Entanglement and objectivity in pure dephasing models

Abstract

We study the relation between the emergence of objectivity and qubit-environment entanglement generation. We find that although entanglement with the unobserved environments is irrelevant (since sufficiently strong decoherence can occur regardless), entanglement with the observed environments is crucial. In fact, the appearance of an objective qubit-observed-environment state is strictly impossible if their joint evolution does not lead to entanglement. Furthermore, if a single observer has access to a single environment (no macrofractions) then the required orthogonality of the observed environmental states comes only as a consequence of the system-enviornment state becoming strongly entangled (maximally entangled for the given initial occupation of the qubit if the environmental state is initially pure).

Measure of qubit-environment entanglement for pure dephasing evolutions

“…This is in concurrence with the requirement for strict orthogonality proven in Ref. [45], that it is only possible for initial environmental states which have at least half of the occupations equal to zero after diagonalization. A qubit state with half of its occupations equal to zero is pure.…”

“…The zero-fidelity requirement means thatR 00 (t ) andR 11 (t ) have orthogonal supports [R 00 (t )R 11 (t ) = 0] or, restating the requirement equivalently, that the eigenstates ofR 00 (t ) with nonzero eigenvalues occupy a different subspace than the eigenstates ofR 11 (t ) with nonzero eigenvalues. Such orthogonality of environmental states, which can naturally occur during PD evolution, is important for the emergence of objectivity [40][41][42][43][44] and has been studied in detail in this context [45], where it has been called strict orthogonality. The upper limit of the measure (9) is obtained for exactly the same set of states for which it would be obtained using any convex-roof entanglement measure, such as entanglement of formation (EOF) [46,47].…”

“…Since the same logic applies to the other conditional state of the environmentR 11 (t ), it is obvious that for the strict orthogonality condition to be fulfilled, the number of eigenstates with nonzero eigenvalues of the initial state of the environment cannot exceed half of the whole dimension of the environment. This in turn limits the initial purity of the environment to at least twice the minimum purity allowed for a given environment size (the strict orthogonality condition cannot be fulfilled for states with lower initial purity) [45]. Using Eqs.…”

How to detect qubit environment entanglement in pure dephasing evolutions

“…The zero-Fidelity requirement means thatR 00 (t) andR 11 (t) have orthogonal supports (R 00 (t)R 11 (t) = 0) or, restating the requirement equivalently, that the eigenstates of R 00 (t) with non-zero eigenvalues occupy a different subspace than the eigenstates ofR 11 (t) with non-zero eigenvalues. Such orthogonality of environmental states, which can naturally occur during QE evolution of pure dephasing type, is important for the emergence of objectivity [37][38][39][40][41] and has been studied in detail in this context [42], where it has been called "strict orthogonality".…”

“…Since the same logic applies to the other conditional state of the environmentR 11 (t), it is obvious that for the strict orthogonality condition to be fulfilled, the number of eigenstates with non-zero eigenvalues of the initial state of the environment cannot exceed half of the whole dimension of the environment. This in turn limits the initial purity of the environment to at least twice the minimum purity allowed for a given environment size (the strict orthogonality condition cannot be fulfilled for states with lower initial purity) [42]. Using eqs.…”