/
entanglement.c
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/
entanglement.c
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//------------------------------------------------------------------------------
// Copyright (C) 2012, Robert Johansson <robert@riken.jp>
// All rights reserved.
//
// This file is part of QDpack, and licensed under the LGPL.
// http://dml.riken.jp/~rob/qdpack.html
//------------------------------------------------------------------------------
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_complex.h>
#include <gsl/gsl_complex_math.h>
#include <gsl/gsl_eigen.h>
#include <qdpack/qdpack.h>
/**
* \breif Calculate the von Neumann Entropy of a system with density matrix rho.
*
* This function takes a density matrix as input and calculates the von Neumann entropy.
* Note that the density matrix is not restricted to be representing any particular
* for of quantum system.
*
* @param rho The density matrix for which the von Neumann entropy is to be calculated.
*
*/
double
qdpack_entanglement_neumann_entropy(qdpack_operator_t *rho)
{
int i;
double e, ne = 0.0;
qdpack_matrix_t *rho_w;
qdpack_matrix_t *eval, *evec;
rho_w = qdpack_matrix_alloc(rho->m, rho->n);
qdpack_matrix_memcpy(rho_w, rho->data);
eval = qdpack_matrix_alloc(rho->m, 1);
evec = qdpack_matrix_alloc(rho->m, rho->n);
qdpack_matrix_eigen_hermv(rho_w, eval, evec, 0);
for (i = 0; i < rho->m; i++)
{
e = QDPACK_REAL(qdpack_matrix_get(eval, i, 0));
//ne += - e * log2(e);
ne += - e * log(e);
}
/* clean-up */
qdpack_matrix_free(rho_w);
qdpack_matrix_free(eval);
qdpack_matrix_free(evec);
return ne;
}
/**
* \breif Calculate the log negativity for the density matrix of two TLS, rho.
*
* Calculate the negativity of a two-qubit system.
*
* @param rho Density matrix for a two-qubit system.
*
*/
double
qdpack_entanglement_log_neg(qdpack_operator_t *rho)
{
int i;
double ln, sum_eval;
qdpack_matrix_t *rho_pt;
qdpack_matrix_t *eval, *evec;
/* -- calculate partial transpose of rho, and store in rho_pt
*
* | x x A B | | x x a b |
* | x x C D | -> | x x c d |
* | a b y y | | A B y y |
* | c d y y | | C D y y |
*
*/
rho_pt = qdpack_matrix_alloc(4,4);
qdpack_matrix_memcpy(rho_pt, rho->data);
// A,B,C,D -> a,b,c,d
qdpack_matrix_set(rho_pt, 0, 2, qdpack_operator_get(rho, 2, 0));
qdpack_matrix_set(rho_pt, 0, 3, qdpack_operator_get(rho, 2, 1));
qdpack_matrix_set(rho_pt, 1, 2, qdpack_operator_get(rho, 3, 0));
qdpack_matrix_set(rho_pt, 1, 3, qdpack_operator_get(rho, 3, 1));
// a,b,c,d -> A,B,C,D
qdpack_matrix_set(rho_pt, 2, 0, qdpack_operator_get(rho, 0, 2));
qdpack_matrix_set(rho_pt, 2, 1, qdpack_operator_get(rho, 0, 3));
qdpack_matrix_set(rho_pt, 3, 0, qdpack_operator_get(rho, 1, 2));
qdpack_matrix_set(rho_pt, 3, 1, qdpack_operator_get(rho, 1, 3));
/* --- diagonalize rho_pt --- */
eval = qdpack_matrix_alloc(4, 1);
evec = qdpack_matrix_alloc(4, 4);
qdpack_matrix_eigen_hermv(rho_pt, eval, evec, 0);
/* --- calculate the log negativity --- */
sum_eval = 0;
for (i = 0; i < 4; i++)
{
sum_eval += qdpack_complex_abs(qdpack_matrix_get(eval, i, 0));
}
ln = log2(sum_eval);
/* clean-up */
qdpack_matrix_free(rho_pt);
qdpack_matrix_free(eval);
qdpack_matrix_free(evec);
return ln;
}
/**
* \breif Calculate the concurrence for the density matrix (two TLS) rho.
*
* Calculate the concurrence for the density matrix of a two-qubit system.
*
* @param qs Data structure that specifies the form of the quantum system.
* @param rho The density matrix for a two-qubit system.
*
*/
double
qdpack_entanglement_concurrence(qdpack_hilbert_space_t *qs, qdpack_operator_t *rho)
{
double c = 0.0;
qdpack_operator_t *y1, *y2, *y, *ws1, *ws2;
qdpack_matrix_t *eval = qdpack_matrix_alloc(4, 1);
qdpack_matrix_t *evec = qdpack_matrix_alloc(4, 4);
y1 = qdpack_operator_alloc(qs);
y2 = qdpack_operator_alloc(qs);
operator_sigma_y(y1, qs, 0);
operator_sigma_y(y2, qs, 1);
y = qdpack_operator_alloc(qs);
ws1 = qdpack_operator_alloc(qs);
ws2 = qdpack_operator_alloc(qs);
// calculate: rho * (y1 * y2) * rho' * (y1 * y2)
// rho * y * rho' * y
// --------------- -----------------
// ws1 ws2
//
qdpack_operator_blas_zgemm(QDpackNoTrans, QDpackNoTrans, QDPACK_COMPLEX_ONE, y1, y2, QDPACK_COMPLEX_ZERO, y);
qdpack_operator_blas_zgemm(QDpackNoTrans, QDpackNoTrans, QDPACK_COMPLEX_ONE, rho, y, QDPACK_COMPLEX_ZERO, ws1);
qdpack_operator_blas_zgemm(QDpackConjTrans, QDpackNoTrans, QDPACK_COMPLEX_ONE, rho, y, QDPACK_COMPLEX_ZERO, ws2);
qdpack_operator_blas_zgemm(QDpackNoTrans, QDpackNoTrans, QDPACK_COMPLEX_ONE, ws1, ws2, QDPACK_COMPLEX_ZERO, y);
//printf("rr_real = \n"); qdpack_operator_print_real(y);
//printf("rr_imag = \n"); qdpack_operator_print_imag(y);
printf("rr = \n"); qdpack_operator_print(y);
// calculate eigenvalues
qdpack_matrix_eigen_zgeev(y->data, eval, evec, 0);
{
int i, j, n = 4;
printf("Eigenvalues = \n");
for (i = 0; i < n; i++)
{
qdpack_complex z;
z = qdpack_matrix_get(eval, i, 0);
printf("\t(%f, %f)", QDPACK_REAL(z), QDPACK_IMAG(z));
}
printf("\n");
printf("evec = \n");
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
{
qdpack_complex z;
z = qdpack_matrix_get(evec, i, j);
printf("\t(%f, %f)", QDPACK_REAL(z), QDPACK_IMAG(z));
}
printf("\n");
}
}
printf("c: eigvals: (%f, %f), (%f, %f), (%f, %f), (%f, %f)\n",
QDPACK_REAL(qdpack_matrix_get(eval, 0, 0)), QDPACK_IMAG(qdpack_matrix_get(eval, 0, 0)),
QDPACK_REAL(qdpack_matrix_get(eval, 1, 0)), QDPACK_IMAG(qdpack_matrix_get(eval, 1, 0)),
QDPACK_REAL(qdpack_matrix_get(eval, 2, 0)), QDPACK_IMAG(qdpack_matrix_get(eval, 2, 0)),
QDPACK_REAL(qdpack_matrix_get(eval, 3, 0)), QDPACK_IMAG(qdpack_matrix_get(eval, 3, 0)));
c = + QDPACK_REAL(qdpack_matrix_get(eval, 0, 0)) - QDPACK_REAL(qdpack_matrix_get(eval, 1, 0))
- QDPACK_REAL(qdpack_matrix_get(eval, 2, 0)) - QDPACK_REAL(qdpack_matrix_get(eval, 3, 0));
c = (c < 0.0) ? 0.0 : c;
qdpack_operator_free(y1);
qdpack_operator_free(y2);
qdpack_operator_free(y);
qdpack_operator_free(ws1);
qdpack_operator_free(ws2);
qdpack_matrix_free(evec);
qdpack_matrix_free(eval);
return c;
}