Local Distinguishability of Generic Unentangled Orthonormal Bases

TL;DR

This paper characterizes the geometric structure of unentangled orthonormal bases in multi-qubit systems, identifying which are distinguishable by local operations and classical communication, and constructs generalized examples for any number of qubits.

Contribution

It provides a geometric and algebraic description of the space of unentangled orthonormal bases and characterizes those that are locally distinguishable, extending previous examples to arbitrary qubit numbers.

Findings

The space of UOB forms a bouquet of products of Riemann spheres.

Maximum dimensional components correspond to locally distinguishable bases.

Constructed generalized 3-qubit UOB examples that are not LOCC distinguishable.

Abstract

An orthonormal basis consisting of unentangled (pure tensor) elements in a tensor product of Hilbert spaces is an Unentangled Orthogonal Basis (UOB). In general, for $n$ qubits, we prove that in its natural structure as a real variety, the space of UOB is a bouquet of products of Riemann spheres parametrized by a class of edge colorings of hypercubes. Its irreducible components of maximum dimension are products of $2^n-1$ two-spheres. Using a theorem of Walgate and Hardy, we observe that the UOB whose elements are distinguishable by local operations and classical communication (called locally distinguishable or LOCC distinguishable UOB) are exactly those in the maximum dimensional components. Bennett et al, in their in-depth study of quantum nonlocality without entanglement, include a specific 3 qubit example UOB which is not LOCC distinguishable; we construct certain generalized counterparts of this UOB in $n$ qubits.