How to ascertain entanglement in Heisenberg picture? It appears to me that the definition of entanglement explicitly refers to the state of the system in the Schrodinger picture, i.e., if a system $\psi\in\mathcal{\otimes_i\mathcal{H}_i}$ is such that $\psi=\otimes_i\phi_i$ is not true $\forall\phi_i\in\mathcal{H}_i$. To ascertain the entanglement of a system at a moment, the validity of this condition needs to be checked at the given moment. Thus, it is not sufficient to ascertain the validity of this condition at $t=0$ to answer the question as to whether the given system is entangled at time $t\neq 0$. Thus, in order to answer the question as to whether the given system is entangled or not in the Heisenberg picture, we need to find the equivalent condition in terms of the observables of the system -- which are the time-dependent entities in the formalism. However, I can't think of any simple way to translate the usual definition in the language of observables.
Instinctively, I think that one possible approach towards the answer might be using the language of entanglement entropy but I am not sure how to go about it. For example, von Neumann entropy is a function of only the density matrix and the density matrix is time-independent in the Heisenberg picture so, treated naively, in the Heisenberg picture, the von Neumann entropy would remain zero throughout the time-evolution if it was initially zero. But clearly, a system can become entangled during the time-evolution even if it started out unentangled.

One qualitative way of describing entanglement in the language of observables is to say that it is not true that for all $\mathcal{H}_i$, there exists a complete set of commuting observables $\{O_{ij}\vert j = 1,...,(\mathrm{dim}(\mathcal{H}_i))^2\}$ that is diagonalized. As one can appreciate, this is what it means to say that there exists at least one $i$ for which it is true that there is no state-vector of the subsystem $\mathcal{H}_i$ that can describe the subsystem. However,...

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*I am not sure if this criterion differs from the usual definition in some subtle ways.

*I am not sure how to quantitatively formulate a measure of entanglement in the Heisenberg picture even if this criterion is correct. How does one accommodate the fact that, for example, von Neumann entropy can change with time?

 A: In the schrödinger picture, entanglement is usually defined with respect to a fixed factorization of the Hilbert space, say $H_A\otimes H_B$. The factors correspond to complementary subsystems. The key to adapting the concept of entanglement to the heisenberg picture is to define subsystems in terms of observables instead, which is arguably more natural anyway.
First consider a fixed time, say $t=0$, so that the distinction between the schrödinger and heisenberg pictures doesn't matter. For a given factorization $H_A\otimes H_B$, let $\Omega_A$ denote the set of all operators that act nontrivially only on $H_A$. This is a subset of all of the operators on the full Hilbert space. Here's the key: the factorization $H_A\otimes H_B$ is uniquely determined by the subset $\Omega_A$. Therefore, given a generic state (density matrix) $\rho$ on the full Hilbert space, and given a subsystem defined by the set $\Omega_A$ of observables, we have all the information we need to define entanglement (and to calculate entanglement entropy) in the usual way.

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*In the schrödinger picture, the state $\rho_t$ carries the time-dependence, and the observables are independent of time. In particular, the set $\Omega_A$ of operators representing the observables for a given subsystem $A$ is independent of time. This set of observables defines a time-independent factorization $H_A\otimes H_B$ of the Hilbert space. The entanglement of the state $\rho_t$ is defined with respect to this factorization in the usual way. The entanglement depends on time because the state $\rho_t$ depends on time and the factorization does not.


*In the heisenberg picture, the state $\rho$ is time-independent, and the observables carry the time-dependence. In particular, the set $\Omega_{A,t}$ of operators representing the observables for a given subsystem $A$ depends on the time $t$. This defines a time-dependent factorization $H_{A,t}\otimes H_{B,t}$ of the Hilbert space. The entanglement of the state $\rho$ is defined with respect to this factorization in the usual way. The entanglement depends on time because the factorization depends on time and the state $\rho$ does not.
In this answer, I assumed that the set of observables corresponding to a given subsystem is such that it defines a factorization of the Hilbert space. That's enough for answering the question, because most discussions of entanglement in the schrödinger picture also make that assumption. However, that assumption is not necessary. The concept of entanglement, or at least the entropy of entanglement, can be defined with respect to any set of observables on a finite-dimensional Hilbert space. The details are explained in appendix A in arXiv:1607.03901, and since it's expressed in terms of observables anyway, it automatically works just as well in the heisenberg picture.
