Eavesdropping on the Decohering Environment: Quantum Darwinism, Amplification, and the Origin of Objective Classical Reality

Solvable Model of Quantum-Darwinism-Encoding Transitions

We propose a solvable model of quantum Darwinism to encoding transitions-abrupt changes in how quantum information spreads in a many-body system under unitary dynamics. We consider a random Clifford circuit on an expanding tree, whose input qubit is entangled with a reference. The model has a quantum Darwinism phase, where one classical bit of information about the reference can be retrieved from an arbitrarily small fraction of the output qubits, and an encoding phase where such retrieval is impossible. The two phases are separated by a mixed phase and two continuous transitions. We compare the exact result to a two-replica calculation. The latter yields a similar "annealed" phase diagram, which applies also to a model with Haar random unitaries. We relate our approach to measurement-induced phase transitions (MIPTs), by solving a modified model where an environment eavesdrops on an encoding system. It has a sharp MIPT only with full access to the environment.

Relationship between Information Scrambling and Quantum Darwinism

A quantum system interacting with a multipartite environment can induce redundant encoding of the information of a system into the environment, which is the essence of quantum Darwinism. At the same time, the environment may scramble the initially localized information about the system. Based on a collision model, we mainly investigate the relationship between information scrambling in an environment and the emergence of quantum Darwinism. Our results show that when the mutual information between the system and environmental fragment is a linear increasing function of the fragment size, the tripartite mutual information (TMI) is zero, which can be proved generally beyond the collision model; when the system exhibits Darwinistic behavior, the TMI is positive (i.e., scrambling does not occur); when we see the behavior of an "encoding" environment, the TMI is negative (i.e., scrambling occurs). Additionally, we give a physical explanation for the above results by considering two simple but illustrative examples. Moreover, depending on the nature of system and environment interactions, it is also shown that the single qubit and two-qubit systems behave differently for the emergence of quantum Darwinism, and hence the scrambling, while their relationship is consistent with the above conclusion.

Pointer States and Quantum Darwinism with Two-Body Interactions

Quantum Darwinism explains the emergence of classical objectivity within a quantum universe. However, to date, most research on quantum Darwinism has focused on specific models and their stationary properties. To further our understanding of the quantum-to-classical transition, it appears desirable to identify the general criteria a Hamiltonian has to fulfill to support classical reality. To this end, we categorize all N-qubit models with two-body interactions, and show that only those with separable interaction of the system and environment can support a pointer basis. We further demonstrate that "perfect" quantum Darwinism can only emerge if there are no intra-environmental interactions. Our analysis is complemented by solving the ensuing dynamics. We find that in systems exhibiting information scrambling, the dynamical emergence of classical objectivity directly competes with the non-local spread of quantum correlations. Our rigorous findings are illustrated through the numerical analysis of four representative models.

Quantum Theory of the Classical: Einselection, Envariance, Quantum Darwinism and Extantons

Core quantum postulates including the superposition principle and the unitarity of evolutions are natural and strikingly simple. I show that-when supplemented with a limited version of predictability (captured in the textbook accounts by the repeatability postulate)-these core postulates can account for all the symptoms of classicality. In particular, both objective classical reality and elusive information about reality arise, via quantum Darwinism, from the quantum substrate. This approach shares with the Relative State Interpretation of Everett the view that collapse of the wavepacket reflects perception of the state of the rest of the Universe relative to the state of observer's records. However, our "let quantum be quantum" approach poses questions absent in Bohr's Copenhagen Interpretation that relied on the preexisting classical domain. Thus, one is now forced to seek preferred, predictable, hence effectively classical but ultimately quantum states that allow observers keep reliable records. Without such (i) preferred basis relative states are simply "too relative", and the ensuing basis ambiguity makes it difficult to identify events (e.g., measurement outcomes). Moreover, universal validity of quantum theory raises the issue of (ii) the origin of Born's rule, pk=|ψk|2, relating probabilities and amplitudes (that is simply postulated in textbooks). Last not least, even preferred pointer states (defined by einselection-environment-induced superselection)-are still quantum. Therefore, unlike classical states that exist objectively, quantum states of an individual system cannot be found out by an initially ignorant observer through direct measurement without being disrupted. So, to complete the 'quantum theory of the classical' one must identify (iii) quantum origin of objective existence and explain how the information about objectively existing states can appear to be essentially inconsequential for them (as it does for states in Newtonian physics) and yet matter in other settings (e.g., thermodynamics). I show how the mathematical structure of quantum theory supplemented by the only uncontroversial measurement postulate (that demands immediate repeatability-hence, predictability) leads to preferred states. These (i) pointer states correspond to measurement outcomes. Their stability is a prerequisite for objective existence of effectively classical states and for events such as quantum jumps. Events at hand, one can now enquire about their probability-the probability of a pointer state (or of a measurement record). I show that the symmetry of entangled states-(ii) entanglement-assisted invariance or envariance-implies Born's rule. Envariance also accounts for the loss of phase coherence between pointer states. Thus, decoherence can be traced to symmetries of entanglement and understood without its usual tool-reduced density matrices. A simple and manifestly noncircular derivation of pk=|ψk|2 follows. Monitoring of the system by its environment in course of decoherence typically leaves behind multiple copies of its pointer states in the environment. Only pointer states can survive decoherence and can spawn such plentiful information-theoretic progeny. This (iii) quantum Darwinism allows observers to use environment as a witness-to find out pointer states indirectly, leaving systems of interest untouched. Quantum Darwinism shows how epistemic and ontic (coexisting in epiontic quantum state) separate into robust objective existence of pointer states and detached information about them, giving rise to extantons-composite objects with system of interest in the core and multiple records of its pointer states in the halo comprising of environment subsystems (e.g., photons) which disseminates that information throughout the Universe.

Composite Discordant States and Quantum Darwinism

The framework of Quantum Darwinism strives at characterizing the quantum-to-classical transition by introducing the concept of redundancy of information-as measured by Mutual Information-that a set of observers would acquire on the state of a physical system of interest. Further development on this concept, in the form of Strong Quantum Darwinism and Spectrum Broadcast Structures, has recently led to a more fine-grained identification of the nature of such information, which should not involve any quantum correlations between observing and observed systems, while the assessment of information proliferation from individual systems has attracted most of the attention so far, the way such mechanism takes place in more complex states is open to exploration. To this end, we shall consider a two-qubit state, sharing initial quantum correlations in the form of Quantum Discord, and different dephasing-like interactions between them and an observing environment. We will focus on the amount of information regarding the subsystem not involved in the interaction that is proliferated to the environment. We shall refer to this as mediated redundancy. We will show that, in some cases, the channel capacity of the subsystems, given these interactions, can exceed that of the fragments.

Redundantly Amplified Information Suppresses Quantum Correlations in Many-Body Systems

We establish bounds on quantum correlations in many-body systems. They reveal what sort of information about a quantum system can be simultaneously recorded in different parts of its environment. Specifically, independent agents who monitor environment fragments can eavesdrop only on amplified and redundantly disseminated-hence, effectively classical-information about the decoherence-resistant pointer observable. We also show that the emergence of classical objectivity is signaled by a distinctive scaling of the conditional mutual information, bypassing hard numerical optimizations. Our results validate the core idea of quantum Darwinism: objective classical reality does not need to be postulated and is not accidental, but rather a compelling emergent feature of quantum theory that otherwise-in the absence of decoherence and amplification-leads to "quantum weirdness." In particular, a lack of consensus between agents that access environment fragments is bounded by the information deficit, a measure of the incompleteness of the information about the system.

Amplification, Inference, and the Manifestation of Objective Classical Information

Our everyday reality is characterized by objective information-information that is selected and amplified by the environment that interacts with quantum systems. Many observers can accurately infer that information indirectly by making measurements on fragments of the environment. The correlations between the system, S, and a fragment, F, of the environment, E, is often quantified by the quantum mutual information, or the Holevo quantity, which bounds the classical information about S transmittable by a quantum channel F. The latter is a quantum mutual information but of a classical-quantum state where measurement has selected outcomes on S. The measurement generically reflects the influence of the remaining environment, E/F, but can also reflect hypothetical questions to deduce the structure of SF correlations. Recently, Touil et al. examined a different Holevo quantity, one from a quantum-classical state (a quantum S to a measured F). As shown here, this quantity upper bounds any accessible classical information about S in F and can yield a tighter bound than the typical Holevo quantity. When good decoherence is present-when the remaining environment, E/F, has effectively measured the pointer states of S-this accessibility bound is the accessible information. For the specific model of Touil et al., the accessible information is related to the error probability for optimal detection and, thus, has the same behavior as the quantum Chernoff bound. The latter reflects amplification and provides a universal approach, as well as a single-shot framework, to quantify records of the missing, classical information about S.

Non-Perfect Propagation of Information to a Noisy Environment with Self-Evolution

We study the non-perfect propagation of information for evolving a low-dimensional environment that includes self-evolution as well as noisy initial states and analyse the interrelations between the degree of objectivization and environment parameters. In particular, we consider an analytical model of three interacting qubits and derive its objectivity parameters. The numerical analysis shows that the quality of the spectrum broadcast structure formed during the interaction may exhibit non-monotonicity both in the speed of self-dynamics of the environment as well as its mixedness. The former effect is particularly strong, showing that—considering part of the environment as a measurement apparatus—an increase of the external magnetic field acting on the environment may turn the vague measurement into close to ideal. The above effects suggest that quantum objectivity may appear after increasing the dynamics of the environment, although not with respect to the pointer basis, but some other, which we call the generalized pointer or indicator basis. Furthermore, it seems also that, when the objectivity is poor, it may be improved, at least by some amount, by increasing the thermal noise. We provide further evidence of this by analysing the upper bounds on distance to the set of states representing perfect objectivity in the case of a higher number of qubits.

Correlations, Information Backflow, and Objectivity in a Class of Pure Dephasing Models

We critically examine the role that correlations established between a system and fragments of its environment play in characterising the ensuing dynamics. We employ a dephasing model with different initial conditions, where the state of the initial environment represents a tunable degree of freedom that qualitatively and quantitatively affects the correlation profiles, but nevertheless results in the same reduced dynamics for the system. We apply recently developed tools for the characterisation of non-Markovianity to carefully assess the role that correlations, as quantified by the (quantum) Jensen-Shannon divergence and relative entropy, as well as changes in the environmental state, play in whether the conditions for classical objectivity within the quantum Darwinism paradigm are met. We demonstrate that for precisely the same non-Markovian reduced dynamics of the system arising from different microscopic models, some exhibit quantum Darwinistic features, while others show that no meaningful notion of classical objectivity is present. Furthermore, our results highlight that the non-Markovian nature of an environment does not a priori prevent a system from redundantly proliferating relevant information, but rather it is the system's ability to establish the requisite correlations that is the crucial factor in the manifestation of classical objectivity.