Entropic trade-off relations for quantum operations

Spectral properties of an arbitrary matrix can be characterized by the entropy of its rescaled singular values. Any quantum operation can be described by the associated dynamical matrix or by the corresponding superoperator. The entropy of the dynamical matrix describes the degree of decoherence introduced by the map, while the entropy of the superoperator characterizes the a priori knowledge of the receiver of the outcome of a quantum channel $\Phi$. We prove that for any map acting on an $N$-dimensional quantum system the sum of both entropies is not smaller than $\ln N$. For any bistochastic map this lower bound reads $2\ln N$. We investigate also the corresponding Rényi entropies, providing an upper bound for their sum, and analyze the entanglement of the bi-partite quantum state associated with the channel.

Figure 1

An image of the Bloch ball under an exemplary one-qubit bistochastic map $\Phi$.

Figure 2

The striped region denotes the allowed set of points representing one-qubit quantum operations characterized by the entropy of the channel $S^{\mathrm{map}}\left(\Phi \right)$ and the receiver entropy $S^{\mathrm{rec}}\left(\Phi \right)$. The gray (shaded) region represents bistochastic channels, while the white striped region corresponds to interval channels. Distinguished points of the allowed set represent ($a$) a completely depolarizing channel; ($b$) an identity channel; ($c$) a coarse-graining channel; and ($d$) a channel completely contracting the entire Bloch ball into a given pure state. The action of these four channels on the Bloch ball is schematically shown in the auxiliary circular plots.

Figure 3

The striped region represents the set of one-qubit operations projected into the plane spanned by the linear entropy of the map, $S_2^{\mathrm{map}}\left(\Phi \right)$, and the linear receiver entropy, $S_2^{\mathrm{rec}}\left(\Phi \right)$, i.e., the Rényi entropies of order $q=2$. The shaded region represents the bistochastic quantum operations. The dashed diagonal line represents the lower bound, (44), which holds for bistochastic operations, while the solid diagonal line denotes the weaker bound, (43), which holds for all quantum operations. The dotted diagonal line represents the upper bound, (50), applied for $N=2$.

Figure 4

One-qubit maps projected onto the entropy plane (striped set) with superimposed bounds given in Proposition 6 concerning the separability of the dynamical matrix. Region $A$ contains no superpositive channels, while region $B$ contains both classes of the maps. Region $C$ (determined by PPT criteria) contains only superpositive channels (all corresponding states are separable). Lower, (43), and upper, (50), bounds imply that there are no one-qubit quantum operations projected into region $D$. The diagonal of the figure contains the reshuffling-invariant channels, in particular, the coarse-graining channel ($c$) and the transition depolarizing channel, ${\Phi }_{1/3}=\frac{1}{3}\mathbb{1}+\frac{2}{3}{\Phi }_{*}$, located at the boundary of the set of superpositive maps ($e$).

Figure 5

The region of allowed values of von Neumann entropies [$S^{\mathrm{map}}\left(\Phi \right),S(\Phi \left(\frac{\mathbb{1}}{N}\right))$] for one-qubit quantum channels $\Phi$. Extremal lines denote the bounds proven in Proposition 8. The four distinguished points represent ($a$) a completely depolarizing channel, ($b$) an identity channel, ($c$) a coarse-graining channel, and ($d$) channels describing spontaneous emission as shown in Fig 2. The shading denotes an estimation of the probability density of [$S^{\mathrm{map}}\left(\Phi \right),S(\Phi \left(\frac{\mathbb{1}}{N}\right))$], when $\Phi$ is chosen randomly [38]. The lower bound on the sum $S^{\mathrm{map}}\left(\Phi \right)+S(\Phi \left(\frac{\mathbb{1}}{N}\right))$ is, in general, not saturated.