multipartite

bound_entangled.multipartite.smolin

https://arxiv.org/abs/quant-ph/0001001

J. A. Smolin, "Four-party unlockable bound entangled state", Phys. Rev. A 63, 032306 (2001). Bound entangled state on C^4 ⊗ C^4 (four qubits) that can be unlocked by classical communication between any two of the four parties.

smolin()

Construct the Smolin bound-entangled state on C^4 ⊗ C^4 (arXiv:quant-ph/0001001).

The Smolin state is the equal mixture of the four two-qubit Bell states tensored with themselves::

rho = (1/4) sum_{i=0}^{3} |phi_i><phi_i| ⊗ |phi_i><phi_i|

It lives on a four-qubit system A⊗B⊗C⊗D. Returns

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np.ndarray 16×16 density matrix representing the Smolin state

bound_entangled.multipartite.generalized_smolin

https://arxiv.org/abs/quant-ph/0411142

A. Kay, "Degree of quantum bound entanglement for a family of mixed states", Phys. Rev. A 71, 032309 (2005). (See also Smolin: arXiv:quant-ph/0001001.) Generalization of the Smolin state to 2n qubits: bound entangled for all even n >= 2, constructed from tensor products of Pauli operators.

generalized_smolin(systems)

Construct the generalized Smolin (GSS) bound-entangled state (arXiv:quant-ph/0411142).

For an even number of qubits systems = 2n, the state is::

rho = (I + (-1)^n * sum_{i in {X,Y,Z}} sigma_i^{⊗2n}) / 2^{2n}

The systems=4 case (n=1 pairs ⊗ 2 copies) reproduces the original Smolin state on C^4 ⊗ C^4. For all even systems >= 4 the state is bound entangled.

Parameters

systems: Total number of qubits. Must be a positive even integer.

Returns

np.ndarray 2^systems × 2^systems density matrix of the GSS state.

Raises

AssertionError If systems is odd.