Coherence

Coherence.FlagPoleState(dim, p)

Provide a flag pole state

\[\ket{\psi} = \sqrt{p} \ket{0} + \sum_{i=1}^{d-1} \sqrt{\frac{1-p}{d-1}}\ket{i}\]
Parameters:

  • dim (numeric) – The dimension of the system.

  • p (numeric) – The amplitude of the flag.

Returns:

The density matrix of the flag state.

Return type:

state (matrix)

Coherence.RobustnessCoherentChannel(channel, varargin)

Provide the robustness of a coherent channel.

\[1 + C_R(\mathcal{N}) = \min_{\mathcal{M}}\{\lambda | \mathcal{N} \le \lambda \mathcal{M}\},\]

where \(\mathcal{M}\) is a maximally incoherent operation (MIO).

Parameters:

  • channel (matrix) – The Choi matrix of the input channel \(\mathcal{N}\).

  • varargin (list) – Dimension of the given channel, default to [d_i,d_o], with d_i=d_o.

Returns:

The robustness of the input channel.

Return type:

C_R (numeric)

Note

Díaz, M. G., Fang, K., Wang, X., Rosati, M., Skotiniotis, M., Calsamiglia, J., & Winter, A. (2018). Using and reusing coherence to realize quantum processes. Quantum, 2, 100.

Coherence.RobustnessCoherentState(rho)

Provide the robustness of a coherent state.

\[1 + C_R(\rho) = \min_{\sigma}\{\lambda | \rho \le \lambda \sigma\},\]

where \(\sigma\) is the incoherent state.

Parameters:

rho (matrix) – The density matrix of quantum state.

Returns:

The robustness of the input state.

Return type:

C_R (numeric)

Note

Napoli, C., Bromley, T. R., Cianciaruso, M., Piani, M., Johnston, N., & Adesso, G. (2016). Robustness of coherence: an operational and observable measure of quantum coherence. Physical review letters, 116(15), 150502.

Coherence.SimulateCoherentChannel(target_channel, resource_state, free_op, varargin)

Find the optimal free operation \(\mathcal{M}\) to minimize diamond norm between the simulated channel \(\mathcal{M}(\omega\otimes\cdot)\) and the target channel \(\mathcal{N}\)

\[\| \mathcal{M}(\omega \otimes \cdot) - \mathcal{N}\|_\diamond,\]

where \(\omega\) is the given resource state.

Parameters:

  • target_channel (matrix) – The Choi matrix of the input channel \(\mathcal{N}\).

  • reousrce_state (matrix) – The density matrix of the given resource state \(\omega\).

  • free_op (numeric) – The choice of free operation \(\mathcal{M}\), we can choose 0 (MIO) or 1 (DIO).

  • varargin (list) – Dimension of the given channel, default to [d_i,d_o], with d_i=d_o.

Returns:

The diamond norm between the simulated channel and the target channel.

Return type:

distance (numeric)

Note

Díaz, M. G., Fang, K., Wang, X., Rosati, M., Skotiniotis, M., Calsamiglia, J., & Winter, A. (2018). Using and reusing coherence to realize quantum processes. Quantum, 2, 100.

Coherence.SimulateCoherentChannelProb(target_channel, resource_state, free_op, error_tolerance, varargin)

Find the maximally success probability \(p\) to simulate the target channel simulation with free operations \(\mathcal{M}\) and resource state \(\omega\) up to error tolerance \(\epsilon\).

\[\max_{\mathcal{M}} \{p| \mathcal{M}(\sigma\otimes\cdot)=p\mathcal{L}(\cdot) + (1-p)\mathcal{R}(\cdot), \|\mathcal{L} - \mathcal{N}\|\le \epsilon\}\]

where \(\mathcal{R}\) is a rubbish channel.

Parameters:

  • target_channel (matrix) – The Choi matrix of the input channel \(\mathcal{N}\).

  • reousrce_state (matrix) – The density matrix of the given resource state \(\omega\).

  • free_op (numeric) – The choice of free operation \(\mathcal{M}\), we can choose 0 (MIO) or 1 (DIO).

  • error_tolerance (numeric) – The maximal simulation error allowed.

  • varargin (list) – Dimension of the given channel, default to [d_i,d_o], with d_i=d_o.

Returns:

The maximally success probability of channel simulation.

Return type:

probability (numeric)

Note

Zhao, B., Ito, K., & Fujii, K. (2024). Probabilistic channel simulation using coherence. arXiv preprint arXiv:2404.06775.