Matrix Decompositions and Entanglement Dynamics

Stephen S. Bullock, Gavin K. Brennen

Div. 891, ITL; Div. 842, PL

Given a closed system of quantum data, quantum computations are mathematically modeled by exponentially large unitary matrices. A matrix decomposition is an algorithm for factoring matrices, and in this context such an algorithm splits a quantum computation into a sequence of smaller, hopefully simpler subcomputations.

This poster describes the concurrence canonical decomposition (**C.C.D.**), a unitary matrix decomposition developed by the authors. The matrix decomposition writes a computation *u* as a product of three factors:

*u=k _{1} a k_{2}*. Only the central factor has the ability to change the concurrence, a quantitative measure of the entanglement of a quantum data state. Thus, we may use the decomposition to study how entanglement changes over the course of a given quantum computation. Such entanglement dynamics are of great interest, since it is the entanglement of quantum data that allows theoretical quantum computers to outperform classical computers.

The construction also leads to a deeper understanding of the concurrence itself, relating it to a time-reversal symmetry of quantum angular-momentum, i.e. a quantum-bit (qubit) flip. Further, the spin interpretation explains a surprising dissimilarity of the even-qubit and odd-qubit **C.C.D.**’s. Specifically, this variation reflects a Kramer’s degeneracy. Such a degeneracy arises since the total spin of the n-qubit system is integral or half-integral as n is even or odd.

Presenting author:

Stephen S. Bullock

MCSD (div 891)

Information Technology Lab

Room 383, mail stop 8910

X4793

Fax: (301) 990-4127

Not sigma chi member

Subject: Mathematics (better: mathematical physics)