toqito.state_opt.bell_inequality_max

Computes max values for Bell inequalities (General and Qubit-specific).

Module Contents

toqito.state_opt.bell_inequality_max.bell_inequality_max(coefficients, desc, notation, mtype='classical', k=1, tol=1e-08)[source]

Compute the maximum value of a Bell inequality.

Calculates the maximum value achievable for a given Bell inequality under classical, quantum, or no-signalling assumptions.

The maximum classical and no-signalling values are computed exactly. The maximum quantum value is upper bounded using the NPA (Navascués-Pironio-Acín) hierarchy `Navascues_2008_AConvergent`[@Navascues_2008_AConvergent].

Examples

The CHSH inequality in Full Correlator (FC) notation. The classical maximum is 2, the quantum maximum (Tsirelson’s bound) is (2sqrt{2}), and the no-signalling maximum is 4.

[

langle A_1 B_1 rangle + langle A_1 B_2 rangle + langle A_2 B_1 rangle - langle A_2 B_2 rangle le V

]

Represented by the coefficient matrix:

[

M_{FC} = begin{pmatrix} 0 & 0 & 0 \ 0 & 1 & 1 \ 0 & 1 & -1 end{pmatrix}

]

`python exec="1" source="above" import numpy as np from toqito.state_opt.bell_inequality_max import bell_inequality_max M_chsh_fc = np.array([[0, 0, 0], [0, 1, 1], [0, 1, -1]]) desc_chsh = [2, 2, 2, 2] bell_inequality_max(M_chsh_fc, desc_chsh, 'fc', 'classical') bell_inequality_max(M_chsh_fc, desc_chsh, 'fc', 'quantum', tol=1e-7) print(bell_inequality_max(M_chsh_fc, desc_chsh, 'fc', 'nosignal', tol=1e-9)) `

The CHSH inequality in Collins-Gisin (CG) notation. The classical maximum is 0, the quantum maximum is (1/sqrt{2} - 1/2), and the no-signalling maximum is 1/2.

[

p(00|11)+p(00|12)+p(00|21)-p(00|22)-p_A(0|1)-p_B(0|1) le V

]

Represented by the coefficient matrix:

[

M_{CG} = begin{pmatrix} 0 & -1 & 0 \ -1 & 1 & 1 \ 0 & 1 & -1 end{pmatrix}

]

`python exec="1" source="above" import numpy as np from toqito.state_opt.bell_inequality_max import bell_inequality_max M_chsh_cg = np.array([[0, -1, 0], [-1, 1, 1], [0, 1, -1]]) desc_chsh = [2, 2, 2, 2] bell_inequality_max(M_chsh_cg, desc_chsh, 'cg', 'classical') bell_inequality_max(M_chsh_cg, desc_chsh, 'cg', 'quantum', tol=1e-7) print(bell_inequality_max(M_chsh_cg, desc_chsh, 'cg', 'nosignal', tol=1e-9)) `

The I3322 inequality in Collins-Gisin (CG) notation. Classical max = 1, No-signalling max = 2. Quantum value is between 1 and 2.

`python exec="1" source="above" import numpy as np from toqito.state_opt.bell_inequality_max import bell_inequality_max M_i3322_cg = np.array([[0, 1, 0, 0], [1, -1, -1, -1], [0, -1, -1, 1], [0, -1, 1, 0]]) desc_i3322 = [2, 2, 3, 3] bell_inequality_max(M_i3322_cg, desc_i3322, 'cg', 'classical') bell_inequality_max(M_i3322_cg, desc_i3322, 'cg', 'quantum', k=1, tol=1e-7) bell_inequality_max(M_i3322_cg, desc_i3322, 'cg', 'quantum', k='1+ab', tol=1e-7) print(bell_inequality_max(M_i3322_cg, desc_i3322, 'cg', 'nosignal', tol=1e-9)) `

Parameters:

  • coefficients (numpy.ndarray) – A matrix or tensor specifying the Bell inequality coefficients in either full probability (FP), full correlator (FC), or Collins-Gisin (CG) notation.

  • desc (list[int]) – A list [(oa), (ob), (ma), (mb)] describing the number of outputs for Alice ((oa)) and Bob ((ob)), and the number of inputs for Alice ((ma)) and Bob ((mb)).

  • notation (str) – A string (‘fp’, ‘fc’, or ‘cg’) indicating the notation of the coefficients.

  • mtype (str) – The type of theory to maximize over (‘classical’, ‘quantum’, or ‘nosignal’). Defaults to ‘classical’. Note: ‘quantum’ computes an upper bound via NPA hierarchy.

  • k (int | str) – The level of the NPA hierarchy to use if mtype='quantum'. Can be an integer (e.g., 1, 2) or a string specifying intermediate levels (e.g., ‘1+ab’, ‘1+aab’). Defaults to 1. Higher levels yield tighter bounds but require more computation. Ignored if mtype is not ‘quantum’.

  • tol (float) – Tolerance for numerical comparisons and solver precision. Defaults to 1e-8.

Returns:

The maximum value (or quantum upper bound) of the Bell inequality.

Raises:

  • ValueError – If the input notation is invalid.

  • ValueError – If the input mtype is invalid.

  • ValueError – If notation conversion fails (e.g., ‘fc’ for non-binary outputs).

  • ValueError – If the NPA level k is invalid.

  • ValueError – If generating NPA constraints fails.

  • cp.error.SolverError – If the cp solver fails.

Return type:

float

toqito.state_opt.bell_inequality_max.bell_inequality_max_qubits(joint_coe, a_coe, b_coe, a_val, b_val, solver_name='SCS')[source]

Return the upper bound for the maximum violation(Tsirelson Bound) for a given bipartite Bell inequality.

This computes the upper bound for the maximum value of a given bipartite Bell inequality using an SDP. The method is from [@Navascues_2014_Characterization] and the implementation is based on [@QETLAB_link]. This is useful for various tasks in device independent quantum information processing.

The function formulates the problem as a SDP problem in the following format for the (W)-state.

[

begin{multline} max operatorname{tr}!Bigl( W cdot sum_{a,b,x,y} B^{xy}_{ab}, M^x_a otimes N^y_b Bigr),\[1ex] text{s.t.} quad operatorname{tr}(W) = 1,quad W ge 0,\[1ex] W^{T_P} ge 0,quad text{for all bipartitions } P. end{multline}

]

Examples

Consider the I3322 Bell inequality from [@Collins_2004].

[

begin{aligned} I_{3322} &= P(A_1 = B_1) + P(B_1 = A_2) + P(A_2 = B_2) + P(B_2 = A_3) \

&quad - P(A_1 = B_2) - P(A_2 = B_3) - P(A_3 = B_1) - P(A_3 = B_3) \ &le 2

end{aligned}

]

The individual and joint coefficents and measurement values are encoded as matrices. The upper bound can then be found in |toqito⟩ as follows.

```python exec=”1” source=”above” import numpy as np from toqito.state_opt.bell_inequality_max import bell_inequality_max_qubits

joint_coe = np.array([

[1, 1, -1], [1, 1, 1], [-1, 1, 0],

]) a_coe = np.array([0, -1, 0]) b_coe = np.array([-1, -2, 0]) a_val = np.array([0, 1]) b_val = np.array([0, 1])

result = bell_inequality_max_qubits(joint_coe, a_coe, b_coe, a_val, b_val) print(f”Bell inequality maximum value: {result:.3f}”) ```

Raises:

ValueError – If a_val or b_val are not length 2.

Parameters:

  • joint_coe (numpy.ndarray) – The coefficents for terms containing both A and B.

  • a_coe (numpy.ndarray) – The coefficent for terms only containing A.

  • b_coe (numpy.ndarray) – The coefficent for terms only containing B.

  • a_val (numpy.ndarray) – The value of each measurement outcome for A.

  • b_val (numpy.ndarray) – The value of each measurement outcome for B.

  • solver_name (str) – The solver used.

Returns:

The upper bound for the maximum violation of the Bell inequality.

Return type:

float