Motivated by the novel applications of the mathematical formalism of quantum theory and its generalizations in cognitive science, psychology, social and political sciences, and economics, we extend the notion of the tensor product and entanglement. We also study the relation between conventional entanglement of complex qubits and our generalized entanglement. Our construction can also be used to describe entanglement in the framework of non-Archimedean physics. It is also possible to construct tensor products of non-Archimedean (e.g., p-adic) and complex Hilbert spaces.
We consider a non-negative integer valued grading function on tensor products which aims to measure the extent of entanglement. This grading, unlike most of the other measures of entanglement, is defined exclusively in terms of the tensor product. It gives a possibility to approach the notion of entanglement in a more refined manner, as the non-entangled elements are those of grade zero or one, while the rest of elements with grade at least two are entangled, and the higher its grade, the more entangled an element of the tensor product is. The problem of computing and reducing the grade is studied in products of arbitrary vector spaces over arbitrary fields.
In the framework of quantum mechanics over a quadratic extension of the ultrametric field of p-adic numbers, we introduce a notion of tensor product of p-adic Hilbert spaces. To this end, following a standard approach, we first consider the algebraic tensor product of p-adic Hilbert spaces. We next define a suitable norm on this linear space. It turns out that, in the p-adic framework, this norm is the analogue of the projective norm associated with the tensor product of real or complex normed spaces. Eventually, by metrically completing the resulting p-adic normed space, and equipping it with a suitable inner product, we obtain the tensor product of p-adic Hilbert spaces. That this is indeed the correct p-adic counterpart of the tensor product of complex Hilbert spaces is also certified by establishing a natural isomorphism between this p-adic Hilbert space and the corresponding Hilbert–Schmidt class. Since the notion of subspace of a p-adic Hilbert space is highly nontrivial, we finally study the tensor product of subspaces, stressing both the analogies and the significant differences with respect to the standard complex case. These findings should provide us with the mathematical foundations necessary to explore quantum entanglement in the p-adic setting, with potential applications in the emerging field of p-adic quantum information theory.
Quantum correlations, crucial for the advantage and advancement of quantum science and technology, arise from the impossibility of expressing a quantum state as a tensor product over a given set of parties. In this work, a generalized notion of correlations via arbitrary products is formulated. Remarkably, as a universal property, the connection between such general products and tensor products is established, allowing one to relate generic non-product states to the common notion of entangled states. We construct the set of free operations for general types of products by extending the local-operation-and-classical-communication paradigm, familiar from standard entanglement theory, thereby establishing a resource theory of correlations for general products. A generalization is provided beyond two factors that can be universally related to multipartite entanglement. Applications that highlight the usefulness of the approach are discussed, such as the factorization of fermionic states, the non-local factorization of multi-photon states into single-photon states, and the interesting possibility of understanding prime numbers as a form of single-party entanglement.
The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state
v of a composite system consisting of
d subsystems with
n levels each is viewed as a vector in the
d-fold tensor product of
n-dimensional Hilbert space and can be identified with a tensor with
d indices, each running from
1 to
n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state
v is called {\sl entangled}, if it {\sl cannot} be written in the product form,
v=v1⊗v2⊗⋯⊗vd, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed..
In the context of
p-adic quantum mechanics, we investigate composite systems of
p-adic qubits and
p-adically controlled quantum logic gates. We build on the notion of a single
p-adic qubit as a two-dimensional irreducible representation of the compact
p-adic special orthogonal group SO(3)
p. We show that the classification of these representations reduces to the finite case, as they all factorise through some finite quotient SO(3)
p mod
pk. Then, we tackle the problem of
p-adic qubit composition and entanglement, fundamental for a
p-adic formulation of quantum information processing. We classify the representations of SO(3)
p mod
p, and analyse tensor products of two
p-adic qubit representations lifted from SO(3)
p mod
p. We solve the Clebsch-Gordan problem for such systems, revealing that the coupled bases decompose into singlet and doublet states. We further study entanglement arising from those stable subsystems. For
p=3, we construct a set of gates from
4-dimensional irreducible representations of SO(3)
p mod
p that we prove to be universal for quantum computation.