Witnessing negative conditional entropy

Quantum states that possess negative conditional von Neumann entropy provide quantum advantage in several information-theoretic protocols including superdense coding, state merging, distributed private randomness distillation, and one-way entanglement distillation. While entanglement is an important resource, only a subset of entangled states have negative conditional von Neumann entropy. In this work, we characterize the class of density matrices having non-negative conditional von Neumann entropy as convex and compact. This allows us to prove the existence of a Hermitian operator (a witness) for the detection of states having negative conditional entropy for bipartite systems in arbitrary dimensions. We show two constructions of such witnesses. For one of the constructions, the expectation value of the witness in a state is an upper bound to the conditional entropy of the state. We pose the problem of obtaining a tight upper bound to the set of conditional entropies of states in which an operator gives the same expectation value. We solve this convex optimization problem numerically for a two qubit case and find that this enhances the usefulness of our witnesses. We also find that for a particular witness, the estimated tight upper bound matches the value of conditional entropy for Werner states. We explicate the utility of our work in the detection of useful states in several protocols.

Figure 1

Schematic diagram of ${\mathcal{F}}_{\mathrm{v}}\left(\mathcal{H}\right)$. The outer class is the set of all Hermitian matrices with trace 1. It extends beyond the figure as indicated by the dotted lines. Within that is the class of density matrices. The pure states (entangled or separable) lie farthest from the maximally mixed state. The density matrix set is the set of all convex combinations of pure state projectors, but since there are infinite such projectors, it does not form a polytope, unlike what it may seem like from the figure. Within the density matrix space is the convex class of separable states. ${\mathcal{F}}_{\mathrm{v}}\left(\mathcal{H}\right)$ is also a convex class and it contains the class of separable states.

Figure 2

The $x$ axis is $\alpha$, the isotropic mixing parameter, and refers to three-dimensional states of the form $\rho =\alpha |{\varphi }^{+}\rangle \langle {\varphi }^{+}|+(1-\alpha )\frac{I}{d^2}$. The $y$ axis represents different quantities for the red (dashed) line, and the blue (solid) line, placed together for comparison. For the red (dashed) line, it represents the value of $\mathrm{Tr}\left(W_{{\rho }_i}\rho \right)$ and for the blue (solid) line, it represents the conditional von Neumann entropy in bits. Notice that the witness gives a positive value whenever the conditional von Neumann entropy is positive.

Figure 3

Conditional entropy of Werner states and the obtained tight upper bound: The $x$ axis refers to the parameter $\alpha$ for states of the form ${\sigma }_{\alpha }=\alpha |{\varphi }^{+}\rangle \langle {\varphi }^{+}|+(1-\alpha )I/4$. The $y$ axis represents different quantities for the red (dashed) line and the blue (solid) line, placed together for comparison. For the red (dashed) line, it represents the value of $\chi \left(\alpha \right)=\mathrm{Tr}\left(W_{\gamma }{\sigma }_{\alpha }\right)$. For the blue (solid) line, it represents $S_{A|B}\left({\sigma }_{\alpha }\right)$ in bits as well the tight upper bound of $T_{W_{\gamma }}^{\chi \left(\alpha \right)}$, which happen to coincide.

Figure 4

Case 1: corresponds to the angle at ${\sigma }^{\prime }$ being obtuse.

Figure 5

Case 2: corresponds to the angle at ${\sigma }^{\prime }$ being acute.

Figure 6

Variation of conditional entropy and the value of $\mathrm{Tr}\left(W\rho \right)$ with the mixing parameter $p$. The $x$ axis is $p$, the Werner mixing parameter, and refers to two-dimensional states of the form $\rho =p|{\varphi }^{+}\rangle \langle {\varphi }^{+}|+(1-p)\frac{I}{d^2}$. The $y$ axis represents different quantities for the red (dashed) line and the blue (solid) line, placed together for comparison. For the red (dashed) line, it represents the value of $\mathrm{Tr}\left(W\rho \right)$ where $W$ is the geometric witness calculated in Eq. (21) and for the blue (solid) line, it represents the conditional entropy in bits.

Figure 7

The dynamics of the system $A$ when sent through the quantum channel $N$ with $P$ as the reference system. $B$ is the output of the system $A$ after the channel transmission.

Figure 8

The inner set represents ${\mathcal{F}}_{\mathrm{v}}\left(\mathrm{H}\right)$—the set of states that give no advantage with quantum memory over the original scenario. Beyond it, labeled $\mathcal{U}\left(\mathcal{H}\right)$, is the set of states that do give an advantage, but still have a degree of uncertainty. And beyond those (outermost) are the states with $S_{A|B}\left({\rho }_{AB}\right)\le {log}_{2}c$, for which the uncertainty bound is trivial.