Local arrows of time in quantum many-body systems

Abstract

We demonstrate that in quantum many-body systems, local arrows of time can differ from the global time $t$ induced by Hamiltonian evolution. That is, within a quantum many-body system, the flow of time can be relative to each observer or by proxy each local subsystem. We provide a definition of local arrows of time in quantum many-body systems, and explain their relation to spacetime quantum entropies. Then we give a variety of numerical and analytical examples which explore different ways in which local arrows of time can manifest in quantum many-body dynamics, including exotic arrows of time arising from quantum thermalization and quantum error correction.

Figures (5)

• Figure 1: (a) A depiction of a spacetime lattice. (b) The inlay shows the calculation of the local arrow of time $\vv{\textsf{AOT}}_{\!t,x}$ (black arrow) as a sum of contributions from neighbors $q_i \in \square_{t,x}$ (Eq. \ref{['eq:MAIN_aot_def']}), indexed $i=1,...,8$. Each dashed, red arrow represents the term corresponding to neighbor $q_i$, weighted by the causal influence $C_i=\overline{\text{CI}}_{q_i,(t,x)}$ from $q_i$ to $(t,x)$. Only six red arrows are depicted because $q_4$ and $q_8$ are spacelike separated from $(t, x)$, and thus cannot contribute to the local arrow of time.
• Figure 2: Vectorfield $\vv{\textsf{AOT}}_{\!t,x}$ under Hamiltonian $H$ with time step $\Delta t$ and an initial state $|{\Psi}\rangle\!$. Heat maps show the single-site von Neumann entropy at each lattice site. In each subplot the evolution runs up to a maximum time $T$, which is $T = 1.2$, $0.65$, $0.30$, and $57$, respectively. All panels except (e) and (f) use the same Hamiltonian $H = \sum_j X_jX_{j+1} + .01\sum_j X_j - .21\sum_j Z_j$ and $\Delta t = 0.005$, varying only $T$ and $|{\Psi}\rangle\!$. Panels (e) and (f) use the PXP Hamiltonian. (a, b) The high-entropy initial state $|{\Psi}\rangle\!$ is generated by evolving a product state backwards in time by $T/2$. Evolving that state forward under $H$ then exactly retraces the backward step, producing the blue fringe at $t = T/2$. The arrow of time vectorfield points away from this fringe, following the local entropy gradient. (c, d) The system begins from a product state whose right half has been evolved backward in time by $T/2$ and whose left half has been evolved forward by $T/2$. Under subsequent evolution by $H$, the left side becomes more entangled while the right side becomes less entangled. By $t = T$, each half’s entropy profile is exactly reversed from its initial profile at $t = 0$. The interaction at their boundary (sites 10 and 11) then gives rise to spatiotemporal $\vv{\textsf{AOT}}_{\!t,x}$ vectors at the interface. (e, f) Simulating the PXP Hamiltonian, the system is initialized in a Néel state, and displays periodic revivals to a low entropy state (blue fringes). Between revivals, the system has higher entropy (red fringes). The arrow of time points away from the low-entropy revivals and towards regions of higher entropy. (g, h) The system is initialized in a right-moving wavepacket centered at the origin. Near the packet's core, the arrow of time vectorfield exhibits combined temporal and spatial causal flow while remaining orthogonal to the quasi-particle's tangent vector. At larger distances from the center, the vectors become purely spacelike and continue to point toward the wavepacket’s center.
• Figure 3: Spacetime lattice illustrating \ref{['thm:acausal']}. The lattice shows the neighborhood $\square_{t,x}$ around the point $(t,x)$ (black dot), with neighbor $q = (t+\Delta t, x+\Delta x)$. The red dashed arrow indicates a causal influence $C_3 = \overline{\text{CI}}_{q,(t,x)}$ that vanishes if and only if the conditions of \ref{['thm:acausal']} are satisfied.
• Figure 4: (a) One error-correction cycle in the 1D $X$-basis repetition code correcting single-qubit $Z$ errors. The initial state is the logical superposition $\alpha|{\overline{0}}\rangle\! + \beta|{\overline{1}}\rangle\! = \alpha|{+\cdots+}\rangle\! + \beta|{-\cdots-}\rangle\!$. Time runs from $t=0$ to $t=1$; the blue band highlights one QEC cycle. Alternating physical and ancilla qubits form two-site unit cells. Each ancilla is prepared in $|+\rangle$, coupled by $\mathsf{CX}$ gates to its two neighboring physical qubits to extract an $X$-parity syndrome, and then measured in the $X$ basis. The orange and green lines indicate pathways along which physical $Z$ and $X$ errors propagate under time evolution. Two operator species are shown: a physical $X$ (i.e., a logical $\overline Z$) commutes through the circuit and generates a strictly temporal arrow of time (it propagates upward, leaving no syndrome record), whereas a physical $Z$ error is detected and removed so that its arrow of time terminates at the syndrome measurement. (b) Visualization of two types of causal influences on the spacetime lattice. The green arrow represents the influence exerted by a physical $X$ perturbation (i.e., a logical $\overline Z$ error) at $t = 0$ on the same physical qubit at $t = 1$. The orange arrows represent the influence exerted by the correctable $Z$ perturbation at $t = 0$ on the ancillas at $t = 1$.
• Figure 5: The quantum circuit that generates the quantum superdensity operator $\varrho_{\text{SDO}}(t)$. The measurement procedure accepts as input a quantum system in an initial state $|{\Psi}\rangle\!$.

Theorems & Definitions (9)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem
• proof
• Lemma 3
• proof