Magic.MagicQudit

Magic.MagicQudit.Chan_Mana(JN, DIM)
\[\mathcal{M}(\mathcal{N}_{A \rightarrow B}) = \log \max_\mathbf{u} \| \mathcal{N}_{A \rightarrow B}(A_A^\mathbf{u})\|_{W,1} = \log \max_u \sum_v |W_{\mathcal{N}}(\mathbf{v}|\mathbf{u})|\]
Parameters:

  • JN (numeric) – The choi matrix of the quantum channel.

  • DIM (int) – The dimension vector [da db], where da and db are the input and output dimensions of the channel.

Returns:

Channel’s mana.

Return type:

numeric

Raises:

error – If the input and output dimension does not match with the channel, an error is raised.

Examples:
mana = Chan_Mana(JN, [3 3]);
% Compute the mana of a qutrit-qutrit channel.

Note

Wang, X., Wilde, M. M., & Su, Y. (2019). Quantifying the magic of quantum channels. New Journal of Physics, 21(10), 103002.

Magic.MagicQudit.Generate_A(dim, num_copy)
\[ \begin{align}\begin{aligned}T_{\mathbf{u}}=\tau^{-a_1 a_2} Z^{a_1} X^{a_2}, \tau=e^{(d+1) \pi i / d}\\A_\mathbf{0}=\frac{1}{d} \sum_{\mathbf{u}} T_{\mathbf{u}}, A_{\mathbf{u}}=T_{\mathbf{u}} A_\mathbf{0} T_{\mathbf{u}}^{\dagger}.\end{aligned}\end{align} \]

Generate Heisenberg-Weyl Operators and n-copy Phase-Space Point Operators

Parameters:

  • dim (numeric) – dimension of the operators.

  • num_copy (numeric) – number of copies for phase-space point operators.

Returns:

cell array containing the generated operators.

Return type:

An (numeric)

Raises:

error – None.

Examples:
An = Generate_A(3, 2);
% Generate 2-copy phase-space point operators for dimension 3.

Note

Emerson, J. (2014). The resource theory of stabilizer computation. Bulletin of the American Physical Society, 59.

Magic.MagicQudit.Mana_state(rho)
\[\mathcal{M}(\rho) = \log \sum_\mathbf{u} |W_{\rho}(\mathbf{u})|,\]

where \(W_{(\rho)}\) is the Wigner representation of \(\rho\).

Parameters:

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

Returns:

Magic mana of the given quantum state.

Return type:

numeric

Raises:

error – If either input/output dimension does not match, an error is raised.

Note

Emerson, J. (2014). The resource theory of stabilizer computation. Bulletin of the American Physical Society, 59.

Magic.MagicQudit.QutritHstate()

Generate H state

Returns:

qudit H state.

Return type:

H (numeric)

Note

Emerson, J. (2014). The resource theory of stabilizer computation. Bulletin of the American Physical Society, 59.

Magic.MagicQudit.QutritNstate()

Generate qutrit N state

Returns:

qudit N state.

Return type:

N (numeric)

Note

Emerson, J. (2014). The resource theory of stabilizer computation. Bulletin of the American Physical Society, 59.

Magic.MagicQudit.QutritSstate()

Generate S state

Returns:

qudit S state.

Return type:

S (numeric)

Note

Emerson, J. (2014). The resource theory of stabilizer computation. Bulletin of the American Physical Society, 59.

Magic.MagicQudit.QutritTstate()

Generate qutrit T state

Returns:

qudit T state.

Return type:

T (numeric)

Note

Emerson, J. (2014). The resource theory of stabilizer computation. Bulletin of the American Physical Society, 59.

Magic.MagicQudit.WTraceNorm(PhaseOp, rho)

Compute the Wigner trace norm of a linear operator with respect to a phase-space frame.

Parameters:

  • PhaseOp (numeric) – Cell array of phase-space point operators (frame).

  • rho (numeric) – Density matrix.

Returns:

Wigner trace norm of the given operator.

Return type:

output (numeric)

Raises:

error – If either input/output dimensions does not match, an error is raised.

Examples:
output = WTraceNorm(PhaseOp, rho);
% Compute the Wigner trace norm of rho given a phase-space frame.

Note

Wang, X., Wilde, M. M., & Su, Y. (2020). Efficiently computable bounds for magic state distillation. Physical review letters, 124(9), 090505.