Appendix E - Density Operator: Extensions

The following are somewhat standard conventions and those contained in Ref.~\cite{Byrd/Khaneja:03}. A density operator on an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\,\!} -dimensional Hilbert space ${\mathcal {H}}_{N}\,\!$ will be represented using a set of traceless, Hermitan matrices $\{\lambda _{i}\}\,\!$, $i=1,2,...,N^{2}-1\,\!$ with the normalization condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Tr}(\lambda_i\lambda_j)=2\delta_{ij}\,\!} . The commutation and anticommutation relations for this set of matrices are given by

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\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

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = \sqrt{(N(N-1)/2)}\,\!} . The ``dot product is a sum over repeated indices,

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{a}\cdot \vec{b} = a_ib_i = \sum_{i=1}^{N^2-1}a_ib_i. \,\!}	(E.4)

Any complete set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^2-1\,\!} mutually trace-orthogonal, Hermitian matrices can serve as a basis and can be chosen to satisfy the conditions given here.

Pure states have the properties that

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

where the "star" product is defined by

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\vec{a}\star\vec{b})_k = \frac{1}{N-2}\sqrt{\frac{N(N-1)}{2}}\;d_{ijk}a_ib_j. \,\!}	(E.6)

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

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\vec{a}\times\vec{b})_k = c_{ijk}a_ib_j. \,\!}	(E.7)