Information Scrambling versus Decoherence—Two Competing Sinks for Entropy

A possible solution of the information paradox can be sought in quantum information scrambling. In this paradigm, it is postulated that all information entering a black hole is rapidly and chaotically distributed across the event horizon, making it impossible to reconstruct the information by means of any local measurement. However, in this scenario, the effects of decoherence are typically ignored, which may render information scrambling moot in cosmological settings. In this work, we develop key steps toward a thermodynamic description of information scrambling in open quantum systems. In particular, we separate the entropy change into contributions arising from scrambling and decoherence, for which we derive statements of the second law. This is complemented with a numerical study of the Sachdev-Ye-Kitaev, Maldacena-Qi, XXX, mixed-field Ising, and Lipkin-Meshkov-Glick models in the presence of decoherence in the energy or in the computational basis.

Popular Summary

One of the major open problems in theoretical physics concerns the reconciliation of quantum mechanics with general relativity. In this context, a hallmark problem is the so-called black-hole information paradox, which addresses what happens to information that passes the event horizon. A possible resolution can be sought in information scrambling, which asserts that any information entering a black hole is rapidly and chaotically “scrambled” across the entirety of the event horizon and thus becomes inaccessible to any local observation. This paper analyzes information scrambling in the presence of decoherence, which is the inevitable loss of quantum information due to the interaction with the rest of the universe.

In a recent paper, the authors have shown that scrambling in isolated quantum systems can be quantified by mutual information. In this new study, they further extend the analysis to open quantum systems. In particular, they show that the mutual information can be separated into unique contributions arising from either scrambling or decoherence. This separation of terms lends itself naturally to the further derivation of statements of the second law of thermodynamics for scrambling of information in open systems. The general findings are then illustrated for a variety of models, including the Sachdev-Ye-Kitaev, the Maldacena-Qi, the XXX, the mixed-field Ising, and the Lipkin-Meshkov-Glick model with decoherence in energy or in the computational basis.

This work may have universal applications, from black holes (as their intense gravitational field is a decoherence channel) to ion-trap systems, for which information scrambling has been observed experimentally.

Figure 1

An illustration of this paper’s narrative.

Figure 2

An illustration of the subsystem structure defining the support of the local operators $O_A$ and $O_B$, in a given spin chain. Under a scrambling unitary, the operator $O_B$ becomes highly nonlocal and spreads into the support of $A$, triggering the decay of the OTOC.

Figure 3

A sketch of the SYK model with 12 Majorana fermions. The network represents the all-to-all interactions between the fermionic sites.

Figure 4

The growth of the average OTOC (2) as a function of time for the SYK model with $N=12$ and an initial “all-up” state (22). The results are obtained as averages over ${10}^2$ realizations.

Figure 5

The evolution of the change of mutual information ($\mathrm{\Delta }\mathcal{I}$) and relative entropy of coherence ($\mathrm{\Delta }\mathcal{C}$) for the SYK model (24) with $N=12$, for an initial “all-up” state (22). The results are obtained as averages over $3\times {10}^2$ realizations.

Figure 6

A sketch of the MQ model with 12 Majorana fermions and $J_{i,j,k,l}^L=J_{i,j,k,l}^R=J_{i,j,k,l}$.

Figure 7

The evolution of the change in mutual information ($\mathrm{\Delta }\mathcal{I}$) and the relative entropy of coherence ($\mathrm{\Delta }\mathcal{C}$) for the MQ model (25) with $\mu =0.1$. Each black hole is composed of six Majorana fermions, which gives a total of 12 fermions living in a Hilbert space of $2^6$ dimensions. The results are obtained as averages over $3\times {10}^2$ realizations.

Figure 8

A sketch of the XXX model with nearest-neighbor interactions and random local fields $h_i$.

Figure 9

The evolution of the change in mutual information ($\mathrm{\Delta }\mathcal{I}$) and the relative entropy of coherence ($\mathrm{\Delta }\mathcal{C}$) for the disordered XXX model (26) in its ergodic phase with $N=6$ and for an initial Néel state (23). The results are obtained as averages over $4\times {10}^2$ realizations.

Figure 10

A sketch of the MFI model with nearest-neighbor interactions (in the $z$ direction), random local fields $h_i$, and a global field in the $x$ direction.

Figure 11

The evolution of the change in mutual information ($\mathrm{\Delta }\mathcal{I}$) and the relative entropy of coherence ($\mathrm{\Delta }\mathcal{C}$) for the MFI model (27) in its ergodic phase, $J=1$ and $g=1.05$, with $N=6$ and for the “all-up” state (22). The results are obtained as averages over $2\times {10}^2$.

Figure 12

A sketch of the LMG model with all-to-all interactions (in the $x$ and $y$ directions), random local fields $h_i$, and a global field in the $x$ direction.

Figure 13

The evolution of the change in mutual information ($\mathrm{\Delta }\mathcal{I}$) and the relative entropy of coherence ($\mathrm{\Delta }\mathcal{C}$) for the LMG model (28) with $N=6$ and an initial Néel state (23).