property
The qq-bit (III): Symmetric q-Jordan-Wigner embeddings
Full text linkWe prove that, replacing the left Jordan-Wigner q-embedding by the symmetric q-embedding described in Sec. 2, the result of the corresponding central limit theorem changes drastically with respect to those obtained in Ref. 5. In fact, in the former case, for any q ∈ C, the limit space is precisely the 1-mode Interacting Fock Space (IFS) that realizes the canonical quantum decomposition of the limit classical random variable. In the latter case, this happens if and only if q = ±1. Furthermore, as shown in Sec. 4, the limit classical random variable turns out to coincide with the 1-mode version of the q Λ -deformed quantum Brownian introduced by Parthasarathy 8,9 and extended to the general context of bi-algebras by Schürman 10,11 . The last section of the paper (Appendix) describes this continuous version in white noise language, leading to a simplification of the original proofs, based on quantum stochastic calculus
On phase transitions in Quantum Markov Chains on Cayley tree
No full textIn the present paper we continue our investigations started in [Accardi L. , Ohno, H. , Mukhamedov, F. , Quantum Markov fields on graphs, Inf. Dim. Analysis, Quantum Probab. Related Topics (accepted) arxi v: 0911 . 1667]. In [Accardi L., Mukhamedov, F., Saburov M. On Quantum Markov chains on Cayley tree and associated chains with XY-model arxiv: 1004.3623] we provided a construction of forward and backward Quantum Markov Chains (QMC) defined on the Cayley
tree, and established uniqueness of QMC associated with XY-model on a Cayley tree order 2. In the present paper we study the same model on a Cayley tree order 3. Surprisingly in this case, we establish a phase transition (i.e. existence of two distinct quantum Markov chains) for the considered model on the Cayley tree order 3
Emergence of Quantum Theories from Classical Probability: Historical Origins, Developments, and Open Problems
Full text linkAfter briefly mentioning some achievements of quantum probability
during the past 20 years, we concentrate our attention on the emergence
of natural extensions of usual quantum theory from the combination of classical
probability with the theory of orthogonal polynomials and we discuss
some implications, both for mathematics and physics of this fact
Quantum Theories Associated to Increasing Hilbert Space Filtrations and Generalized Jacobi 3–Diagonal Relation
Full text linkWe prove that the quantum decomposition of a classical random
variable, or random field, is a very general phenomenon involving only an
increasing filtration of Hilbert spaces and a family of Hermitean operators
increasing by 1 the filtration. The creation, annihilation and preservation
operators (CAP operators), defining the quantum decomposition of these
Hermitean operators, satisfy commutation relations that generalize those of
usual quantum mechanics. In fact there are two types of commutation relations
(Type I and Type II). In Type I commutation relations the commutator
is given by an operator–valued sesqui–linear form. The case when this
operator–valued sesqui–linear form is scalar valued (multiples of the identity)
characterizes the non–relativistic free Bose field and the associated commutation
relations reduce to the Heisenberg ones. Type II commutation relations
did not appear up to now because they are identically satisfied when the
probability distribution of the random field is a product measure. In this
sense they encode information on the self–interaction of the random field
The n-Dimensional Quadratic Heisenberg Algebra as a “Non–Commutative” sl(2,C)
Full text linkWe prove that the commutation relations among the generators
of the quadratic Heisenberg algebra of dimension , look like a kind of non-commutative extension of sl(2,C) (more precisely of its unique 1–
dimensional central extension), denoted and called the complex
n–dimensional quadratic Boson algebra. This non-commutativity has a dif-
ferent nature from the one considered in quantum groups. We prove the
exponentiability of these algebras (for any n) in the Fock representation. We
obtain the group multiplication law, in coordinates of the first and second
kind, for the quadratic Boson group and we show that, in the case of the
adjoint representation, these multiplication laws can be expressed in terms of
a generalization of the Jordan multiplication. We investigate the connections
between these two types of coordinates (disentangling formulas). From this
we deduce a new proof of the expression of the vacuum characteristic function
of homogeneous quadratic boson fields
The Vacuum Distributions of the Truncated Virasoro Fields are Products of Gamma Distributions
No full textIn a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables
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