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Register a new userQuantum Mereology: Factorizing Hilbert Space into Subsystems with Quasi-Classical Dynamics
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We study the question of how to decompose Hilbert space into a preferred tensor-product factorization without any pre-existing structure other than a Hamiltonian operator, in particular the case of a bipartite decomposition into system and environment. Such a decomposition can be defined by looking for subsystems that exhibit quasi-classical behavior. The correct decomposition is one in which pointer states of the system are relatively robust against environmental monitoring (their entanglement with the environment does not continually and dramatically increase) and remain localized around approximately-classical trajectories. We present an in-principle algorithm for finding such a decomposition by minimizing a combination of entanglement growth and internal spreading of the system. Both of these properties are related to locality in different ways. This formalism could be relevant to the emergence of spacetime from quantum entanglement.
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It is shown that when properly analyzed using principles consistent with the use of a Hilbert space to describe microscopic properties, quantum mechanics is a local theory: one system cannot influence another system with which it does not interact. Claims to the contrary based on quantum violations of Bell inequalities are shown to be incorrect. A specific example traces a violation of the CHSH Bell inequality in the case of a spin-3/2 particle to the noncommutation of certain quantum operators in a situation where (non)locality is not an issue. A consistent histories analysis of what quantum measurements measure, in terms of quantum properties, is used to identify the basic problem with derivations of Bell inequalities: the use of classical concepts (hidden variables) rather than a probabilistic structure appropriate to the quantum domain. A difficulty with the original Einstein-Podolsky-Rosen (EPR) argument for the incompleteness of quantum mechanics is the use of a counterfactual argument which is not valid if one assumes that Hilbert-space quantum mechanics is complete; locality is not an issue. The quantum correlations that violate Bell inequalities can be understood using local quantum common causes. Wavefunction collapse and Schrodinger steering are calculational procedures, not physical processes. A general Principle of Einstein Locality rules out nonlocal influences between noninteracting quantum systems. Some suggestions are made for changes in terminology that could clarify discussions of quantum foundations and be less confusing to students.
We consider the quantum-to-classical transition for macroscopic systems coupled to their environments. By applying Borns Rule, we are led to a particular set of quantum trajectories, or an unravelling, that describes the state of the system from the frame of reference of the subsystem. The unravelling involves a branch dependent Schmidt decomposition of the total state vector. The state in the subsystem frame, the conditioned state, is described by a Poisson process that involves a non-linear deterministic effective Schrodinger equation interspersed with quantum jumps into orthogonal states. We then consider a system whose classical analogue is a generic chaotic system. Although the state spreads out exponentially over phase space, the state in the frame of the subsystem localizes onto a narrow wave packet that follows the classical trajectory due to Ehrenfests Theorem. Quantum jumps occur with a rate that is the order of the effective Lyapunov exponent of the classical chaotic system and imply that the wave packet undergoes random kicks described by the classical Langevin equation of Brownian motion. The implication of the analysis is that this theory can explain in detail how classical mechanics arises from quantum mechanics by using only unitary evolution and Borns Rule applied to a subsystem.
We introduce and analyse the problem of encoding classical information into different resources of a quantum state. More precisely, we consider a general class of communication scenarios characterised by encoding operations that commute with a unique resource destroying map and leave free states invariant. Our motivating example is given by encoding information into coherences of a quantum system with respect to a fixed basis (with unitaries diagonal in that basis as encodings and the decoherence channel as a resource destroying map), but the generality of the framework allows us to explore applications ranging from super-dense coding to thermodynamics. For any state, we find that the number of messages that can be encoded into it using such operations in a one-shot scenario is upper-bounded in terms of the information spectrum relative entropy between the given state and its version with erased resources. Furthermore, if the resource destroying map is a twirling channel over some unitary group, we find matching one-shot lower-bounds as well. In the asymptotic setting where we encode into many copies of the resource state, our bounds yield an operational interpretation of resource monotones such as the relative entropy of coherence and its corresponding relative entropy variance.
We propose a new formalism of quantum subsystems which allows to unify the existing and new methods of reduced description of quantum systems. The main mathematical ingredients are completely positive maps and correlation functions. In this formalism generalized quantum systems can be composed and there is a notion of generalized entanglement. Models of fermionic and bosonic systems and also quantum systems described by the SU(2) symmetry are studied.
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