The Quantum Entanglement of Bipolar Sequences

Classication of dieren t forms of quan- tum entanglement is an active area of research, central to the development of eectiv e quantum computers, and similar to classication of error-correction codes, where the concept of code duality is broadened to equivalence under all 'local' unitary transforms. We examine links between entanglement and coding the- ory by forming Algebraic Normal Form (ANF) de- scriptions for bipolar indicator sequences to describe binary codes with codewords which occur with bipolar probabilities. Quadratic entanglement is the basis of particle-entangling arrays found in recent literature. I. Definition of Entanglement Recent interest in Quantum Computation has fuelled a desire to understand Quantum Entanglement. Entanglement exists between any two or more systems if their joint probability state cannot be factorised using the tensor product. Consider two qubits, x0 and x1. Their joint probability state is given by, s = (s0; s1; s2; s3) where the si are complex and P 3=0 jsij 2 = 1. There is a prob- ability of js0j 2 ,js1j 2 ,js2j 2 ,js3j 2 of measuring the two qubits in states 00; 01; 10; 11, respectively. If s can be written as (a0; b0) (a1; b1), then s is tensor-factorisable and the two qubits are not entangled. Conversely, if we cannot write s in the above form then the two qubits are entangled. This idea generalises in an obvious manner to m qubits. Conven- tional (classical) computers only ever use tensor-factorisable space of physical matter. But the existence of entanglement between quantum particles allows us to store and operate on exponentially larger data vectors than possible classically. II. Entanglement and Error-Correction Codes Here and in the rest of the paper normalisation of the joint- state vector is omitted for clarity. Normalisation would ensure P2n 1 i=0 jsij