Equation for quantum entanglement
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Equation for Quantum Entanglement
Quantum entanglement is a fundamental phenomenon in quantum mechanics where the quantum states of two or more particles become interconnected such that the state of one particle cannot be described independently of the state of the other(s). This article synthesizes the key findings from recent research papers on the equations and dynamics governing quantum entanglement.
Evolution Equation for Quantum Entanglement
The time evolution of quantum entanglement, especially under the influence of environmental decoherence, is a critical area of study. A general factorization law has been proven for quantum systems shared by two parties, which describes the time evolution of entanglement as one component passes through an arbitrary noisy channel. This factorization law simplifies the characterization of the robustness of entanglement-based quantum information processing protocols by reducing it to a single quantity1.
Entanglement of Formation
The entanglement of formation is a measure used to quantify the amount of entanglement in a mixed state of a bipartite quantum system. For mixed states of two qubits with no more than two nonzero eigenvalues, an exact formula for the entanglement of formation has been derived. This formula is suggested to be valid for all states of this system, providing a precise method to calculate the minimum number of singlets needed to create an ensemble of pure states representing the mixed state2.
Real-Space Entanglement
A new method has been introduced to analytically determine the entanglement entropy between configurations of a quantum field at two distinct spatial locations. This method allows for the derivation of explicit and exact formulas for entanglement entropy, mutual information, and quantum discord in terms of the Fourier-space power spectra of the field. This approach contrasts with previous numerical studies and provides exact expressions for these quantities, particularly for massless fields in flat space3.
Quantum Entanglement in Nanocavity Arrays
Theoretical studies have shown that quantum interference between linearly coupled modes with weak local nonlinearity can generate continuous variable entanglement. By solving the quantum master equation for the density matrix, it has been demonstrated that this entanglement can survive realistic levels of pure dephasing, forming a paradigm for entanglement generation in arrays of coupled quantum modes4.
Entanglement Dynamics in Integrable Systems
The entanglement dynamics after a quench from a piecewise homogeneous initial state in integrable systems have been investigated. A formula for the entanglement production rate and the steady-state entanglement entropy of a finite subregion has been conjectured. These quantities are determined by the quasiparticles created in the Non-Equilibrium Steady State (NESS) at the interface between two reservoirs, with the steady-state entropy coinciding with the thermodynamic entropy of the NESS5.
Markovian Entanglement Dynamics
In the context of locally scrambled quantum dynamics, the average entanglement entropy follows Markovian dynamics. This means that the entanglement properties of the future state can be predicted based on the current state and the unitary operator at each step. The entanglement feature formulation organizes the entanglement entropies over all subsystems into a many-body wave function, allowing the description of entanglement dynamics using an imaginary-time Schrödinger equation6.
Entanglement in Continuous-Variable Quantum Networks
In continuous-variable quantum networks, entanglement dynamics can be mapped to a random-walk process on a graph. Squeezing is identified as the source of entanglement generation, leading to a diffusive spread of entanglement with a "parabolic light cone." Despite the nonlinear nature of entanglement dynamics, a surprising linear superposition law in entanglement growth has been predicted and numerically verified7.
Conclusion
The study of quantum entanglement equations and dynamics is crucial for advancing quantum information technology. From factorization laws and exact formulas for entanglement of formation to new methods for determining entanglement entropy and understanding entanglement dynamics in various systems, these research findings provide a comprehensive understanding of how entanglement evolves and can be harnessed in practical applications.
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Most relevant research papers on this topic
Evolution equation for quantum entanglement
This paper presents a simple factorization law for quantum systems shared by two parties, describing the time evolution of entanglement on passage of either component through an arbitrary noisy channel, improving the robustness of entanglement-based quantum information processing protocols.
Highly CitedEntanglement of a Pair of Quantum Bits
The entanglement of formation formula is valid for all mixed states of two qubits with no more than two nonzero eigenvalues, suggesting it is a universal formula for bipartite quantum systems.
Highly CitedReal-space entanglement of quantum fields
Our method allows for the analytical determination of entanglement entropy, mutual information, and quantum discord in quantum fields, with exact formulas for arbitrary distances.
Quantum entanglement in nanocavity arrays
Quantum entanglement can be generated in nanocavity arrays through weak local nonlinearity, surviving realistic levels of pure dephasing.
Entanglement and quantum transport in integrable systems
Entanglement production rate and steady-state entanglement entropy in integrable systems are determined by quasiparticles created in the Non Equilibrium steady State, with their spreading reflecting the bulk of the two reservoirs.
Markovian entanglement dynamics under locally scrambled quantum evolution
The entanglement feature approach effectively predicts future entanglement properties in locally scrambled quantum dynamics, enabling investigation of various random quantum dynamics beyond Haar random circuits and Brownian circuits.
Entanglement formation in continuous-variable random quantum networks
Squeezing is the source of entanglement generation in continuous-variable quantum networks, leading to a linear superposition law and various shapes of equilibrium entanglement distributions.
Entanglement in quantum critical phenomena.
Critical entanglement in spin systems is analogous to entropy in conformal field theories, connecting quantum information, condensed matter physics, and quantum field theory.
Highly CitedMeasuring entanglement entropy in a quantum many-body system
This study successfully measures entanglement in a quantum many-body system using quantum interference of many-body twins, enabling the study of quantum phases and dynamics in strongly correlated systems.
Rigorous JournalHighly CitedEntanglement Equilibrium and the Einstein Equation.
The Einstein equation links the maximal vacuum entanglement hypothesis and the stability of vacuum entanglement in conformal quantum fields.
Highly Cited