Entanglement measures#

Set of functions to calculate entanglement measures.

Concurrence#

qibo.quantum_info.concurrence(state, bipartition, check_purity=True, backend=None)[source]#

Calculates concurrence of a pure bipartite quantum state \(\rho \in \mathcal{H}_{A} \otimes \mathcal{H}_{B}\) as

\[C(\rho) = \sqrt{2 \, (\text{tr}^{2}(\rho) - \text{tr}(\rho_{A}^{2}))} \, ,\]

where \(\rho_{A} = \text{tr}_{B}(\rho)\) is the reduced density operator obtained by tracing out the qubits in the bipartition \(B\).

Parameters:

  • state (ArrayLike) – statevector or density matrix.

  • bipartition (list or tuple) – qubits in the subsystem to be traced out.

  • check_purity (bool, optional) – if True, checks if state is pure. If False, it assumes state is pure. Defaults to True.

  • backend (qibo.backends.abstract.Backend, optional) – backend to be used in the execution. If None, it uses the current backend. Defaults to None.

Returns:

Concurrence of \(\rho\).

Return type:

float

Entanglement of formation#

qibo.quantum_info.entanglement_of_formation(state, bipartition, base=2, check_purity=True, backend=None)[source]#

Calculates the entanglement of formation \(E_{f}\) of a pure bipartite quantum state \(\rho\), which is given by

\[E_{f} = H([1 - x, x]) \, ,\]

where

\[x = \frac{1 + \sqrt{1 - C^{2}(\rho)}}{2} \, ,\]

\(C(\rho)\) is the qibo.quantum_info.concurrence() of \(\rho\), and \(H\) is the qibo.quantum_info.entropies.shannon_entropy().

Note

This expression is valid only if at least one of the two subsystems is a single qubit. For pure states of larger subsystems, see qibo.quantum_info.entanglement_entropy().

Parameters:

  • state (ArrayLike) – statevector or density matrix.

  • bipartition (list or tuple) – qubits in the subsystem to be traced out.

  • base (float) – the base of the log in qibo.quantum_info.entropies.shannon_entropy(). Defaults to \(2\).

  • check_purity (bool, optional) – if True, checks if state is pure. If False, it assumes state is pure. Defaults to True.

  • backend (qibo.backends.abstract.Backend, optional) – backend to be used in the execution. If None, it uses the current backend. Defaults to None.

Returns:

entanglement of formation of state \(\rho\).

Return type:

float

References

  1. W. K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits, Phys. Rev. Lett. 80, 2245 (1998).

Negativity#

qibo.quantum_info.negativity(state, bipartition, backend=None)[source]#

Calculates the negativity of a bipartite quantum state.

Given a bipartite state \(\rho \in \mathcal{H}_{A} \otimes \mathcal{H}_{B}\), the negativity \(\operatorname{Neg}(\rho)\) is given by

\[\operatorname{Neg}(\rho) = \frac{1}{2} \, \left( \|\rho^{T_{B}}\|_{1} - 1 \right) \, ,\]

where \(\rho^{T_{B}}\) is the partial transpose of \(\rho\) with respect to partition \(B\), and \(\|\cdot\|_{1}\) is the Schatten \(1\)-norm (also known as nuclear norm or trace norm).

Parameters:

  • state (ArrayLike) – statevector or density matrix.

  • bipartition (list or tuple) – indices of qubits in partition \(B\), which is partially transposed.

  • backend (qibo.backends.abstract.Backend, optional) – backend to be used in the execution. If None, it uses the current backend. Defaults to None.

Returns:

Negativity \(\operatorname{Neg}(\rho)\) of state \(\rho\).

Return type:

float

References

  1. G. Vidal, R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).

Entanglement fidelity#

qibo.quantum_info.entanglement_fidelity(channel, nqubits, state=None, backend=None)[source]#

Entanglement fidelity \(F_{\mathcal{E}}\) of a channel \(\mathcal{E}\) on state \(\rho\) is given by

\[F_{\mathcal{E}}(\rho) = F(\rho_{f}, \rho)\]

where \(F\) is the qibo.quantum_info.fidelity() function for states, and \(\rho_{f} = \mathcal{E}_{A} \otimes I_{B}(\rho)\) is the state after the channel \(\mathcal{E}\) was applied to partition \(A\).

Parameters:

  • channel (qibo.gates.channels.Channel) – quantum channel acting on partition \(A\).

  • nqubits (int) – total number of qubits in state.

  • state (ArrayLike, optional) – statevector or density matrix to be evolved by channel. If None, defaults to the maximally entangled state \(\frac{1}{\sqrt{d}} \, \sum_{k = 0}^{d - 1} \, \ket{k}\ket{k}\), where \(d = 2^{n / 2}\) and \(n\) is nqubits, which has to be even. The first \(n / 2\) qubits are paired with the last \(n / 2\) qubits. Defaults to None.

  • backend (qibo.backends.abstract.Backend, optional) – backend to be used in the execution. If None, it uses the current backend. Defaults to None.

Returns:

Entanglement fidelity \(F_{\mathcal{E}}\).

Return type:

float

Meyer-Wallach entanglement#

qibo.quantum_info.meyer_wallach_entanglement(state, backend=None)[source]#

Compute the Meyer-Wallach entanglement \(Q\) of a state,

\[Q(\rho) = 2\left(1 - \frac{1}{N} \, \sum_{k} \, \text{tr}\left(\rho_{k}^{2}\right)\right) \, ,\]

where \(\rho_{k}\) is the reduced density matrix of qubit \(k\), and \(N\) is the total number of qubits in state. We use the definition of the Meyer-Wallach entanglement as the average purity proposed in Brennen (2003), which is equivalent to the definition introduced in Meyer and Wallach (2002).

Parameters:

  • state (ArrayLike) – statevector or density matrix of a pure state.

  • backend (qibo.backends.abstract.Backend, optional) – backend to be used in the execution. If None, it uses the current backend. Defaults to None.

Returns:

Meyer-Wallach entanglement \(Q\).

Return type:

float

References

  1. G. K. Brennen, An observable measure of entanglement for pure states of multi-qubit systems, Quantum Information and Computation, vol. 3 (6), 619-626 (2003).

  2. D. A. Meyer and N. R. Wallach, Global entanglement in multiparticle systems, J. Math. Phys. 43, 4273–4278 (2002).

Entanglement capability#

qibo.quantum_info.entangling_capability(circuit, samples, seed=None, backend=None)[source]#

Return the entangling capability \(\text{Ent}\) of a parametrized circuit.

It is defined as the average Meyer-Wallach entanglement \(Q\) (qibo.quantum_info.meyer_wallach_entanglement()) of the circuit, i.e.

\[\text{Ent} = \frac{1}{|\mathcal{S}|}\sum_{\theta_{k} \in \mathcal{S}} \, Q(\rho_{k}) \, ,\]

where \(\mathcal{S}\) is the set of sampled circuit parameters, and \(\rho_{k}\) is the state prepared by the circuit with uniformly-sampled parameters \(\theta_{k}\).

Note

Currently, function does not work with circuit that contains noisy channels.

Note

The parameters of circuit are overwritten by the sampling. After the call, circuit holds the last sampled parameters.

Parameters:

  • circuit (qibo.models.Circuit) – Parametrized circuit.

  • samples (int) – number of sampled circuit parameter vectors \(|\mathcal{S}|\).

  • seed (int or numpy.random.Generator, optional) – Either a generator of random numbers or a fixed seed to initialize a generator. If None, initializes a generator with a random seed. Default: None.

  • backend (qibo.backends.abstract.Backend, optional) – backend to be used in the execution. If None, it uses the current backend. Defaults to None.

Returns:

Entangling capability \(\text{Ent}\).

Return type:

float

References

  1. S. Sim, P. D. Johnson, and A. Aspuru-Guzik, Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms, Adv. Quantum Technol. 2, 1900070 (2019).