Overview¶

Many entanglement detection methods have been proposed. Here, we summarize the pros and cons of each method.

1. Positive Partial Transpose (PPT) criterion
   • pros: computationally efficient
   • cons: false positive (bound entangled states)
2. Convex Hull Approximation (CHA) method
   • pros: computationally efficient
   • cons: non-convex optimization
3. Bosonic extension semi-definite programming (SDP)
   • pros: accurate
   • cons: computationally heavy
4. Pure bosonic extension (PureB)
   • pros: computationally efficient
   • cons: correct for generic states, but not for all separable states, nor for all entangled states

To check whether a density matrix $\rho$ is entangled or not, the following is my personal recommendation

Above, small loss means the minimized loss is smaller than some predefined threshold (e.g. 1e-12 for double precision), and large loss means the minimized loss is larger than this threshold.

In [1]:

import numpy as np

try:
    import numqi
except ImportError:
    %pip install numqi
    import numqi

import numpy as np try: import numqi except ImportError: %pip install numqi import numqi

In [2]:

dimA = 3
dimB = 3

dm0 = numqi.random.rand_density_matrix(dimA*dimB, k=2*dimA*dimB)

tag_ppt = numqi.entangle.is_ppt(dm0, (dimA,dimB))
if not tag_ppt:
    print('[entangled] dm0 is not PPT')
else:
    eps = 1e-10
    model_cha = numqi.entangle.AutodiffCHAREE((dimA, dimB))
    model_cha.set_dm_target(dm0)
    loss_cha = numqi.optimize.minimize(model_cha, num_repeat=3, tol=1e-12, print_every_round=0).fun
    tag_cha = loss_cha < eps
    if tag_cha:
        print(f'[separable] dm0 has seperable decomposition, loss={loss_cha}')
    else:
        tag_kext = numqi.entangle.is_ABk_symmetric_ext(dm0, (dimA,dimB), kext=4, use_ppt=True, use_boson=True)
        if not tag_kext:
            print('[entangled] dm0 is not bosonic extendable')
        else:
            model_pureb = numqi.entangle.PureBosonicExt(dimA, dimB, kext=32)
            model_pureb.set_dm_target(dm0)
            loss_pureb = numqi.optimize.minimize(model_pureb, num_repeat=3, tol=1e-12, print_every_round=0).fun

dimA = 3 dimB = 3 dm0 = numqi.random.rand_density_matrix(dimA*dimB, k=2*dimA*dimB) tag_ppt = numqi.entangle.is_ppt(dm0, (dimA,dimB)) if not tag_ppt: print('[entangled] dm0 is not PPT') else: eps = 1e-10 model_cha = numqi.entangle.AutodiffCHAREE((dimA, dimB)) model_cha.set_dm_target(dm0) loss_cha = numqi.optimize.minimize(model_cha, num_repeat=3, tol=1e-12, print_every_round=0).fun tag_cha = loss_cha < eps if tag_cha: print(f'[separable] dm0 has seperable decomposition, loss={loss_cha}') else: tag_kext = numqi.entangle.is_ABk_symmetric_ext(dm0, (dimA,dimB), kext=4, use_ppt=True, use_boson=True) if not tag_kext: print('[entangled] dm0 is not bosonic extendable') else: model_pureb = numqi.entangle.PureBosonicExt(dimA, dimB, kext=32) model_pureb.set_dm_target(dm0) loss_pureb = numqi.optimize.minimize(model_pureb, num_repeat=3, tol=1e-12, print_every_round=0).fun

[entangled] dm0 is not PPT