# 8.54: Expressing Bell and GHZ States in Vector Format Using Mathcad

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

Mathcad provides the kronecker command for matrix tensor multiplication. It requires square matrices for its arguments and therefore cannot be used directly for vector tensor multiplication. However, if a vector is augmented with a null vector (or matrix) to produce a square matrix, vector tensor multiplication can be carried out using kronecker and a submatrix command that discards everything except the first column of the product matrix. This technique is illustrated by putting the Bell and GHZ states in vector format.

The z- and x-direction spin eigenfunctions and the appropriate null vector are required.

$\begin{matrix} z_u = \begin{pmatrix} 1 \\ 0 \end{pmatrix} & z_d = \begin{pmatrix} 0 \\ 1 \end{pmatrix} & x_u = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} & x_d = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} & n = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \end{matrix} \nonumber$

The Mathcad syntax for tensor multiplication of two 2-dimensional vectors.

$\psi \text{(a, b)} = \text{submatrix(kronecker(augment(a, n), augment(b, n)), 1, 4, 1, 1)} \nonumber$

The four maximally entangled Bell states will be expressed in both the z- and x-basis.

$| \Phi_p \rangle = \frac{1}{ \sqrt{2}} \left[ | \uparrow_1 \rangle | \uparrow_2 \rangle + | \downarrow_1 \rangle | \downarrow_2 \rangle \right] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} \nonumber$

$\begin{matrix} \Phi_p = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_u) + \psi (z_d,~z_d) \right) & \Phi_p = \begin{pmatrix} 0.707 \\ 0 \\ 0 \\ 0.707 \end{pmatrix} & \Phi_p = \frac{1}{ \sqrt{2}} \left( \psi (x_u,~x_u) \right) + \left( \psi (x_d,~x_d) \right) & \Phi_p = \begin{pmatrix} 0.707 \\ 0 \\ 0 \\ 0.707 \end{pmatrix} \end{matrix} \nonumber$

$| \Phi_m \rangle = \frac{1}{ \sqrt{2}} \left[ | \uparrow_1 \rangle | \uparrow_2 \rangle - | \downarrow_1 \rangle | \downarrow_2 \rangle \right] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} - \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix} \nonumber$

$\begin{matrix} \Phi_m = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_u) - \psi (z_d,~z_d) \right) & \Phi_m = \begin{pmatrix} 0.707 \\ 0 \\ 0 \\ -0.707 \end{pmatrix} & \Phi_m = \frac{1}{ \sqrt{2}} \left( \psi (x_u,~x_u) \right) + \left( \psi (x_d,~x_d) \right) & \Phi_m = \begin{pmatrix} 0.707 \\ 0 \\ 0 \\ -0.707 \end{pmatrix} \end{matrix} \nonumber$

$| \Psi_p \rangle = \frac{1}{ \sqrt{2}} \left[ | \uparrow_1 \rangle | \uparrow_2 \rangle + | \downarrow_1 \rangle | \downarrow_2 \rangle \right] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix} \nonumber$

$\begin{matrix} \Psi_p = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_u) + \psi (z_d,~z_d) \right) & \Phi_p = \begin{pmatrix} 0 \\ 0.707 \\ 0.707 \\ 0 \end{pmatrix} & \Psi_p = \frac{1}{ \sqrt{2}} \left( \psi (x_u,~x_u) \right) - \left( \psi (x_d,~x_d) \right) & \Psi_p = \begin{pmatrix} 0 \\ 0.707 \\ 0.707 \\ 0 \end{pmatrix} \end{matrix} \nonumber$

$| \Psi_m \rangle = \frac{1}{ \sqrt{2}} \left[ | \uparrow_1 \rangle | \uparrow_2 \rangle - | \downarrow_1 \rangle | \downarrow_2 \rangle \right] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} - \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 0 \\ 1 \\ -1 \\ 0 \end{pmatrix} \nonumber$

$\begin{matrix} \Psi_m = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_u) - \psi (z_d,~z_d) \right) & \Psi_m = \begin{pmatrix} 0 \\ 0.707 \\ -0.707 \\ 0 \end{pmatrix} & \Psi_m = \frac{1}{ \sqrt{2}} \left( \psi (x_u,~x_u) \right) - \left( \psi (x_d,~x_d) \right) & \Psi_m = \begin{pmatrix} 0 \\ 0.707 \\ -0.707 \\ 0 \end{pmatrix} \end{matrix} \nonumber$

The Mathcad syntax for tensor multiplication of three 2-dimensional vectors.

$\Psi \text{(a, b, c)} = \text{submatrix(kronecker(augment(a, n), kronecker(augment(b, n), augment(c, n))), 1, 8, 1, 1)} \nonumber$

$| \Psi \rangle = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \pm \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & \pm 1 \end{pmatrix}^T \nonumber$

$\begin{matrix} \Psi_1 = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_u,~z_u) + \psi (z_d,~z_d,~z_d) \right) & \Psi_1^T = \begin{pmatrix} 0.707 & 0 & 0 & 0 & 0 & 0 & 0 & 0.707 \end{pmatrix} \\ \Psi_2 = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_u,~z_u) - \psi (z_d,~z_d,~z_d) \right) & \Psi_2^T = \begin{pmatrix} 0.707 & 0 & 0 & 0 & 0 & 0 & 0 & -0.707 \end{pmatrix} \end{matrix} \nonumber$

$| \Psi \rangle = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \pm \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 & \pm 1 & 0 \end{pmatrix}^T \nonumber$

$\begin{matrix} \Psi_3 = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_u,~z_d) + \psi (z_d,~z_d,~z_u) \right) & \Psi_3^T = \begin{pmatrix} 0 & 0.707 & 0 & 0 & 0 & 0 & 0.707 & 0 \end{pmatrix} \\ \Psi_4 = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_u,~z_d) - \psi (z_d,~z_d,~z_u) \right) & \Psi_4^T = \begin{pmatrix} 0 & 0.707 & 0 & 0 & 0 & 0 & -0.707 & 0 \end{pmatrix} \end{matrix} \nonumber$

$| \Psi \rangle = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \pm \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 0 & 0 & 1 & 0 & 0 & \pm 1 & 0 & 0 \end{pmatrix}^T \nonumber$

$\begin{matrix} \Psi_5 = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_d,~z_u) + \psi (z_d,~z_u,~z_d) \right) & \Psi_5^T = \begin{pmatrix} 0 & 0 & 0.707 & 0 & 0 & 0.707 & 0 & 0 \end{pmatrix} \\ \Psi_6 = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_d,~z_u) - \psi (z_d,~z_u,~z_d) \right) & \Psi_6^T = \begin{pmatrix} 0 & 0 & 0.707 & 0 & 0 & -0.707 & 0 & 0 \end{pmatrix} \end{matrix} \nonumber$

$| \Psi \rangle = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \pm \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 0 & 0 & 0 & 1 & \pm 1 & 0 & 0 & 0 \end{pmatrix}^T \nonumber$

$\begin{matrix} \Psi_7 = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_d,~z_d) + \psi (z_d,~z_u,~z_u) \right) & \Psi_7^T = \begin{pmatrix} 0 & 0 & 0 & 0.707 & 0.707 & 0 & 0 & 0 \end{pmatrix} \\ \Psi_8 = \frac{1}{ \sqrt{2}} \left( \psi (z_u,~z_d,~z_d) - \psi (z_d,~z_u,~z_u) \right) & \Psi_8^T = \begin{pmatrix} 0 & 0 & 0 & 0.707 & -0.707 & 0 & 0 & 0 \end{pmatrix} \end{matrix} \nonumber$

This page titled 8.54: Expressing Bell and GHZ States in Vector Format Using Mathcad is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform.