# Is there a name for $I(A:B)-I(A:C)-I(A:BC)$ in (quantum) information theory?

Let $I(X:Y)=S(X)+S(Y)-S(XY)$ be the mutual information between $X$ and $Y$, where $S(X)$ is the entropy of $X$ (say the entanglement entropy of subsystem $X$ in quantum information). To characterize multi-party entanglement, the concept of tripartite information has been proposed:

$$\begin{split}I_3(A:B:C)&=I(A:B)+I(A:C)-I(A:BC)\\ &=S(A)+S(B)+S(C)-S(AB)-S(BC)-S(AC)+S(ABC).\end{split}$$

Recently I have encountered another three-party information, differed from the tripartite information by a minus sign of $I(A:C)$:

$$???=I(A:B)-I(A:C)-I(A:BC).$$

My question is: Is there a name for the above quantity in information theory (either classical or quantum information)?

Background: the motivation to study this quantity is to characterize the entanglement property of two-qubit quantum gates. Suppose a two-qubit gate takes the qubits $A$ and $D$ as input and returns $B$ and $C$ as output, described by the unitary operator

$$U=\sum_{abcd}U_{bc,ad}|b \rangle|c \rangle\langle a|\langle d|.$$

If we treat the unitary operator as a quantum state $|U\rangle$ by bending the output legs backwards:

$$|U\rangle=\sum_{abcd}U_{bc,ad}|a\rangle |b \rangle|c \rangle |d\rangle,$$

one can study the entanglement properties of the gate state $| U\rangle$ (such as the entanglement entropies over all different parts of the circuit). These entanglement properties will characterize the entanglement generation, information propagation and scrambling in quantum circuits.

In particular, for two-qubit gates, there are only two independent entanglement properties:

• the tripartite information,
• the above mentioned "unnamed" three-party information.

The (negative) tripartite information measures the amount of information about the input $A$ that is shared between the outputs $CD$ that can not be told from the separate measurement on either $C$ or $D$ alone, which characterizes the information scrambling. The (negative) unnamed information measures the amount of information that is swapped between the channels $A$-$B$ and $D$-$C$, which characterizes the information propagation. That is why I am wondering if there is a name for this quantity $I(A:B)-I(A:C)-I(A:BC)$ in the literature.

• I don’t know of a name, but this quantity has a B|AC symmetry, since it is equal to $I_3(A:B:C)-2I(A:C)$. – Frédéric Grosshans Dec 14 '17 at 17:14
• @FrédéricGrosshans Thanks for your comment. I am aware of this relation. I am not sticking to this naming question now. – Everett You Dec 15 '17 at 16:37