Foundations of Quantum Discord

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I defined classical follows that the quantum mutual information and entan- correlations as C(A, B) = S(A/B) and I quantified en- glement are monotones (non-increasing) under LO (all tanglement using the relative entropy of entanglement the relevant proofs can be found in [2]). E(A, B). However, monotonicity under LO is not true for dis- My then student, Leah Henderson, and I discovered cord! We can start with a state with no discord and ac- that the sum C + E is mostly smaller than IQ for mixed tually create some by LO. A simple example is a stating states [4]. In other words, there is more to quantum cor- state which is an equal mixture of |00i and |11i, which relations than just entanglement when it comes to mixed can be converted by LO into a mixture of of |00i and states. For pure states, entanglement and classical cor- |1+i. It is therefore hard to think of discord as a form relations are equal to one another and the sum is then of correlations. Also, given that it can be created by exactly equal to the quantum mutual information which local means, it is questionable if we can think of (all) explains why quantum mutual information is twice as discordant states as useful for quantum information pro- big as the classical mutual information (as anticipated cessing. Having said this, there are examples of protocols by Lindblad). where discord has an operational meaning [7, 8]. It is also A few months later, Ollivier and Zurek [5] wrote a pa- still an open question if universal quantum computation per where they named this difference between the two can be done without entanglmenet in the general case of ways of defining quantum mutual information quantum mixed states. Maybe not all, but certainly some kind of discord. They defined it slightly differently as they had discord could be of importance. the open system setting in mind, but I do not wish to This does not mean that we cannot quantify discord enter any subtleties in this introductory article (an inter- using the quantum relative entropy. We can take the rel- ested reader is encouraged to consult the review in [9]). ative entropy from a given state to the closest classically Physically, quantum discord, according to Zurek, rep- correlated state. This set, however, is not invarient under resents the difference between the efficiency of classical LOs which is why this measure fails to be a monotone (as and quantum Maxwell’s demons, while in other interpre- examplified in the previous paragraph). tations it has also been linked to the fidelity of remote state preparation as well as to the difference in informa- tion extraction by local and global means (mathemati- III. OUTLOOK cally, at least, a protocol that is somewhat related to the Maxwell’s demon interpretation). One should emphasise that though this article has So discord seems to measure quantum correlations that dealt with bipartite systems only (for clarity, as well as go beyond just entanglement. Disentangled states can for historical reasons), correlation measures can be gen- actually possess non-zero quantum discord. But is dis- eralized to many partite systems (see e.g. [13] for entan- cord really a form of correlation? To answer that, we glement in many-body systems and [9] for discord and need to discuss an important property of any measure of related measures). A way to do that is using the same correlation. relative entropy based logic outlines above (see also [12] One of the features of correlations is that they cannot for a unified view of all correlations based of the quantum increase by local operations (LO). If we do something relative entropy). to A alone, and, independently, to B, we should not be Also, we did not discuss how we can tell if a given state able to correlate them to a higher degree than we started has discord. The method is simple and it boils down to with. The intuition behind this is clear: we cannot cor- showing that correlations are non-vanishing in more than relate things more unless we are allowed to act on them one basis [14]. Classical correlations, according to this jointly. Any separate action can only degrade the initial logic, are the ones that exist only in one basis (though correlation (or, at best, preserve it). this basis could be different for different subsystems, de- Both mutual information and entanglement are de- pending on how they couple to their environments, for creasing under LO (entanglement, in fact, under an even instance). more general class, but this need not concern us here In conclusion, discord without entanglement can be [10]). This is actually straightforward to prove if we seen as a form of classical correlation aided with quan- express both in terms of the quantum relative entropy tum coherence (superpositions) at the level of individual S(σ||ρ)= tr(σ log σ − σ log ρ). Entanglement is then the subsystems. This is why the research on discord has nat- relative entropy to the closest disentangled (separable) urally led to the research on quantifying quantum coher- n n state [10, 11] (whose form is Pn pnρA ⊗ ρB , where pn is ence. any probability distribution), while the quantum mutual Acknowledgments: The author acknowledges funding information is the relative entropy to the closest product from the John Templeton Foundation, the National Re- states (which happens to be the product of the reduced search Foundation (Singapore), the Ministry of Edu- states σA ⊗σB ). Since quantum relative entropy is mono- cation (Singapore), the Engineering and Physical Sci- tonic under completely positive maps and as the sets of ences Research Council (UK), the Leverhulme Trust, product states and separable states are invarient under the Oxford Martin School, and Wolfson College, Uni- LO (which is a special set of completely positive maps), it versity of Oxford. This research is also supported by the 3

National Research Foundation, Prime Ministers Office, istered by Centre for Quantum Technologies, National Singapore under its Competitive Research Programme University of Singapore. (CRP Award No. NRF- CRP14-2014-02) and admin-

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