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arXiv:2104.10597 · Linear and Multilinear Algebra

Entanglement and products

Tatyana Barron, Noah Wheatley

Published Mar 5, 2022• 0 citations
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Abstract

We address a general question whether geometry of submanifolds of an integral compact Kähler manifold is characterized by invariants that come from analysis. In geometric quantization, we have an integral compact Kähler manifold M and a holomorphic line bundle L on this manifold. There is a known procedure how to associate a sequence of mixed states (ρN), N=1,2,3,…, to a submanifold Λ of M. Do analytic properties of this sequence reflect the geometry of Λ ? In this paper, we consider the case when M is a product of two integral compact Kähler manifolds. We show that, when Λ is a product submanifold of M, then the entanglement of formation of ρN is zero for all sufficiently large N.

DOI
10.1080/03081087.2022.2043817

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We address a general question whether geometry of submanifolds of an integral compact Kähler manifold is characterized by invariants that come from analysis. In geometric quantization, we have an integral compact Kähler manifold M and a holomorphic line bundle L on this manifold. There is a known procedure how to associate a sequence of mixed states (ρN), N=1,2,3,…, to a submanifold Λ of M. Do analytic properties of this sequence reflect the geometry of Λ ? In this paper, we consider the case when M is a product of two integral compact Kähler manifolds. We show that, when Λ is a product submanifold of M, then the entanglement of formation of ρN is zero for all sufficiently large N.
We associate quantum states with subsets of a product of two compact connected Kähler manifolds M1M_1M1​ and M2M_2M2​. To associate the quantum state with the subset, we use the map that restricts holomorphic sections of the quantum line bundle over the product of the two Kähler manifolds to the subset. We present a description of the kernel of this restriction map when the subset is a finite union of products. This in turn shows that the quantum states associated with the finite union of products are separable. Finally, for every pure state and certain mixed state, we construct subsets of M1×M2M_1\times M_2M1​×M2​ such that the states associated with these subsets are the original states, to begin with.
Various problems concerning the geometry of the space u ∗ (H) of Hermitian operators on a Hilbert space H are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into Kähler submanifolds. It is also shown that the space D(H) of density states on an n-dimensional Hilbert space H is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space D k (H) of rank-k states, k = 1,...,n, is a smooth manifold of (real) dimension 2nk −k 2 −1 and this stratification is maximal in the sense that every smooth curve in D(H), viewed as a subset of the dual u ∗ (H) to the Lie algebra of the unitary group U(H), at every point must be tangent to the strata D k (H) it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition H = H 1 ⊗ H 2, an abstract criterion of entanglement is proved.
Various problems concerning the geometry of the space of Hermitian operators on a Hilbert space are addressed. In particular, we study the canonical Poisson and Riemann–Jordan tensors and the corresponding foliations into Kähler submanifolds. It is also shown that the space of density states on an n -dimensional Hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space of rank- k states, k = 1, ..., n , is a smooth manifold of (real) dimension 2 nk − k 2 − 1 and this stratification is maximal in the sense that every smooth curve in , viewed as a subset of the dual to the Lie algebra of the unitary group , at every point must be tangent to the strata it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition , an abstract criterion of entanglement is proved.
Quantum mechanics, gauge theory, and general relativity have recently been derived as regime-dependent limits of gradient flow on a compact Kähler manifold. In that framework, six structural conditions identified the product geometry M = Fl₁,₂(ℂ³) × ℂP¹ as the unique candidate consistent with physical observations. This paper shows that those six conditions reduce to two axioms: (1) Physical dynamics is the negative gradient flow of a moment map component on a compact Kähler manifold.(2) Each irreversible recording event is a single, irreducible binary act on ℂP¹. From these axioms, three results follow. First, the admissible geometry is uniquely determined. The irreducibility requirement forces the tangent weight lattice to be non-decomposable, eliminating all compact Kähler threefolds except the flag manifold Fl₁,₂(ℂ³). Combined with the ℂP¹ recording sector, this yields the unique solution M = Fl₁,₂(ℂ³) × ℂP¹. Second, quantum entanglement is identified as a topological property rather than a dynamical one. The cohomology ring of Fl₁,₂(ℂ³) satisfies the Weyl relation x² + xy + y² = 0, which prevents factorization of internal quantum numbers. Entanglement therefore originates from geometric irreducibility of the configuration space, not from state preparation. Third, the recording sphere ℂP¹_rec and the canonical fiber ℂP¹_fib of the flag manifold are shown to be isomorphic as Hamiltonian U(1) spaces. Measurement is thus internal to the geometry: the recorder is realized as a fiber of the configuration space itself. The observer–system distinction is topologically obstructed. The framework eliminates the landscape of possible geometries and introduces no free parameters. The Standard Model geometry emerges as the unique fixed point of a self-consistency condition in which the recording structure determines the manifold and the manifold returns the recording structure.
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In this paper, we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is as follows: we embed the smooth manifold of real dimension [Formula: see text] into [Formula: see text] and induce the quantizations from there. The standard way by which reproducing kernel Hilbert spaces are defined on submanifolds gives a way to define the pullback coherent states. In Berezin-type quantization, the Hilbert space of quantization is the pullback (by the embedding) of the Hilbert space of geometric quantization of [Formula: see text]. In the Odzijewicz-type quantization, one has to consider a tensor product of the geometric quantization line bundle with holomorphic [Formula: see text]-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space [Formula: see text]. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Ghosh. In both Berezin- and Odzijewicz-type quantizations, we first exhibit this quantization explicitly on [Formula: see text] and we induce the quantization on the smooth compact embedded manifold from [Formula: see text].