Operational Meaning of Quantum Discord

September 30, 2010 at 11:55 pm Leave a comment

After we posted our paper, Operational interpretations of quantum discord by D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter (arXiv:1008.3205), on the arXiv a similar note, Interpreting quantum discord through quantum state merging by V. Madhok and A. Datta (arXiv:1008.4135), appeared. Recently someone asked me to discuss the differences in the two papers.

So here it goes.

Let’s start by defining quantum discord. To do so, we have to first discuss bipartite mutual information. Some folks noticed, about ten years, the two classical equivalent definitions of mutual information are not the same in quantum mechanics. Suppose we have a shared (between Alice and Bob) probability distribution $p_{ab}$ with marginal distribution $p_a=\sum_b p_{ab}$ and $p_b=\sum_a p_{ab}$ . The mutual information in this distribution is given by $I(a:b)=H_a+H_b-H_{ab}$ . $H_A$ is the Shannon’s entropy, which quantifies the ignorance of Alice and is defined as $H_a=\sum_a -p_a\log(p_a)$ . Now, there is an equivalent quantity called conditional information, $J(b|a)=H_b-H(b|a)$ , where the conditional entropy if defined as $H(b|a)=\sum_a p_a H(p_{ab}/p_a)$ . The conditional entropy quantifies the ignorance of Bob given some knowledge about Alice.

Classical $\rightarrow$ quantum: $H \rightarrow S$ , $p \rightarrow \rho$ , $a,b\rightarrow A,B$ .

Once again classically we have $I(a:b)=J(b|a)$ and there is no definition for the conditional entropy, $H(b|a)$ , in quantum mechanics, i.e. $S(B|A)$ . Well there two options.

1. Stick with $I(a:b)=J(b|a)$ even for the quantum case.

$S(B|A)=S_{AB}-S_A$ .

This quantity itself has a rich history. Classically this quantity is always positive but quantum mechanically it can be negative when $AB$ is entangled. It also has a nice operational interpretation for the task known as quantum state merging.

2. Define conditional entropy by measurements on Alice’s side.

$S(B|\Pi)=\sum_i p_i S(\Pi_i \rho_{AB} \Pi_i/p_i),$

where $p_i=\mbox{Tr}[\Pi_i\rho_{AB}]$ . The only restrictions are $\Pi_i \geq 0$ and $\sum\Pi_i =I$ . In this case one can prove that $I(A:B)\geq J(B|\Pi)$ for any $\Pi$ and quantum discord is defined as

$D(B|A)=I(A:B)-\max_\Pi J(B|\Pi).$

It is this quantity that we hope to give some operational meaning to.

What both papers noticed was that quantum discord is not only the difference in two definitions of (classically equivalent) mutual information, it is also the difference in the two conditional information.

$D(B|A)=S(B|\Pi)-S(B|A)$ .

What our team was able to show was the following. We considered purification of the system $AB$ to system $ABC$ . In that case we found that $D(B|A)=\Gamma(B\rangle C)$ . Where $\Gamma(B\rangle C)$ total cost of entanglement in (extended) state merging between Bob and Charlie with the final state with Charlie.

Madhok and Datta showed that the state merging resource (cost), $S(B|A)$ , is depleted (inflated) by measurements on $A$ is equal to quantum discord. That is, $D(B|A)=S(B|\Pi)-S_A-S(B|A)+S_A$ .

At the superficial level that is the difference in two papers. We are both on to very similar ideas but the results look different, at least for now. I will try to write up something on state merging and extended state merging before I expand the discussion above.

Entry filed under: classical, discord, quantum. Tags: .

‘Don’t become a scientist’

irreversible