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Boris Ischi

Publications and source records attributed to Boris Ischi.

6 recordsLinked to original sources
arXiv2011

Orthocomplemented weak tensor products

Let L_1 and L_2 be complete atomistic lattices. In a previous paper, we have defined a set S=S(L_1,L_2) of complete atomistic lattices, the elements of which are called weak tensor products of L_1 and L_2. S is defined by means of three axioms, natural regarding the description of some compound systems in quantum logic. It has been proved that S is a complete lattice. The top element of S, denoted by L_1 v L_2, is the tensor product of Fraser whereas the bottom element, denoted by L_1 ^ L_2, is the box product of Graetzer and Wehrung. With some additional hypotheses on L_1 and L_2 (true for instance if L_1 and L_2 are moreover orthomodular with the covering property) we prove that S is a singleton if and only if L_1 or L_2 is distributive, if and only if L_1 v L_2 has the covering property. Our main result reads: L in S admits an orthocomplementation if and only if L=L_1 ^ L_2. At the end, we construct an example in S which has the covering property.

math.LO↗
arXiv2005

Orthocomplementation and compound systems

In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice L_sep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different ``rooms'' of the lab, and before any interaction takes place. In that case the state of the compound system is necessarily a product state. As a consequence, Dirac's superposition principle fails, and therefore L_sep cannot satisfy all Piron's axioms. In previous works, assuming that L_sep is orthocomplemented, it was argued that L_sep is not orthomodular and fails to have the covering property. Here we prove that L_sep cannot admit and orthocomplementation. Moreover, we propose a natural model for L_sep which has the covering property.

quant-ph↗
arXiv2005

A characterization of the Aerts product of Hilbertian lattices

Let H_1 and H_2 be complex Hilbert spaces, L_1=P(H_1) and L_2=P(H_2) the lattices of closed subspaces, and let L be a complete atomistic lattice. We prove under some weak assumptions relating L_i and L, that if L admits an orthocomplementation, then L is isomorphic to the separated product of L_1 and L_2 defined by Aerts. Our assumptions are minimal requirements for L to describe the experimental propositions concerning a compound system consisting of so called separated quantum systems. The proof does not require any assumption on the orthocomplementation of L.

math-ph↗
arXiv2003

Weak tensor products of complete atomistic lattices

Given two complete atomistic lattices L_1 and L_2, we define a set S(L_1,L_2) of complete atomistic lattices by means of three axioms (natural regarding the description of separated quantum compound systems), or in terms of a universal property with respect to a given class of bimorphisms. We call the elements of S(L_1,L_2) weak tensor products of L_1 and L_2. We prove that S(L_1,L_2) is a complete lattice. We compare the bottom element with the separated product of Aerts and with the box product of Graetzer and Wehrung. Similarly, we compare the top element with the tensor products of Fraser, Chu and Shmuely. With some additional hypotheses on L_1 and L_2 (true for instance if L_1 and L_2 are moreover irreducible, orthocomplemented and with the covering property), we characterize the automorphisms of weak tensor products in terms of those of L_1 and L_2.

math.LO↗
arXiv2003

Property lattices for independent quantum systems

We consider the description of two independent quantum systems by a complete atomistic ortho-lattice (cao-lattice) L. It is known that since the two systems are independent, no Hilbert space description is possible, i.e. $L\ne P(H)$, the lattice of closed subspaces of a Hilbert space (theorem 1). We impose five conditions on L. Four of them are shown to be physically necessary. The last one relates the orthogonality between states in each system to the ortho-complementation of L. It can be justified if one assumes that the orthogonality between states in the total system induces the ortho-complementation of L. We prove that if L satisfies these five conditions, then L is the separated product proposed by Aerts in 1982 to describe independent quantum systems (theorem 2). Finally, we give strong arguments to exclude the separated product and therefore our last condition. As a consequence, we ask whether among the ca-lattices that satisfy our first four basic necessary conditions, there exists an ortho-complemented one different from the separated product.

quant-ph↗