Non-negativity of conditional von Neumann entropy and global unitary operations

Conditional von Neumann entropy is an intriguing concept in quantum information theory. In the present work, we examine the effect of global unitary operations on the conditional entropy of the system. We start with a set containing states with a non-negative conditional entropy and find that some states preserve the non-negativity under unitary operations on the composite system. We call this class of states the absolute conditional von Neumann entropy non-negative (ACVENN) class. We characterize such states for $2\otimes 2$–dimensional systems. From a different perspective the characterization accentuates the detection of states whose conditional entropy becomes negative after the global unitary action. Interestingly, we show that this ACVENN class of states forms a set which is convex and compact. This feature enables the existence of Hermitian witness operators. With these we can distinguish the unknown states which will have a negative conditional entropy after the global unitary operation. We also show that this has immediate application to superdense coding and state merging, as the negativity of the conditional entropy plays a key role in both these information processing tasks. Some illustrations followed by analysis are also provided to probe the connection of such states with absolutely separable states and absolutely local states.

Figure 1

von Neumann entropy of the Werner state ${\sigma }_{\mathrm{wer}}$ versus the classical mixing parameter $p$.

Figure 2

von Neumann entropy of the Bell diagonal state ${\sigma }_{\mathrm{bell}}$ versus the parameter $c_i$.

Figure 3

The set ACVENN is convex and compact, and using the Hahn-Banach theorem [26] it follows that any state not belonging to ACVENN can be separated from the states that belong to ACVENN by a hyperplane, thus providing for the existence of a witness.

Figure 4

Approximate size of the ACVENN class.

Figure 5

Relation between ACVENN, AS, and separable classes of states.

Figure 6

Role of the witness in detecting states which are useful for superdense coding and state merging after global unitary transformation.