On Quantum States with a Finite-Dimensional Approximation Property

Abstract

We consider a class (convex set) of quantum states containing all finite rank states and infinite rank states with the sufficient rate of decreasing of eigenvalues (in particular, all Gaussian states). Quantum states from this class are characterized by the property (called the FA-property) that allows to obtain several results concerning finite-dimensional approximation of basic entropic and information characteristics of quantum systems and channels.

We obtain a simple sufficient condition of the FA-property. We show that all the states with the FA-property form a face of the convex set of all quantum states that is contained within the face of all states with finite von Neumann entropy (the non-coincidence of these two faces follows from the recent result of S. Becker, N. Datta and M.G. Jabbour).

We obtain uniform approximation results for characteristics depending on a pair (channel, input state) and for characteristics depending on a pair (channel, input ensemble). We establish the uniform continuity of the above characteristics as functions of a channel w.r.t. the strong convergence provided that the FA-property holds either for the input state or for the average state of input ensemble.

APPENDIX

Lemma 1. Let $f$ be a function on $\mathfrak{S}(\mathcal{H})$ satisfying inequality (20) with some parameters $a_{f}$ and $b_{f}$ then for any positive trace-non-increasing linear map $\Phi:\mathfrak{T}(\mathcal{H})\rightarrow\mathfrak{T}(\mathcal{H})$ the function¹¹

$$f_{\Phi}(\rho)=||\Phi(\rho)||_{1}f\left(\frac{\Phi(\rho)}{||\Phi(\rho)||_{1}}\right)$$

on $\mathfrak{S}(\mathcal{H})$ satisfies inequality (20) with the same parameters $a_{f}$ and $b_{f}$.

Proof. Let $\rho$ and $\sigma$ be arbitrary states and $\lambda$ any number in $(0,1)$. Let $p=||\Phi(\rho)||_{1}$ and $q=||\Phi(\sigma)||_{1}$. If $pq=0$ then inequality (20) trivially holds for the function $f_{\Phi}$. If $pq\neq 0$ then the validity of the l.h.s. of inequality (20) for the function $f$ implies

$$\displaystyle f_{\Phi}(\lambda\rho+\bar{\lambda}\sigma)\displaystyle=(\lambda p+\bar{\lambda}q)f\left(\frac{\lambda p}{\lambda p+\bar{\lambda}q}\frac{\Phi(\rho)}{p}+\frac{\bar{\lambda}q}{\lambda p+\bar{\lambda}q}\frac{\Phi(\sigma)}{q}\right)$$

$${}\geq\lambda pf\left(\frac{\Phi(\rho)}{p}\right)+\bar{\lambda}qf\left(\frac{\Phi(\sigma)}{q}\right)-a_{f}(\lambda p+\bar{\lambda}q)h_{2}\left(\frac{\lambda p}{\lambda p+\bar{\lambda}q}\right)$$

$${}=\lambda f_{\Phi}(\rho)+\bar{\lambda}f_{\Phi}(\sigma)-a_{f}(\lambda p+\bar{\lambda}q)h_{2}\left(\frac{\lambda p}{\lambda p+\bar{\lambda}q}\right),\quad\bar{\lambda}=1-\lambda.$$

So, to prove the l.h.s. of inequality (20) for the function $f_{\Phi}$ it suffices to show that

$$(x+y)h_{2}\left(\frac{x}{x+y}\right)\leq h_{2}(x)$$

for all positive numbers $x$ and $y$ such that $x+y\leq 1$. This inequality follows from the concavity of the binary entropy, since the probability distribution $\{x,1-x\}$ is the convex mixture of the probability distributions $\{x/(x+y),y/(x+y)\}$ and $\{0,1\}$ with the coefficients $x+y$ and $1-x-y$.

The r.h.s. of inequality (20) for the function $f_{\Phi}$ is proved similarly. $\Box$