Coherence

Coherence.FlagPoleState(dim, p)

Construct a flag-pole state in the computational basis.

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

so that \(\rho = \ket{\psi}\!\bra{\psi}\) is a pure state.

Parameters:

  • dim (numeric) – System dimension \(d \ge 2\).

  • p (numeric) – Flag amplitude \(p \in [0,1]\).

Returns:

Density matrix \(\rho = \ket{\psi}\!\bra{\psi}\) of the flag-pole state.

Return type:

state (matrix)

Coherence.RobustnessCoherentChannel(channel, varargin)

Compute the robustness of coherence of a quantum channel in Choi form.

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

where \(\mathcal{M}\) ranges over maximally incoherent operations (MIO).

Parameters:

  • channel (matrix) – Choi matrix \(J_\mathcal{N} \in \mathbb{C}^{(d_i d_o)\times(d_i d_o)}\) of the input channel \(\mathcal{N}\).

  • varargin (list) – Channel dimensions, default [d_i, d_o] with d_i = d_o = sqrt{text{size}(J_mathcal{N},1)}.

Returns:

Robustness of coherence \(C_R(\mathcal{N}) \ge 0\) for 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)

Compute the robustness of coherence of a quantum state \(\rho\).

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

where \(\sigma\) is an incoherent (diagonal) state in the reference basis.

Parameters:

rho (matrix) – Density matrix \(\rho \in \mathbb{C}^{d\times d}\) (positive semidefinite, \(\mathrm{Tr}\,\rho=1\)).

Returns:

Robustness of coherence \(C_R(\rho) \ge 0\).

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 \(\mathcal{M}\) that minimizes the diamond-norm distance between the simulated channel \(\mathcal{M}(\omega \otimes \cdot)\) and the target channel \(\mathcal{N}\).

\[\min_{\mathcal{M}}\; \bigl\| \mathcal{M}(\omega \otimes \cdot) - \mathcal{N} \bigr\|_\diamond\]

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

Parameters:

  • target_channel (matrix) – Choi matrix \(J_\mathcal{N} \in \mathbb{C}^{(d_i d_o)\times(d_i d_o)}\) of the target channel \(\mathcal{N}\).

  • resource_state (matrix) – Density matrix \(\omega \in \mathbb{C}^{d_r\times d_r}\) of the resource state.

  • free_op (numeric) – Choice of free operation \(\mathcal{M}\); 0 for MIO, 1 for DIO.

  • varargin (list) – Channel dimensions, default [d_i, d_o] with d_i = d_o = sqrt{text{size}(J_mathcal{N},1)}.

Returns:

Diamond-norm distance 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 maximal success probability \(p\) to simulate the target channel using free operations \(\mathcal{M}\) and a resource state \(\omega\), up to an allowed diamond-norm error \(\epsilon\).

\[\max_{\mathcal{M}} \left\{ p \;\middle|\; \mathcal{M}(\omega \otimes \cdot) = p\,\mathcal{L}(\cdot) + (1-p)\,\mathcal{R}(\cdot),\; \|\mathcal{L} - \mathcal{N}\|_\diamond \le \epsilon \right\}\]

where \(\mathcal{N}\) is the target channel, \(\mathcal{L}\) is the implemented channel, and \(\mathcal{R}\) is an arbitrary CPTP “rubbish” channel.

Parameters:

  • target_channel (matrix) – Choi matrix \(J_\mathcal{N} \in \mathbb{C}^{(d_i d_o)\times(d_i d_o)}\) of the target channel \(\mathcal{N}\).

  • resource_state (matrix) – Density matrix \(\omega \in \mathbb{C}^{d_r\times d_r}\) of the given resource state.

  • free_op (numeric) – Choice of free operation \(\mathcal{M}\); 0 for MIO, 1 for DIO.

  • error_tolerance (numeric) – Allowed error \(\epsilon \ge 0\) measured in diamond norm.

  • varargin (list) – Channel dimensions, default [d_i, d_o] with d_i = d_o = sqrt{text{size}(J_mathcal{N},1)}.

Returns:

Maximal success probability \(p \in [0,1]\) for the simulation.

Return type:

probability (numeric)

Note

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