Universal entanglement-inspired correlations

Abstract

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.

Figures (2)

• Figure 1: Universal property as commuting diagram. The factors combined with the bilinear product $\circ$ result in the same outcome as the linear map $L^\circ$ acting on their tensor product $\otimes$.
• Figure 2: Product state for product $|n_A\rangle\circ|n_B\rangle=|n_A\cdot n_B\rangle$, where we consider unnormalized states $|\psi_A\rangle=|\psi_B\rangle=\sum_{n\geq 2} 1 |n\rangle$ as factors describing Alice's and Bob's state. The resulting components of $|\psi_A\circ\psi_B\rangle=\sum_{n\geq 2} c_n |n\rangle$ are shown, where $c_n=\langle n|\psi_A\circ\psi_B\rangle$. Vertical lines indicate prime $n$, occurring with a zero amplitudes in product states. In general, the amplitude tells us the number of ways one can write as $n=n_A\cdot n_B$, with $n_A,n_B\geq 2$.