Random Unitary Dynamics Research Articles

Entanglement Domain Walls in Monitored Quantum Circuits and the Directed Polymer in a Random Environment

Monitored quantum dynamics reveal quantum state trajectories which exhibit a rich phenomenology of entanglement structures, including a transition from a weakly-monitored volume law entangled phase to a strongly-monitored area law phase. For one-dimensional hybrid circuits with both random unitary dynamics and interspersed measurements, we combine analytic mappings to an effective statistical mechanics model with extensive numerical simulations on hybrid Clifford circuits to demonstrate that the universal entanglement properties of the volume law phase can be quantitatively described by a fluctuating entanglement domain wall that is equivalent to a "directed polymer in a random environment" (DPRE). This relationship improves upon a qualitative "mean-field" statistical mechanics of the volume-law-entangled phase [1, 2]. For the Clifford circuit in various geometries, we obtain agreement between the subleading entanglement entropies and error correcting properties of the volume-law phase (which quantify its stability to projective measurements) with predictions of the DPRE. We further demonstrate that depolarizing noise in the hybrid dynamics near the final circuit time can drive a continuous phase transition to a non-error correcting volume law phase that is not immune to the disentangling action of projective measurements. We observe this transition in hybrid Clifford dynamics, and obtain quantitative agreement with critical exponents for a "pinning" phase transition of the DPRE in the presence of an attractive interface.

Global correlation and local information flows in controllable non-Markovian open quantum dynamics

In a fully-controllable experiment platform for studying non-Markovian open quantum dynamics, we show that the non-Markovianity could be investigated from the global and local aspects. By mixing random unitary dynamics, we demonstrate non-Markovian and Markovian open quantum dynamics. From the global point of view, by tuning the base frequency we demonstrate the transition from the Markovianity to the non-Markovianity as measured by the quantum mutual information (QMI). In a Markovian open quantum process, the QMI decays monotonically, while it may rise temporarily in a non-Markovian process. However, under some circumstances, it is not sufficient to globally investigate the non-Markovianity of the open quantum dynamics. As an essential supplement, we further utilize the quantum Fisher information (QFI) flow to locally characterize the non-Markovianity in different channels. We demonstrate that the QMI in combination with the QFI flow are capable of measuring the non-Markovianity for a multi-channel open quantum dynamics.

Measurement-induced topological entanglement transitions in symmetric random quantum circuits

We study a class of (1+1)D symmetric random quantum circuits with two competing types of measurements in addition to random unitary dynamics. The circuit exhibits a rich phase diagram involving robust symmetry-protected topological (SPT), trivial, and volume law entangled phases, where the transitions are hidden to expectation values of operators and can only be accessed through the entanglement entropy averaged over quantum trajectories. In the absence of unitary dynamics, we find a purely measurement-induced critical point with logarithmic scaling of the entanglement entropy, which we map exactly to two copies of a classical 2D percolation problem. We perform numerical simulations that indicate this transition is a tricritical point that splits into two critical lines in the presence of arbitrarily sparse unitary dynamics with an intervening volume law entangled phase. Our results show how measurements alone are sufficient to induce criticality and logarithmic entanglement scaling, and how arbitrarily sparse unitary dynamics can be sufficient to stabilize volume law entangled phases in the presence of rapid yet competing measurements.

Growth of genuine multipartite entanglement in random unitary circuits

We study the growth of genuine multipartite entanglement in random quantum circuit models, which include random unitary circuit models and the random Clifford circuit. We find that for the random Clifford circuit, the growth of multipartite entanglement remains slower in comparison to the random unitary case. However, the final saturation value of multipartite entanglement is almost the same in both cases. The behavior is then compared to the genuine multipartite entanglement obtained in random matrix product states with a moderately high bond dimension. We then relate the behavior of multipartite entanglement to other global properties of the system, viz. the delocalization of the many-body wavefunctions in Hilbert space. Along with this, we analyze the robustness of such highly entangled quantum states obtained through random unitary dynamics under weak measurements.

Quantumness-generating capability of quantum dynamics

We study quantumness-generating capability of quantum dynamics, where quantumness refers to the noncommutativity between the initial state and the evolving state. In terms of the commutator of the square roots of the initial state and the evolving state, we define a measure to quantify the quantumness-generating capability of quantum dynamics with respect to initial states. Quantumness-generating capability is absent in classical dynamics and hence is a fundamental characteristic of quantum dynamics. For qubit systems, we present an analytical form for this measure, by virtue of which we analyze several prototypical dynamics such as unitary dynamics, phase damping dynamics, amplitude damping dynamics, and random unitary dynamics (Pauli channels). Necessary and sufficient conditions for the monotonicity of quantumness-generating capability are also identified. Finally, we compare these conditions for the monotonicity of quantumness-generating capability with those for various Markovianities and illustrate that quantumness-generating capability and quantum Markovianity are closely related, although they capture different aspects of quantum dynamics.

Open System Model for Quantum Dynamical Maps with Classical Noise and Corresponding Master Equations

We show how random unitary dynamics arise from the coupling of an open quantum system to a static environment. Subsequently, we derive a master equation for the reduced system random unitary dynamics and study three specific cases: commuting system and interaction Hamiltonians, the short-time limit, and the Markov approximation.

Eternal non-Markovianity: from random unitary to Markov chain realisations

The theoretical description of quantum dynamics in an intriguing way does not necessarily imply the underlying dynamics is indeed intriguing. Here we show how a known very interesting master equation with an always negative decay rate [eternal non-Markovianity (ENM)] arises from simple stochastic Schrödinger dynamics (random unitary dynamics). Equivalently, it may be seen as arising from a mixture of Markov (semi-group) open system dynamics. Both these approaches lead to a more general family of CPT maps, characterized by a point within a parameter triangle. Our results show how ENM quantum dynamics can be realised easily in the laboratory. Moreover, we find a quantum time-continuously measured (quantum trajectory) realisation of the dynamics of the ENM master equation based on unitary transformations and projective measurements in an extended Hilbert space, guided by a classical Markov process. Furthermore, a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) representation of the dynamics in an extended Hilbert space can be found, with a remarkable property: there is no dynamics in the ancilla state. Finally, analogous constructions for two qubits extend these results from non-CP-divisible to non-P-divisible dynamics.

Quantum Entanglement Growth under Random Unitary Dynamics

Characterizing how entanglement grows with time in a many-body system, for example after a quantum quench, is a key problem in non-equilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time--dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the `entanglement tsunami' in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated Kardar--Parisi--Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like $(\text{time})^{1/3}$ and are spatially correlated over a distance $\propto (\text{time})^{2/3}$. We derive KPZ universal behaviour in three complementary ways, by mapping random entanglement growth to: (i) a stochastic model of a growing surface; (ii) a `minimal cut' picture, reminiscent of the Ryu--Takayanagi formula in holography; and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple `minimal cut' picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the `velocity' of entanglement growth in the 1D `entanglement tsunami'. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder.

Robustness and fragility of Markovian dynamics in a qubit dephasing channel

The Markovian dynamics of a qubit is investigated in the scheme of random unitary dynamics, where Kraus operators are changed by an extra noise. The behavior of Markovianity is explored in the perturbed scenario. We provide a new algorithm for checking CP-divisibility (Markovianity) of a dynamical map.

Investigation of a Two-Qubit Decoherence by Using Bloch Vectors

The loss of the coherence happens due to the interaction between the desired system and its surroundings. Addressing decoherence is one of the main concepts in the study of quantum channels to access their potential for various scale. In this process the information of the system is penetrated into the environment. Therefore, the measurement of the environment could be used as the error correction method. Phase damping channels principally belong to random unitary channels and can be corrected by classical information. This paper presents a method to generate non-random unitary phase damping channels based on the Bloch vectors in two qubits systems. A phase damping channel which consisted of a two qubits system and a single qubit environment was investigated. The results demonstrated that the phase damping channels belong to random unitary dynamics if the three-dimensional tetrahedron volume spanned by the Bloch vectors in R is not zero, or, the same hyperplane in R was not pointed by the Bloch vectors. It is found that the Bell state belongs to random unitary class and can be corrected based on classical information obtained from environmental measurements.

Non-Markovian random unitary qubit dynamics

We compare two approaches to non-Markovian quantum evolution: one based on the concept of divisible maps and the other one based on distinguishability of quantum states. The former concept is fully characterized in terms of local generator whereas it is in general not true for the latter one. A simple example of random unitary dynamics of a qubit shows the intricate difference between those approaches. Moreover, in this case both approaches are fully characterized in terms of local decoherence rates.

Quantum decoherence of two qubits

It is commonly stated that decoherence in open quantum systems is due to growing entanglement with an environment. In practice, however, surprisingly often decoherence may equally well be described by random unitary dynamics without invoking a quantum environment at all. For a single qubit, for instance, pure decoherence (or phase damping) is always of random unitary type. Here, we construct a simple example of true quantum decoherence of two qubits: we present a feasible phase damping channel of which we show that it cannot be understood in terms of random unitary dynamics. We give a very intuitive geometrical measure for the positive distance of our channel to the convex set of random unitary channels and find remarkable agreement with the so-called Birkhoff defect based on the norm of complete boundedness.