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.

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 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 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 {\mathcal{H}}_N\,\!} will be represented using a set of traceless, Hermitan matrices 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\}\,\!} , 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 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

	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

	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 \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.

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

	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)