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.Comment: Reports on Mathematical Physics, Vol. 50 no. 2 (2002), p. 155-16

Weak tensor products of complete atomistic lattices

Abstract.: Given two complete atomistic lattices $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ , we define a set $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ 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 $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ weak tensor products of $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ . We prove that $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ is a complete lattice. We compare the bottom element $${\mathcal{L}}_{1}$$ $${\mathcal{L}}_{2}$$ with the separated product of Aerts and with the box product of Grätzer and Wehrung. Similarly, we compare the top element $${\mathcal{L}}_{1}$$ $${\mathcal{L}}_{2}$$ with the tensor products of Fraser, Chu and Shmuely. With some additional hypotheses on $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ (true for instance if $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ are moreover irreducible, orthocomplemented and with the covering property), we characterize the automorphisms of weak tensor products in terms of those of $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$

Decoherence in a N-qubit solid-state quantum computer

We investigate the decoherence process for a quantum register composed of N qubits coupled to an environment. We consider an environment composed of one common phonon bath and several electronic baths. This environment is relevant to the implementation of a charge based solid-state quantum computer. We explicitly compute the time evolution of all off-diagonal terms of the register's reduced density matrix. We find that in realistic configurations, "superdecoherence" and "decoherence free subspaces" do not exist for an N-qubit system. This means that all off-diagonal terms decay like exp(-q(t)N), where q(t) is of the same order as the decay function of a single qubit.Comment: 11 page

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.Comment: Submitted for the proceedings of the 2004 IQSA's conference in Denver. Revised versio