$J_{f,\\sigma }^{p}$ Functions

Jfpsigma(Y,sigma,p,f,f0,fpinf)

This function computes $\mathbf{J}_{f,\sigma}^{p}(Y)$. It expects the input in the computational basis and returns the matrix in the computational basis.

Jfpsigmachoi(σ,p,f,f0,fpinf)

This function returns the Choi operator of $\mathbf{J}_{f,\sigma}^{p}.$ We note this has a specific function for obtaining the Choi operator to force the user to consider $p,f,f(0+),f'(+\inf).$

innerproductf(X,Y,sigma,p,f,f0,fpinf)

This function computes $\langle X , Y \rangle_{\mathbf{J}^{p}_{f,\sigma}}$. Note that it does not check the function $f$ is an operator monotone function.

SchReversalMap(X,Ak,Bk,σ,f,f0,fpinf)

Applies the Schrodinger reversal map to X according to $f$,$\mathcal{E}$, and $σ.$ $\mathcal{E}$ is presumed to be provided in its Kraus operator form. It is assumed all inputs are expressed in the computational basis.

getcontractioncoeff(Ak, Bk, σ, f, f0, fpinf)

This returns the contraction coefficient $\eta_{\chi^{2}_{f}}(\mathcal{E},\sigma)$ for a full rank input state $\sigma$ and symmetric-inducing operator monotone function $f.$

qmaxcorrcoeff(ρA::Matrix, Ak::Vector, Bk::Vector, f, f0, fpinf)

This function computes the maximal correlation coefficient $\mu_{f}(\rho_{AB})$ when given $\rho_{A}$ and the Kraus operators of $\mathcal{E}$ such that $\rho_{AB} = (\text{id}_{A} \otimes \mathcal{E})(\psi_{\rho_{A}})$ where $\psi_{\rho_{A}}$ is the canonical purification of $\rho_{A}$.

qmaxlincorrcoeff(ρA::Matrix, Ak::Vector, Bk::Vector,k)

This function computes the maximal correlation coefficient $\mu_{f_{k}}(\rho_{AB})$ for $f_{k}(x) = x^{k}.$ Currently it requires that it is given $\rho_{A}$ and the Kraus operators of $\mathcal{E}$ such that $\rho_{AB} = (\text{id}_{A} \otimes \mathcal{E})(\psi_{\rho_{A}})$ where $\psi_{\rho_{A}}$ is the canonical purification of $\rho_{A}$.

perspective(x,y,f,f0,fpinf)

For a given function f, this computes the perspective function

\[ P_{f}(x,y) \coloneqq \begin{cases} yf(x/y) & x,y > 0 , \\ yf(0^{+}) & x = 0 , \\ xf'(+\infty) & y = 0 \end{cases}\]

where $0f(0/0) \coloneqq 0$, $0\cdot \infty \coloneqq 0$,

\[ f(0^{+}) \coloneqq \lim_{x \downarrow 0} f(x) , \quad \text{and} \quad f'(+\infty) \coloneqq \lim_{x \to +\infty} \frac{f(x)}{x} . \]

We note that we allow one to control f0 and fpinf. The function will not work if these values are wrong. We assume you will put Inf (resp. -Inf) if f0 or fpinf is infinite.

getONB(σ,p,f,f0,fpinf)

This function performs the (modified) Gram Schmidt process for the inner product spaces $\langle X,Y \rangle_{\mathbf{J}_{f,\sigma}^{p}}$ considered in the paper.

Note inputs need to be in computational basis and are returned in computational basis as the inner product value is a number and thus does not change the basis here.