An introduction to quantum discord and non-classical correlations beyond entanglement

What is quantum? As researchers of quantum physics, we are constantly bombarded with attributes like “non-classical” and “super-classical”. We strive to track down the elusive quantum-classical boundary, to understand what makes quantum mechanics so powerful yet counter-intuitive. But to do this, we must first have a firm understanding of the classical world and the laws that classical mechanics imposes. There are in fact many ways to think about classicality. One facet of the classical world is that any system is always in a fixed and predetermined state. Take for example a bit: it can be either 0 or 1. How does this compare with what is predicted from the rulebook of quantum mechanics? Here we can have systems existing in a superposition of both 0 and 1, called quantum bits or qubits. This form of non-classicality is what is known as quantum coherence [1]. It is also interesting to consider systems of spatially separated parties and the correlations between them. We can try to identify the states that are describable by classical mechanics and the states that are not. You are probably now thinking that this sounds a lot like entanglement [2], and that the classically correlated states are just separable states. However, things are not so simple: it turns out that even separable mixed states can exhibit some quantumness in their correlations! In this manuscript we will explore these manifestations of quantum correlations beyond entanglement [3, 4, 5]. We begin by introducing and motivating the classically correlated states and then showing how to quantify the quantum correlations using an entropic approach, arriving at a well known measure called the quantum discord [6, 7]. Quantum correlations and discord are then operationally linked with the task of local broadcasting [8]. We conclude by providing some alternative perspectives on quantum correlations and how to measure them. Finally, before proceeding it is important to note that there are many layers of quantumness in composite systems. As well as entanglement and discord-type quantum correlations, one can identify e.g. steering and Bell non-locality. For pure composite states, all of these signatures of quantumness become equivalent, yet for mixed states they are different, showing a strict hierarchy. Each form of quantumness is of independent interest, but here we focus on the most general form of quantum correlations, leaving the interested reader to consult Ref. [2] for more information on entanglement and Refs. [9, 10] for steering and non-locality.