Birman—Wenzl—Murakami Algebra and Topological Basis

Abstract

In this paper, we use entangled states to construct 9 × 9-matrix representations of Temperley-Lieb algebra (TLA), then a family of 9 × 9-matrix representations of Birman-Wenzl-Murakami algebra (BWMA) have been presented. Based on which, three topological basis states have been found.And we apply topological basis states to recast nine-dimensional BWMA into its three-dimensional counterpart. Finally, we find the topological basis states are spin singlet states in special case.

State Sampling Dependence of Hopfield Network Inference

“…In this case, another larger mean Hamming distance appears during the state sampling. Since each free energy valley is visited by the Glauber dynamics with a probability proportional to its thermodynamical weight [16], when state transitions occur, the computed average of σ i σ j or σ i over all 2 × 10 4 sampled configurations amounts to the weighted sum, i.e., σ i σ j data = γ w γ σ i σ j γ , σ i data = γ w γ σ i γ where only a few states are considered depending on the actual state transitions in the sampling process and γ is the state index the dynamics visits and w γ is the associated thermodynamical weight and proportional to the exponential of minus its scaled (by β) free energy [12,17]. That is to say, we have now access to the correlations as well as magnetizations in the form of weighted sum.…”

Quantum teleportation and Birman–Murakami–Wenzl algebra

“…It is well-known that how to construct an interesting braid representation (2) (or an interesting solution of the Yang-Baxter equation [10]) is always a meaningful and challenge problem in the research frontier, refer to [9,10,11,12,13,21,34], and the same is true for the construction of an interesting representation of the BMW algebra [16], refer to [9,17,18,19,22,23]. In the previous research [21], a method of constructing the Yang-Baxter gate in the extended Temperley-Lieb diagrammatical approach has been proposed, and hence in this section we want to exploit the extended Temperley-Lieb diagrammatical approach to construct interesting representations of the BMW algebra different from the representation using the Temperley-Lieb projector E (8) and the Yang-Baxter gate B (9).…”

“…where the factor 1 4 is due to the vanishing of two pairs of the cap and cup configurations in the topological straightening deformation and before such the deformation all relevant single-qubit gates have to be moved with the matrix transpose rule (23) of flowing a single-qubit gate.…”

“…In the two spin-1/2 representation of the BMW algebra [22,23], we view the Temperley-Lieb projector as a two-qubit quantum measurement operator and the unitary braid representation as a two-qubit entangling gate (the Yang-Baxter gate). We show that the Temperley-Lieb projector and the Yang-Baxter gate are capable of performing the quantum teleportation protocol, respectively.…”

“…Furthermore, we realize that the tangle relations defining the BMW algebra involving both the Temperley-Lieb projector and the Yang-Baxter gate give rise to the teleportation protocol directly. Moreover, we are able to construct interesting representations of the BMW algebra in the extended Temperley-Lieb diagrammatical approach [20,21] different from the representation in [22,23].…”