Objective past of a quantum universe: Redundant records of consistent histories

Motivated by the advances of quantum Darwinism and recognizing the role played by redundancy in identifying the small subset of quantum states with resilience characteristic of objective classical reality, we explore the implications of redundant records for consistent histories. The consistent histories formalism is a tool for describing sequences of events taking place in an evolving closed quantum system. A set of histories is consistent when one can reason about them using Boolean logic, i.e., when probabilities of sequences of events that define histories are additive. However, the vast majority of the sets of histories that are merely consistent are flagrantly nonclassical in other respects. This embarras de richesses (known as the set selection problem) suggests that one must go beyond consistency to identify how the classical past arises in our quantum universe. The key intuition we follow is that the records of events that define the familiar objective past are inscribed in many distinct systems, e.g., subsystems of the environment, and are accessible locally in space and time to observers. We identify histories that are not just consistent but redundantly consistent using the partial-trace condition introduced by Finkelstein as a bridge between histories and decoherence. The existence of redundant records is a sufficient condition for redundant consistency. It selects, from the multitude of the alternative sets of consistent histories, a small subset endowed with redundant records characteristic of the objective classical past. The information about an objective history of the past is then simultaneously within reach of many, who can independently reconstruct it and arrive at compatible conclusions in the present.

Figure 1

(a) In the original decoherence paradigm, the system $\mathcal{S}$ is immersed in a monolithic environment $\mathcal{E}$ which is traced over. (b) Quantum Darwinism is based on recognizing the natural decomposition of the environment into parts ${\varepsilon }_k$ ($k=1,...,{}^{♯}\mathcal{E}$), where ${}^{♯}\mathcal{E}$ is the total number of parts of the environment. (c) Fragments ${\mathcal{F}}_{\left\{n\right\}}$ partition $\mathcal{E}$ into multiple pieces that may be accessed by different observers.

Figure 2

When the global state of a consistent set of histories $\left\{\alpha \right\}$ is pure, $\rho =|\psi \rangle \langle \psi |$, the branches $|{\psi }_{\alpha }\rangle =C_{\alpha }|\psi \rangle$ take on a simple tree structure. The top (zeroth) level is just the global state $|\psi \rangle$ while the first level contains those branches formed by the projectors $P_{a_1}^{\left(1\right)}$ at time $t_1$: $|{\psi }_{\left(a_1\right)}\rangle =C_{\left(a_1\right)}|\psi \rangle =P_{a_1}^{\left(1\right)}\left(t_1\right)|\psi \rangle$. Across each level, the branches are mutually orthogonal and together sum to $|\psi \rangle$. Each branch is also the sum of those branches connected directly below it. This structure is a result of consistency, $\mathfrak{D}(\alpha ,\beta )=0=\langle {\psi }_{\alpha }|{\psi }_{\beta }\rangle$ for $\alpha \ne \beta$. It does not follow just from Eq. (21).

Figure 3

Evolution in the cnot example. The system starts out in the state ${|0\rangle }_{\mathcal{S}}$ and then undergoes three discrete branching and recording events. During each recording event, the system is perfectly decohered in the $|0\rangle , |1\rangle$ basis. The environment that acquires the record at each event $m, {\mathcal{E}}_m$, will be composed of many spins (even though it is shown for simplicity as a single spin), and thus redundant records are generated.

Figure 4

Branching structure of the evolution in the cnot example. An initial product state of the system and all the environment subsystems evolves according to sequential branching and recording events. Here the records are into a single two-level subsystem of the environment for ease of depiction. However, each recording event can be redundant, making many copies of the system's state after each branching event.