qom.solvers.measure module

Module to solve for various classical and quantum signatures and system measures.

References

[1] (1,2)

18. Simon, Peres-Horodechi Separability Criterion for Continuous Variable Systems, Phys. Rev. Lett. 84, 2726 (2000).

[2] (1,2)

4. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger and M. Aspelmeyer, Quantum Entanglement between a Movable Mirror and a Cavity Field, Phys. Rev. Lett. 98, 030405 (2007).

[3] (1,2)

19. Olivares, Quantum Optics in Phase Space: A Tutorial on Gaussian States, Eur. Phys. J. Special Topics 203, 3 (2012).

[4] (1,2,3)

13. Ludwig and F. Marquardt, Quantum Many-body Dynamics in Optomechanical Arrays, Phys. Rev. Lett. 111, 073603 (2013).

[5] (1,2)

1. Mari, A. Farace, N. Didier, V. Giovannetti and R. Fazio, Measures of Quantum Synchronization in Continuous Variable Systems, Phys. Rev. Lett. 111, 103605 (2013).

[6] (1,2)

6. Galve, G. L. Giorgi and R. Zambrini, Quantum Correlations and Synchronization Measures, Lectures on General Quantum Correlations and their Applications, Quantum Science and Technology, Springer (2017).

[7]

14. Yang, A. Miranowicz, Y.-C. Liu, K. Xia and F. Nori, Chaotic Synchronization of Two Optical Cavity Modes in Optomechanical Systems, Sci. Rep. *9*, 15874 (2019).

[8]

20. 6. Roque, F. Marquardt and O. M. Yevtushenko, Nonlinear Dynamics of Weakly Dissipative Optomechanical Systems, New J. Phys. 22, 013049 (2020).

class qom.solvers.measure.QCMSolver(Modes, Corrs, params: dict, cb_update=None)

Bases: object

Class to solve for quantum correlation measures.

Initializes Modes, Corrs, Omega_s (symplectic matrix), params and updater.

Parameters:

• Modes (numpy.ndarray) – Classical modes with shape (dim, num_modes).

• Corrs (numpy.ndarray) – Quadrature quadrature correlations with shape (dim, 2 * num_modes, 2 * num_modes).

• params (dict) –

  Parameters for the solver. Available options are:

  key	value
  ’show_progress’	(bool) option to display the progress of the solver. Default is False.
  ’measure_codes’	(list or str) codenames of the measures to calculate. Options are 'discord_G' for Gaussian quantum discord [3], 'entan_ln' for quantum entanglement (using matrix multiplications, fallback) [1], 'entan_ln_2' for quantum entanglement (using analytical expressions) [2], 'sync_c' for complete quantum synchronization [4], 'sync_p' for quantum phase synchronization [4]]). Default is ['entan_ln'].
  ’indices’	(list or tuple) indices of the modes as a list or tuple of two integers. Default is (0, 1).

• cb_update (callable, optional) – Callback function to update status and progress, formatted as cb_update(status, progress, reset), where status is a string, progress is a float and reset is a boolean.

desc = 'Quantum Correlations Measure Solver'

Description of the solver.

Type:

str

get_correlation_Pearson(pos_i: int, pos_j: int)

Method to obtain the Pearson correlation coefficient.

The implementation measure reads as [6],

\[C_{ij} = \frac{\Sigma_{t} \langle \delta \mathcal{O}_{i} (t) \delta \mathcal{O}_{j} (t) \rangle}{\sqrt{\Sigma_{t} \langle \delta \mathcal{O}_{i}^{2} (t) \rangle} \sqrt{\Sigma_{t} \langle \delta \mathcal{O}_{j}^{2} (t) \rangle}}\]

where \(\delta \mathcal{O}_{i}\) and \(\delta \mathcal{O}_{j}\) are the corresponding quadratures of quantum fluctuations.

Parameters:

• pos_i (int) – Index of ith quadrature.

• pos_j (int) – Index of jth quadrature.

Returns:

Corr_P – Pearson correlation coefficients.

Return type:

float

get_discord_Gaussian(pos_i: int, pos_j: int)

Method to obtain Gaussian quantum discord values [3].

Parameters:

• pos_i (int) – Index of ith quadrature.

• pos_j (int) – Index of jth quadrature.

Returns:

Discord_G – Gaussian quantum discord values.

Return type:

numpy.ndarray

get_entanglement_logarithmic_negativity(pos_i: int, pos_j: int)

Method to obtain the logarithmic negativity entanglement values using matrices [1].

Parameters:

• pos_i (int) – Index of ith quadrature.

• pos_j (int) – Index of jth quadrature.

Returns:

Entan_lns – Logarithmic negativity entanglement values using matrices.

Return type:

numpy.ndarray

get_entanglement_logarithmic_negativity_2(pos_i: int, pos_j: int)

Method to obtain the logarithmic negativity entanglement values using analytical expression [2].

Parameters:

• pos_i (int) – Index of ith quadrature.

• pos_j (int) – Index of jth quadrature.

Returns:

Entan_ln – Logarithmic negativity entanglement values using analytical expression.

Return type:

numpy.ndarray

get_invariants(pos_i: int, pos_j: int)

Helper function to calculate symplectic invariants for two modes given the correlation matrices of their quadratures.

Parameters:

• pos_i (int) – Index of ith quadrature.

• pos_j (int8) – Index of jth quadrature.

Returns:

• I_1s (numpy.ndarray) – Determinants of A.

• I_2s (numpy.ndarray) – Determinants of B.

• I_3s (numpy.ndarray) – Determinants of C.

• I_4s (numpy.ndarray) – Determinants of corrs_modes.

get_measures()

Method to obtain the each measure.

Returns:

Measures – Measures calculated with shape (dim, num_measure_codes).

Return type:

numpy.ndarray

get_submatrices(pos_i: int, pos_j: int)

Helper function to obtain the block matrices of the required modes and its components.

Parameters:

• pos_i (int) – Index of ith quadrature.

• pos_j (int) – Index of jth quadrature.

Returns:

• Corrs_modes (numpy.ndarray) – Correlation matrix of the required modes.

• A (numpy.ndarray) – Correlation matrix of the first mode.

• B (numpy.ndarray) – Correlation matrix of the first mode.

• C (numpy.ndarray) – Correlation matrix of the cross mode.

get_synchronization_complete(pos_i: int, pos_j: int)

Method to obtain the complete quantum synchronization values [5].

Parameters:

• pos_i (int) – Index of ith quadrature.

• pos_j (int) – Index of jth quadrature.

Returns:

Sync_c – Complete quantum synchronization values.

Return type:

numpy.ndarray

get_synchronization_phase(pos_i: int, pos_j: int)

Method to obtain the quantum phase synchronization values [5].

Parameters:

• pos_i (int) – Index of ith quadrature.

• pos_j (int) – Index of jth quadrature.

Returns:

Sync_p – Quantum phase synchronization values.

Return type:

numpy.ndarray

method_codes = {'corrs_P_p': 'get_correlation_Pearson', 'corrs_P_q': 'get_correlation_Pearson', 'discord_G': 'get_discord_Gaussian', 'entan_ln': 'get_entanglement_logarithmic_negativity', 'entan_ln_2': 'get_entanglement_logarithmic_negativity_2', 'sync_c': 'get_synchronization_complete', 'sync_p': 'get_synchronization_phase'}

Codenames of available methods.

Type:

dict

name = 'QCMSolver'

Name of the solver.

Type:

str

set_params(params: dict)

Method to validate and set the solver parameters.

Parameters:

params (dict) – Parameters of the solver.

solver_defaults = {'indices': (0, 1), 'measure_codes': ['entan_ln'], 'show_progress': False}

Default parameters of the solver.

Type:

dict

qom.solvers.measure.get_Lyapunov_exponents(system, modes, t=None, params: dict = {}, cb_update=None)

Method to obtain the Lyapunov exponents [8].

Parameters:

• system (qom.systems.*) – Instance of the system. Requires predefined system method get_ivc and get_A.

• modes (numpy.ndarray) – Final classical modes with shape (num_modes, ).

• t (numpy.ndarray, optional) – Final time of the evolution.

• params (dict) –

  Parameters for the solver. Refer to qom.solvers.deterministic.ODESolver for all available options. Additionally available options are:

  key	value
  ’show_progress’	(bool) option to display the progress of the solver. Default is False.
  ’num_steps’	(int) number of additional time steps to calculate the deviations for Lyapunov exponents. Default value is 1000.
  ’step_size’	(float) step size of each time step. Default value is 0.1.
  ’use_svd’	(bool) option to use the singular value decomposition method to calculate the Lyapunov exponents. If False, the Gram-Schmidt orthonormalization method is used. Default is True.

• cb_update (callable, optional) – Callback function to update status and progress, formatted as cb_update(status, progress, reset), where status is a string, progress is a float and reset is a boolean.

Returns:

lambdas – Lyapunov exponents with shape (2 * num_modes, ).

Return type:

numpy.ndarray

qom.solvers.measure.get_Wigner_distributions_single_mode(Corrs, params, cb_update=None)

Method to obtain single-mode Wigner distribitions.

Parameters:

• Corrs (numpy.ndarray) – Quadrature quadrature correlations with shape (dim, 2 * num_modes, 2 * num_modes).

• params (dict) –

  Parameters of the solver. Available options are:

  key	value
  ’show_progress’	(bool) option to display the progress of the solver. Default is False.
  ’indices’	(list or tuple) indices of the modes as a list. Default is [0].
  ’wigner_xs’	(list) X-axis values.
  ’wigner_ys’	(list) Y-axis values.

• cb_update (callable, optional) – Callback function to update status and progress, formatted as cb_update(status, progress, reset), where status is a string, progress is a float and reset is a boolean.

Returns:

Wigners – Single-mode Wigner distributions of shape (dim_1, dim_0, p_dim, q_dim), where dim_1 and dim_0 are the first dimensions of the correlations and the indices respectively.

Return type:

numpy.ndarray

qom.solvers.measure.get_Wigner_distributions_two_mode(Corrs, params, cb_update=None)

Method to obtain two-mode Wigner distribitions.

Parameters:

• Corrs (numpy.ndarray) – Quadrature quadrature correlations with shape (dim, 2 * num_modes, 2 * num_modes).

• params (dict) –

  Parameters of the solver. Available options are:

  key	value
  ’show_progress’	(bool) option to display the progress of the solver. Default is False.
  ’indices’	(list or tuple) list of indices of the modes and their quadratures as tuples or lists. Default is [(0, 0), (1, 0)].
  ’wigner_xs’	(list) X-axis values.
  ’wigner_ys’	(list) Y-axis values.

• cb_update (callable, optional) – Callback function to update status and progress, formatted as cb_update(status, progress, reset), where status is a string, progress is a float and reset is a boolean.

Returns:

Wigners – Two-mode Wigner distributions of shape (dim_0, p_dim, q_dim), where dim_0 is the first dimension of the correlations.

Return type:

numpy.ndarray

qom.solvers.measure.get_average_amplitude_difference(Modes)

Method to obtain the average amplitude differences for two specific modes [7].

Parameters:

Modes (numpy.ndarray) – The two specific modes with shape (dim, 2).

Returns:

diff_a – The average amplitude difference.

Return type:

float

qom.solvers.measure.get_average_phase_difference(Modes)

Method to obtain the average phase differences for two specific modes [4].

Parameters:

Modes (numpy.ndarray) – The two specific modes with shape (dim, 2).

Returns:

diff_p – The average phase difference.

Return type:

float

qom.solvers.measure.get_bifurcation_amplitudes(Modes)

Method to obtain the bifurcation amplitudes of the modes.

Parameters:

Modes (numpy.ndarray) – The mode amplitudes in the trajectory with shape (dim, num_modes).

Returns:

Amps – The bifurcation amplitudes of the modes. The first num_modes arrays contain the bifurcation amplitudes of the real parts of the modes; the next num_modes arrays contain those of the imaginary parts.

Return type:

list

qom.solvers.measure.get_correlation_Pearson(Modes)

Method to obtain the Pearson correlation coefficient for two specific modes [6].

\[C_{ij} = \frac{\Sigma_{t} \langle \mathcal{O}_{i} (t) \mathcal{O}_{j} (t) \rangle}{\sqrt{\Sigma_{t} \langle \mathcal{O}_{i}^{2} (t) \rangle} \sqrt{\Sigma_{t} \langle \mathcal{O}_{j}^{2} (t) \rangle}}\]

where \(\mathcal{O}_{i}\) and \(\mathcal{O}_{j}\) are the corresponding modes.

Parameters:

Modes (numpy.ndarray) – The two specific modes with shape (dim, 2).

Returns:

corr_P – Pearson correlation coefficient.

Return type:

float

qom.solvers.measure.get_stability_zone(counts)

Function to obtain the stability zone given the number of unstable roots.

Parameters:

counts (list or numpy.ndarray) – Array of number of eigenvalues with positive real parts.

Returns:

stability_zone –

Stability zone indicator of the optical steady state. The indicator is calculated in the following pattern:

value	meaning
0	one unstable root.
1	one stable root.
2	three unstable roots.
3	one stable root and two unstable roots.
4	two stable roots and one unstable root.
5	three stable roots.
6	five unstable roots.
7	one stable root and four unstable roots.
8	two stable roots and three unstable root.
9	three stable roots and two unstable root.
10	four stable roots and one unstable root.
11	five stable roots.

Return type:

int

qom.solvers.measure.get_system_measures(system, Modes, T=None, params: dict = {}, cb_update=None)

Method to obtain the measures from a system method.

Parameters:

• system (qom.systems.*) – Instance of the system. Requires predefined system method get_ivc and the getter for the system measure.

• Modes (numpy.ndarray) – Classical modes with shape (dim, num_modes).

• T (numpy.ndarray, optional) – Times with shape (dim, ).

• params (dict) –

  Parameters for the solver. Available options are:

  key	value
  ’show_progress’	(bool) option to display the progress of the solver. Default is False.
  ’system_measure_name’	(str) name of the system measure to calculate. Requires a callable formatted with the prefix 'get_' and arguments modes, c and t. For e.g., get_A(modes, c, t).

• cb_update (callable, optional) – Callback function to update status and progress, formatted as cb_update(status, progress, reset), where status is a string, progress is a float and reset is a boolean.

Returns:

measures – Measures obtained with shape (dim, ) plus the shape of each measure.

Return type:

numpy.ndarray