Furthermore, we prove an anti-symmetry relation between discord and the Holevo quantity. We show that the Holevo quantity for other observables decreases depending on the degree of “misalignment” between them and the eigenstates of the pointer observable. This division of the mutual information between the Holevo quantity and the quantum discord allows one to understand why the data about the system accessible to observers are effectively limited to the pointer observable. Thus, under very general conditions this yields a conservation law: While the classically accessible information and quantum discord depend on the observable of the system, their sum does not. Quantum discord makes up the difference between the mutual information (that remains constant) and the Holevo quantity. The Holevo quantity is largest for the pointer observable and decreases for other observables, nearly vanishing for observables that are complementary. The information that is there in principle, but cannot be found out from the environment alone is given by the quantum discord 27, 28, 29 that characterizes the quantumness of correlations. Rather, it leads to a precise split of the quantum mutual information between the system and the environment into two components in proportions that depend on the observable of the system: The (maximum) amount of the accessible information about an observable is given by the Holevo quantity that sets an upper limit on the capacity of a quantum channel to transmit classical data 26. Here we show that recognizing the environment as a communication channel is far more than an allegory. Thus, the decohering environment serves not just as a disposal for uncomfortably quantum evidence, but plays a role analogous to a communication channel, an advertising medium in which multiple copies of selected states of the system are present. The focus of Quantum Darwinism is the redundancy – the presence of multiple copies – of data about certain observables achieved at the expense of the information about complementary observables. We obtain most of our data from the photon environment. Quantum Darwinism 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 recognizes that the same environment that is responsible for decoherence serves also as a channel through which information about systems reaches observers, see Fig. The stability of pointer states fulfills the expectation of predictability built on the daily experience of the classical realm, but it does not address the obvious question: Why is it that observers choose to measure the Universe in a way that reveals pointer states? The key premise of this paper can be summed-up by saying that the choice is made not by observers, but by the medium through which we perceive the Universe. Yet, the underlying question – “Why do we, observers, perceive pointer states?” – remains unanswered even after recognizing the role of decoherence in suppressing non-local superpositions. Their nature – in particular, their persistence – made them obvious candidates for “classical states”: It was natural to expect that predictably evolving states are good candidates for our everyday “classical reality”. This contrast between quantum expectations and everyday classical reality sets up the problem that puzzled Bohr, Einstein and many others since the inception of quantum physics 1, 2, 3, 4, 5.ĭecoherence 6, 7, 8 changed our view of the quantum-classical correspondence by explaining the stability of pointer states that are selected in the presence of the environment 9, 10. Yet, the world we perceive is resolutely classical. This suggests that we should routinely encounter superpositions. There is now overwhelming evidence that the Universe we inhabit is made out of quantum “stuff” and therefore quantum to the core. The resulting complementarity explains why, in a quantum Universe, we perceive objective classical reality while flagrantly quantum superpositions are out of reach. It shows that information becomes objective – accessible to many observers – only as quantum information is relegated to correlations with the global environment, and, therefore, locally inaccessible. We also prove an anti-symmetry property relating accessible information and discord. Other observables are accessible only via correlations with the pointer observable. This split naturally delineates information about quantum systems accessible to observers – information that is redundantly transmitted by the environment – while showing that it is maximized for the quasi-classical pointer observable. The sum of the Holevo quantity (that bounds the capacity of quantum channels to transmit classical information about an observable) and the quantum discord (a measure of the quantumness of correlations of that observable) yields an observable-independent total given by the quantum mutual information.
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