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import jax
import jax.numpy as jnp
from functools import partial
import numpy as np
from utils import find_permutation
from tensors import partial_trace_with_identity, partial_trace_with_identity_numpy
@partial(jax.jit, static_argnums=(1, 2, 3, 4))
def Lv_projector_process_2parties(W, dim_A_I, dim_A_O, dim_B_I, dim_B_O):
# 2T + 4T − 24T − 34T + 234T − 12T + 124T
dim_W = dim_A_I * dim_A_O * dim_B_I * dim_B_O
AO_W = partial_trace_with_identity(W, (1,), (dim_A_I, dim_A_O, dim_B_I, dim_B_O))
BO_W = partial_trace_with_identity(W, (3,), (dim_A_I, dim_A_O, dim_B_I, dim_B_O))
AOBO_W = partial_trace_with_identity(W, (1, 3), (dim_A_I, dim_A_O, dim_B_I, dim_B_O))
BIBO_W = partial_trace_with_identity(W, (2, 3), (dim_A_I, dim_A_O, dim_B_I, dim_B_O))
AOBIBO_W = partial_trace_with_identity(W, (1, 2, 3), (dim_A_I, dim_A_O, dim_B_I, dim_B_O))
AIAO_W = partial_trace_with_identity(W, (0, 1), (dim_A_I, dim_A_O, dim_B_I, dim_B_O))
AIAOBO_W = partial_trace_with_identity(W, (0, 1, 3), (dim_A_I, dim_A_O, dim_B_I, dim_B_O))
W_projected = AO_W + BO_W - AOBO_W - BIBO_W + AOBIBO_W - AIAO_W + AIAOBO_W
return (W_projected + W_projected.conj().T) / 2
@partial(jax.jit, static_argnums=(1, 2, 3, 4, 5, 6))
def Lv_projector_process_3parties(W, dim_A_I, dim_A_O, dim_B_I, dim_B_O, dim_C_I, dim_C_O):
"""Projector L(W) for the process function of a 3-party system. This
implements equation B.24 from arXiv:1603.00043, which is a projector onto
the space of valid process matrices.
Parameters
----------
W : jax array
Input matrix to be projected onto the linear subspace.
dim_A_I : int
Input dimension of system A.
dim_A_O : int
Output dimension of system A.
dim_B_I : int
Input dimension of system B.
dim_B_O : int
Output dimension of system B.
dim_C_I : int
Input dimension of system C.
dim_C_O : int
Output dimension of system C.
Returns
-------
jax array
Returns the projected matrix L(W) forced to be Hermitian to machine
precision, (L(W)+L(W).conj().T)/2.
"""
# Eq. B.24 from arXiv:1603.00043 expanded with SymPy
# Lv =
# AI*AO*BI*BO*CI - AI*AO*BI*BO + AI*AO*BO*CI*CO - AI*AO*BO*CI + AI*AO*BO -
# AI*AO*CI*CO + AI*AO*CI - AI*AO + AO*BI*BO*CI*CO - AO*BI*BO*CI + AO*BI*BO -
# AO*BO*CI*CO + AO*BO*CI - AO*BO + AO*CI*CO - AO*CI + AO - BI*BO*CI*CO +
# BI*BO*CI - BI*BO + BO*CI*CO - BO*CI + BO - CI*CO + CI
_dims = (dim_A_I, dim_A_O, dim_B_I, dim_B_O, dim_C_I, dim_C_O)
# dim indices: 0 1 2 3 4 5
# first line in the comment above
AIAOBIBOCI_W = partial_trace_with_identity(W, (0, 1, 2, 3, 4), _dims)
AIAOBIBO_W = partial_trace_with_identity(W, (0, 1, 2, 3), _dims)
AIAOBOCICO_W = partial_trace_with_identity(W, (0, 1, 3, 4, 5), _dims)
AIAOBOCI_W = partial_trace_with_identity(W, (0, 1, 3, 4), _dims)
AIAOBO_W = partial_trace_with_identity(W, (0, 1, 3), _dims)
# second line in the comment above
AIAOCICO_W = partial_trace_with_identity(W, (0, 1, 4, 5), _dims)
AIAOCI_W = partial_trace_with_identity(W, (0, 1, 4), _dims)
AIAO_W = partial_trace_with_identity(W, (0, 1), _dims)
AOBIBOCICO_W = partial_trace_with_identity(W, (1, 2, 3, 4, 5), _dims)
AOBIBOCI_W = partial_trace_with_identity(W, (1, 2, 3, 4), _dims)
AOBIBO_W = partial_trace_with_identity(W, (1, 2, 3), _dims)
# third line in the comment above
AOBOCICO_W = partial_trace_with_identity(W, (1, 3, 4, 5), _dims)
AOBOCI_W = partial_trace_with_identity(W, (1, 3, 4), _dims)
AOBO_W = partial_trace_with_identity(W, (1, 3), _dims)
AOCICO_W = partial_trace_with_identity(W, (1, 4, 5), _dims)
AOCI_W = partial_trace_with_identity(W, (1, 4), _dims)
AO_W = partial_trace_with_identity(W, (1,), _dims)
BIBOCICO_W = partial_trace_with_identity(W, (2, 3, 4, 5), _dims)
# fourth line in the comment above
BIBOCI_W = partial_trace_with_identity(W, (2, 3, 4), _dims)
BIBO_W = partial_trace_with_identity(W, (2, 3), _dims)
BOCICO_W = partial_trace_with_identity(W, (3, 4, 5), _dims)
BOCI_W = partial_trace_with_identity(W, (3, 4), _dims)
BO_W = partial_trace_with_identity(W, (3,), _dims)
CICO_W = partial_trace_with_identity(W, (4, 5), _dims)
CI_W = partial_trace_with_identity(W, (4,), _dims)
# Apply the projection
W_projected = (AIAOBIBOCI_W - AIAOBIBO_W + AIAOBOCICO_W - AIAOBOCI_W +
AIAOBO_W - AIAOCICO_W + AIAOCI_W - AIAO_W + AOBIBOCICO_W -
AOBIBOCI_W + AOBIBO_W - AOBOCICO_W + AOBOCI_W - AOBO_W +
AOCICO_W - AOCI_W + AO_W - BIBOCICO_W + BIBOCI_W - BIBO_W +
BOCICO_W - BOCI_W + BO_W - CICO_W + CI_W)
# Force the result to be Hermitian as close as possible to machine precision
W_projected = (W_projected + W_projected.conj().T) / 2
return W_projected
def Lv_projector_channel_numpy(C, dim_I, dim_O):
O_C = partial_trace_with_identity_numpy(C, (1,), (dim_I, dim_O))
# IO_C = partial_trace_with_identity_numpy(C, (0, 1), (dim_I, dim_O))
IO_C = np.trace(C) * np.eye(dim_I*dim_O, dtype=C.dtype) / (dim_I*dim_O)
return C - O_C + IO_C
@partial(jax.jit, static_argnums=(1, 2))
def Lv_projector_channel(C, dim_I, dim_O):
O_C = partial_trace_with_identity(C, (1,), (dim_I, dim_O))
IO_C = partial_trace_with_identity(C, (0, 1), (dim_I, dim_O))
C_projected = C - O_C + IO_C
# Force the result to be Hermitian as close as possible to machine precision
# C_projected = (C_projected + C_projected.conj().T) / 2
# # renormalise the trace with a jax conditional which checks if trace_C is zero
# # The desired trace for a channel from dim_I -> dim_O is 'dim_I'.
# trace_C_x_value = dim_I
# # Compute the trace of C
# trace_C = jnp.trace(C)
# # Conditionally renormalize only if trace_C != 0.
# def renormalize(mat):
# return (trace_C_x_value / trace_C) * mat
# def no_op(mat):
# return mat
# C_projected = jax.lax.cond(
# jnp.isclose(trace_C, 0.0),
# no_op,
# renormalize,
# C_projected
# )
return C_projected