$\newcommand{\ket}[1]{|{#1}\rangle}\newcommand{\bra}[1]{\langle{#1}|}$Considere un sistema cuántico compuesto formado por dos estados independientes \begin{align*} \delta^{AB} = \rho^A \otimes \sigma^B = \left[\begin{matrix} \rho_{11}\sigma^B& \rho_{12}\sigma^B \\ \rho_{21}\sigma^B & \rho_{22}\sigma^B \end{matrix}\right] = \left[\begin{matrix} \rho_{11}\sigma_{11} & \rho_{11}\sigma_{12} & \rho_{12}\sigma_{11} & \rho_{12}\sigma_{12} \\ \rho_{11}\sigma_{21} & \rho_{11}\sigma_{22} & \rho_{12}\sigma_{21} & \rho_{12}\sigma_{22} \\ \rho_{21}\sigma_{11} & \rho_{21}\sigma_{12} & \rho_{22}\sigma_{11} & \rho_{22}\sigma_{12} \\ \rho_{21}\sigma_{21} & \rho_{21}\sigma_{22} & \rho_{22}\sigma_{21} & \rho_{22}\sigma_{22} \end{matrix}\right]. \end{align*}
fprintf('\nDefinimos un estado compuesto independiente:\n\n')
rhoA = [[3/4, 0]; [0, 1/4]]
sigmaB = [[1/2, 1/2]; [1/2, 1/2]]
deltaAB = kron(rhoA, sigmaB)
Definimos un estado compuesto independiente:
rhoA =
0.7500 0
0 0.2500
sigmaB =
0.5000 0.5000
0.5000 0.5000
deltaAB =
0.3750 0.3750 0 0
0.3750 0.3750 0 0
0 0 0.1250 0.1250
0 0 0.1250 0.1250
Si descartamos la parte $A$ del sistema obtenemos \begin{align*} \text{Tr}_{A} \left[ \delta^{AB} \right] &= \left[\begin{matrix} \rho_{11}\sigma_{11} + \rho_{22}\sigma_{11} & \rho_{11}\sigma_{12} + \rho_{22}\sigma_{12} \\ \rho_{11}\sigma_{21} + \rho_{22}\sigma_{21} & \rho_{11}\sigma_{22} + \rho_{22}\sigma_{22} \end{matrix}\right] = \underbrace{\text{Tr}[\rho^A]}_{=1} \left[\begin{matrix} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{matrix}\right] = \sigma^B. \end{align*}
function [sigma] = traceA(delta) % traces out part A of a bipartite system
sigma = [delta(1,1)+delta(3,3) delta(1,2)+delta(3,4); ...
delta(2,1)+delta(4,3) delta(2,2)+delta(4,4)];
end
fprintf('\nDescartamos la parte A del estado compuesto:\n\n')
sigmaB = traceA(deltaAB)
Descartamos la parte A del estado compuesto: sigmaB = 0.5000 0.5000 0.5000 0.5000
Si descartamos la parte $B$ del mismo queda \begin{align*} \text{Tr}_{B} \left[ \delta^{AB} \right] &= \left[\begin{matrix} \rho_{11}\sigma_{11} + \rho_{11}\sigma_{22} & \rho_{12}\sigma_{11} + \rho_{12}\sigma_{22} \\ \rho_{21}\sigma_{11} + \rho_{21}\sigma_{22} & \rho_{22}\sigma_{11} + \rho_{22}\sigma_{22} \end{matrix}\right] = \underbrace{\text{Tr}[\sigma^B]}_{=1} \left[\begin{matrix} \rho_{11} & \rho_{12} \\ \rho_{21} & \rho_{22} \end{matrix}\right] = \rho^A. \end{align*}
function [rho] = traceB(delta) % traces out part B of a bipartite system
rho = [delta(1,1)+delta(2,2) delta(1,3)+delta(2,4); ...
delta(3,1)+delta(4,2) delta(3,3)+delta(4,4)];
end
fprintf('\nDescartamos la parte B del estado compuesto:\n\n')
rhoA = traceB(deltaAB)
Descartamos la parte B del estado compuesto:
rhoA =
0.7500 0
0 0.2500