Appendix E - Density Operator: Extensions

Introduction

The Bloch sphere picture for two state systems given in Section 3.5.4 is quite useful. This Appendix presents a generalization to higher dimensions. This generalization can be found in different references and in different forms. Here the conventions are those found in Byrd and Khaneja. (References are needed and the following should be added (1)regions of positivity, (2) affine maps of the polarization vector, (3) the orthogonal adjoint elements are orthogonal when trace is -1, (4) add the two qubit density op and reference, (5).)

An N-dimensional Generalization of the Polarization Vector

The following are somewhat standard conventions and those contained in Ref.~\cite{Byrd/Khaneja:03}. A density operator on an $N\,\!$-dimensional Hilbert space ${\mathcal {H}}_{N}\,\!$ will be represented using a set of traceless Hermitian matrices $\{\lambda _{i}\}\,\!$, $i=1,2,...,N^{2}-1\,\!$ with the normalization condition ${\mbox{Tr}}(\lambda _{i}\lambda _{j})=2\delta _{ij}\,\!$. The commutation and anticommutation relations for this set of matrices are given by

	$[\lambda _{i},\lambda _{j}]=2ic_{ijk}\lambda _{k},\;\;\;\{\lambda _{i},\lambda _{j}\}={\frac {4}{N}}I\delta _{ij}+2d_{ijk}\lambda _{k},\,\!$	(E.1)

where the sum over repeated indices is to be understood unless otherwise stated. (In some cases the sum is displayed explicitly for emphasis.) These relations can be summarized using the trace, antisymmetric, and symmetric combinations of the following equation:

	$\lambda _{i}\lambda _{j}={\frac {2}{N}}I\delta _{ij}+d_{ijk}\lambda _{k}+ic_{ijk}\lambda _{k}.\,\!$	(E.2)

The density operator can now be written as

	$\rho ={\frac {1}{N}}\left(I+b{\vec {n}}\cdot {\vec {\lambda }}\right)\,\!$	(E.3)

where $b={\sqrt {(N(N-1)/2)}}\,\!$. The dot product is a sum over repeated indices,

	${\vec {a}}\cdot {\vec {b}}=a_{i}b_{i}=\sum _{i=1}^{N^{2}-1}a_{i}b_{i}.\,\!$	(E.4)

Any complete set of $N^{2}-1\,\!$ mutually trace-orthogonal, Hermitian matrices can serve as a basis and can be chosen to satisfy the conditions given here.

Using the condition that pure states satisfy $\rho ^{2}=\rho \,\!$, we find that for pure states,

	${\vec {n}}\cdot {\vec {n}}=1,\;\;\;{\mbox{and}}\;\;\;{\vec {n}}\star {\vec {n}}={\vec {n}},\,\!$	(E.5)

where the "star" product is defined by

	$({\vec {a}}\star {\vec {b}})_{k}={\frac {1}{N-2}}{\sqrt {\frac {N(N-1)}{2}}}\;d_{ijk}a_{i}b_{j}.\,\!$	(E.6)

For later use, a "cross" product between two coherence vectors can also be defined by

	$({\vec {a}}\times {\vec {b}})_{k}=c_{ijk}a_{i}b_{j}.\,\!$	(E.7)