Scrambling of Quantum Information in Quantum Many-Body Systems

We systematically investigate scrambling (or delocalizing) processes of quantum information encoded in quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling, we adopt the tripartite mutual information (TMI) that becomes negative when quantum information is delocalized. We clarify that scrambling is an independent property of integrability of Hamiltonians; TMI can be negative or positive for both integrable and non-integrable systems. This implies that scrambling is a separate concept from conventional quantum chaos characterized by non-integrability. Furthermore, we calculate TMI in disordered systems such as many-body localized (MBL) systems and the Sachdev-Ye-Kitaev (SYK) model. We find that scrambling occurs but is slow in a MBL phase, while disorder in the SYK model does not make scrambling slower but makes it smoother. Introduction. Whether an isolated system thermalizes or not is a fundamental issue in statistical mechanics, which is related to non-integrability of Hamiltonians. In classical systems, thermalization has been discussed in terms of ergodicity of chaotic systems [1]. In quantum systems, a counterpart of classical chaos is not immediately obvious, because the Schrödinger equation is linear. Nevertheless, it has been established that there are some indicators of chaotic behaviors in quantum systems, such as the level statistics of Hamiltonians [2–4] and decay of the Loschmidt echo [5, 6]. More recently, the eigenstatethermalization hypothesis (ETH) [7–11] has attracted attention as another indicator of quantum chaos in many-body systems, which states that even a single energy eigenstate is thermal. All these indicators of quantum chaos are directly related to integrability of Hamiltonians; non-integrable quantum systems exhibit chaos. Such a chaotic behavior in isolated quantum systems is also a topic of active researches in real experiments with ultracold atoms [12–14], ion traps [15], NMR [5], and superconducting qubits [16]. In order to investigate “chaotic” properties of quantum many-body systems beyond the conventional concept of quantum chaos, it is significant to focus on dynamics of quantum information encoded in quantum many-body systems. How does locally-encoded quantum information spread out over the entire system by unitary dynamics? Such delocalization of quantum information is referred to as scrambling [17– 21]. Investigating scrambling is important not only for understanding relaxation dynamics of experimental systems at hand, but also in terms of information paradox of black holes [17], where it has been argued that black holes are the fastest scramblers in the universe [18]. However, the fundamental relationship between scrambling and conventional quantum chaos has not been comprehensively understood. Scrambling can be quantified by the tripartite mutual information (TMI) [21, 22], which becomes negative if quantum information is scrambled. There is also another measure of scrambling, named the out-of-time-ordered correlator (OTOC) [20,21,23–28]. TMI and OTOC capture essentially the same feature of scrambling [21], where OTOC depends on a choice of observables but TMI does not. In the context of the holographic theory of quantum gravity, TMI is shown negative [29] if the Ryu-Takayanagi formula [30] 1 ar X iv :1 70 4. 04 85 0v 1 [ co nd -m at .s ta tm ec h] 1 7 A pr 2 01 7 is applied, suggesting that gravity has a scrambling property. This is consistent with fast scrambling in the Sachdev-Ye-Kitaev (SYK) model [31–38], a toy model of a quantum black hole. Then, a natural question raised is to what extent such a property of quantum gravity is intrinsic to gravity or can be valid for general quantum many-body systems. In this Letter, we perform systematic numerical calculations of real-time dynamics of TMI in quantum many-body systems under unitary dynamics, by using exact diagonalization of Hamiltonians. We consider a small system (say, a qubit) and a quantum many-body system (say, a spin chain). The information of the small system is initially encoded in the many-body system through entanglement. The many-body system then evolves unitarily, and we observe how the locally encoded information is scrambled over the entire many-body system. We note that temporal TMI has been investigated by using the channel-state duality in Ref. [21], while we here calculate instantaneous TMI, with which we can study the role of initial states. By studying quantum spin chains such as the XXX model and the transverse-field Ising (TFI) model with and without integrability breaking terms, we find that scrambling occurs (i.e., TMI becomes negative) for both the integrable and non-integrable systems for a majority of initial states. On the other hand, for special initial states, scrambling does not occur (i.e., TMI becomes positive) for both the integrable and non-integrable cases of the XXX model. These results clarify that scrambling is an independent property of integrability of Hamiltonians, as is consistent with recent observations on OTOC [27, 28]. This shows that scrambling does not straightforwardly correspond to conventional quantum chaos, making a sharp contrast to the level statistics and ETH. We also consider many-body localized (MBL) systems [39–46] and the SYK model in order to investigate the role of disorder. It is known that a MBL phase is not chaotic, where ETH does not hold at least in one dimension [39]. In the MBL phase of the disordered XXX model, we show that scrambling can occur but is quite slow, which is consistent with recent results on OTOC [43–46]. In contrast, for the SYK model with four-body interaction of complex fermions, we find that disorder does not lead to slow dynamics but instead makes scrambling smoother than a clean case. This reveals a special characteristic of a quantumgravitational model in contrast to ordinary spin chains. Setup. We consider either a spin-1/2 or a fermionic system on a lattice, which consists of small system A on a single site and a many-body system on L sites (Fig. 1). The many-body system is divided into three subsystems B, C, D, whose sizes (the numbers of the lattice sites) are respectively given by 1, l, and L− l− 1. The lattice structure BCD is supposed to be one-dimensional for spin chains or all-connected for the SYK model. For a single site of a spin (fermion) system, we write |0〉 as the spin-up (particle-occupied) state, and |1〉 as the spin-down (particle-empty) state. In any case, a single qubit is on a single site. We first prepare a product state 1 √ 2 (|0〉A + |1〉A)⊗ |Ξ〉BCD, (1) where |Ξ〉BCD is a product state with the state of each qubit being |0〉 or |1〉 (e.g., the Néel state |0〉|1〉|0〉 · · · |0〉|1〉 or the all-up state |0〉|0〉 · · · |0〉, etc). We then apply the CNOT gate on the state (1), where the control qubit is A and the target qubit is B. By this CNOT gate, information about A is locally encoded in B through entanglement. Then, only BCD obeys a unitary time evolution with a Hamiltonian. We calculate the time dependence of TMI between A, B, C, which characterizes scrambling of the information about A that was initially encoded in B. We note that the foregoing setup is associated with a thought experiment that one of qubits of an EPR pair is thrown into a black hole and then scrambled [17].