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arXiv:1402.3768

Quantum entanglement, Calabi-Yau manifolds, and noncommutative algebraic geometry

Shinnosuke Okawa, Kazushi Ueda

Published Feb 16, 2014• 3 citations
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Abstract

We relate SLOCC equivalence classes of qudit states to moduli spaces of Calabi-Yau manifolds equipped with a collection of line bundles. The cases of 3 qutrits and 4 qubits are also related to noncommutative algebraic geometry.
DOI
10.48550/arxiv.1402.3768

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We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by Stochastic Local Operations with Classical Communication (SLOCC) invariant algebraic varieties. The normal forms of the four-qubit classification of Verstraete et al. are interpreted as dense subsets of components of the dual variety of the set of separable states and an algorithm based on the invariants/covariants of the four-qubit quantum states is proposed to identify a state with a SLOCC equivalent normal form (up to qubits permutation).
In the following we discuss how algebraic group actions can be applied to the study of quantum entanglement. The following facts are new: • A method for calculating equations for the SLOCC class of any pure state, with no restrictions on dimensions other than those imposed by computational speed. • A calculation of the equations defining the closure of one 4-partite state, as an example to illustrate the method. • To my knowledge, it has not previously been observed that SLOCC classes equal if their closure is equal, although the fact about algebraic group actions is known. • Similarly, the fact that a SLOCC class inside the closure of another must be of strictly smaller dimension. We also discuss quotients of the space of states by the action, but do not actually compute any or prove new theorems in this area.
We study planar openbooks obtained by lifting braids through branched covers of D^2, together with the quantum operations in the Ising representation. We give a criterion for the Stein fillings to be Calabi-Yau (CY). Among positive factorizations of a fixed monodromy, a CY one has minimal length and its filling minimizes χand b_2. Applied to Baykur's recent examples, this gives a planar contact 3-manifold with infinitely many non-homeomorphic CY fillings, the cover there having degree \ge 6. At degree 4 a single CY filling forces every filling to be CY, and in one case the filling is unique; at degree \le 3 it is unique, and CY under a mild condition. Admissible cuts of D^2 decompose the openbook into subopenbooks. Every positive factorization then localizes, and the CY condition holds exactly when it holds locally. The state space is then a direct sum of tensor products indexed by the compatible parity choices, with at most two qubits per factor when the pieces have degree ≤4\le 4≤4, the same range in which the CY condition depends only on the monodromy. For one degree 4 cover, the points, lines and flags of the two-qubit doily are realized by its subopenbooks. Fifteen liftable braids there share the same quantum operation and Stein filling, and are separated only by the subopenbook systems they admit. A choice of tensor-product structure is thus carried by the lift, not by the braid group representation, suggesting a link between contact topology and quantum entanglement.
In our earlier posting "Matrix Pencils and Entanglement Classification", arXiv:0911.1803, we gave a polynomial-time algorithm for deciding if two states in a space of dimension 2⊗m⊗n2\otimes m\otimes n2⊗m⊗n are SLOCC equivalent. In this note, we point out that a straightforward modification of the algorithm gives a simple enumeration of all SLOCC equivalence classes in the same space, with the class representatives expressed in the Kronecker canonical normal form of matrix pencils. Thus, two states are equivalent if and only if they have the same canonical form. As an example, we present representatives in canonical form for each of the 26 equivalence classes in 2⊗3⊗n2\otimes 3\otimes n2⊗3⊗n systems.
Taking the A₂root system as the sole core symmetric carrier, this paper constructs a cross-disciplinary self-consistent loop of "algebraic coding-geometric realization-quantum stabilization-topological verification". Through structure-preserving mappings and Qiskit quantum simulations, it achieves in-depth integration and empirical verification of 3-dimensional Calabi-Yau (CY) manifolds, quantum stable systems, and the Poincaré conjecture. The core results include: 1) A constructive proof of the exclusive correspondence between the A₂ root system and 3-dimensional CY manifolds—the rank 2 of the root system strictly determines the Hodge number , and the Cartan subalgebra forms an inner product-preserving mapping with the Kähler moduli space; 2) Establishment of the A₂ root system-driven quantum system theory: algebraic preparation of 4 Bell states via the structure-preserving mapping , realization of noise-resistant stabilization with fidelity ≈1.0 in noisy environments based on the dual reflection mechanism of polarized root systems, and further construction of a 2-qubit time crystal (TIME crystal), verifying period doubling (response period 4.0 = 2× driving period 2.0) and spontaneous time-translation symmetry breaking; 3) Proof of the equivalence between supersymmetric score locking () and the simply connectedness of CY manifolds (). Combined with the convergence of Ricci flow curvature (error < 10⁻⁴), it provides a specific instance verification of the Poincaré conjecture under "CY manifold + quantum constraints".