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Register a new userLocal Distinguishability of Generic Unentangled Orthonormal Bases
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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.
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Gleasons theorem asserts the equivalence of von Neumanns density operator formalism of quantum mechanics and frame functions, which are functions on the pure states that sum to 1 on any orthonormal basis of Hilbert space of dimension at least 3. The unentangled frame functions are initially only defined on unentangled (that is, product) states in a multi-partite system. The third authors Unentangled Gleasons Theorem shows that unentangled frame functions determine unique density operators if and only if each subsystem is at least 3-dimensional. In this paper, we determine the structure of unentangled frame functions in general. We first classify them for multi-qubit systems, and then extend the results to factors of varying dimensions including countably infinite dimensions (separable Hilbert spaces). A remarkable combinatorial structure emerges, suggesting possible fundamental interpretations.
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