state_opt.symmetric_extension_hierarchy

PPT symmetric extension hierarchy.

Module Contents

Functions

symmetric_extension_hierarchy(states[, probs, level, dim])

Compute optimal value of the symmetric extension hierarchy SDP [1].
state_opt.symmetric_extension_hierarchy.symmetric_extension_hierarchy(states, probs=None, level=2, dim=None)

Compute optimal value of the symmetric extension hierarchy SDP [1].

The probability of distinguishing a given set of states via PPT measurements serves as a natural upper bound to the value of obtaining via separable measurements. Due to the nature of separable measurements, it is not possible to optimize directly over these objects via semidefinite programming techniques.

We can, however, construct a hierarchy of semidefinite programs that attains closer and closer approximations at the separable value via the techniques described in [1].

The mathematical form of this hierarchy implemented here is explicitly given from equation 4.55 in [2].

\[\begin{split}\begin{equation} \begin{aligned} \text{maximize:} \quad & \sum_{k=1}^N p_k \langle \rho_k, \mu(k) \rangle, \\ \text{subject to:} \quad & \sum_{k=1}^N \mu(k) = \mathbb{I}_{\mathcal{X} \otimes \mathcal{Y}}, \\ & \text{Tr}_{\mathcal{Y}_2 \otimes \ldots \otimes \mathcal{Y}_s}(X_k) = \mu(k), \\ & \left( \mathbb{I}_{\mathcal{X}} \otimes \Pi_{\mathcal{Y} \ovee \mathcal{Y}_2 \ovee \ldots \ovee \mathcal{Y}_s} \right) X_k \left(\mathbb{I}_{\mathcal{X}} \otimes \Pi_{\mathcal{Y} \ovee \mathcal{Y}_2 \ovee \ldots \ovee \mathcal{Y}_s} \right) = X_k \\ & \text{T}_{\mathcal{X}}(X_k) \in \text{Pos}\left( \mathcal{X} \otimes \mathcal{Y} \otimes \mathcal{Y}_2 \otimes \ldots \otimes \mathcal{Y}_s \right), \\ & \text{T}_{\mathcal{Y}_2 \otimes \ldots \otimes \mathcal{Y}_s}(X_k) \in \text{Pos}\left( \mathcal{X} \otimes \mathcal{Y} \otimes \mathcal{Y}_2 \otimes \ldots \otimes \mathcal{Y}_s \right), \\ & X_1, \ldots, X_N \in \text{Pos}\left(\mathcal{X} \otimes \mathcal{Y} \otimes \mathcal{Y}_2 \otimes \ldots \otimes \mathcal{Y}_s \right). \end{aligned} \end{equation}\end{split}\]

Examples

It is known from [2] that distinguishing three Bell states along with a resource state \(|\tau_{\epsilon}\rangle\) via separable measurements has the following closed form

\[\frac{1}{3} \left(2 + \sqrt{1 - \epsilon^2} \right)\]

where the resource state is defined as

\[|\tau_{\epsilon} \rangle = \sqrt{\frac{1+\epsilon}{2}} |00\rangle + \sqrt{\frac{1-\epsilon}{2}} |11\rangle.\]

The value of optimally distinguishing these states via PPT measurements is strictly larger than the value one obtains from separable measurements. Calculating the first level of the hierarchy provides for us the optimal value of PPT measurements.

Consider a fixed value of \(\epsilon = 0.5\).

>>> from toqito.states import basis, bell
>>> from toqito.perms import swap
>>> import numpy as np
>>> from toqito.state_opt import symmetric_extension_hierarchy
>>> e_0, e_1 = basis(2, 0), basis(2, 1)
>>> e_00, e_11 = np.kron(e_0, e_0), np.kron(e_1, e_1)
>>>
>>> # Define the resource state.
>>> eps = 0.5
>>> eps_state = np.sqrt((1+eps)/2) * e_00 + np.sqrt((1-eps)/2) * e_11
>>> eps_dm = eps_state * eps_state.conj().T
>>>
>>> # Define the ensemble of states to be distinguished.
>>> states = [
...     np.kron(bell(0) * bell(0).conj().T, eps_dm),
...     np.kron(bell(1) * bell(1).conj().T, eps_dm),
...     np.kron(bell(2) * bell(2).conj().T, eps_dm),
...     np.kron(bell(3) * bell(3).conj().T, eps_dm),
... ]
>>>
>>> # Ensure the distinguishability is conducted on the proper spaces.
>>> states = [
...     swap(states[0], [2, 3], [2, 2, 2, 2]),
...     swap(states[1], [2, 3], [2, 2, 2, 2]),
...     swap(states[2], [2, 3], [2, 2, 2, 2]),
... ]
>>>
>>> # Calculate the first level of the symmetric extension hierarchy. This
>>> # is simply the value of optimally distinguishing via PPT measurements.
>>> '%.2f' % symmetric_extension_hierarchy(states=states, probs=None, level=1)
'0.99'
>>>
>>> # Calculating the second value gets closer to the separable value.
>>> '%.2f' % symmetric_extension_hierarchy(states=states, probs=None, level=2)
'0.96'
>>>
>>> # As proven in :cite:`Cosentino_2015_QuantumState`, the true separable value of distinguishing the
>>> # three Bell states is:
>>> '%.2f' % (1/3 * (2 + np.sqrt(1 - eps**2)))
'0.96'
>>>
>>> # Computing further levels of the hierarchy would eventually converge to
>>> # this value, however, the higher the level, the more computationally
>>> # demanding the SDP becomes.

Note

You do not need to use ‘%.2f’ % when you use this function.

We use this to format our output such that doctest compares the calculated output to the expected output upto two decimal points only. The accuracy of the solvers can calculate the float output to a certain amount of precision such that the value deviates after a few digits of accuracy.

References

[1] (1,2,3)

Miguel Navascués. Pure state estimation and the characterization of entanglement. Physical Review Letters, Feb 2008. URL: http://dx.doi.org/10.1103/PhysRevLett.100.070503, doi:10.1103/physrevlett.100.070503.

[2] (1,2)

Alessandro Cosentino. Quantum state local distinguishability via convex optimization. https://uwspace.uwaterloo.ca/handle/10012/9572, 2015.

Parameters:

  • states (list[numpy.ndarray]) – A list of states provided as either matrices or vectors.

  • probs (list[float]) – Respective list of probabilities each state is selected.

  • level (int) – Level of the hierarchy to compute.

  • dim (int | list[int]) – The default has both subsystems of equal dimension.

Returns:

The optimal probability of the symmetric extension hierarchy SDP for level level.

Return type:

float