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Background

The density matrix of a closed quantum system with Hilbert space $\mathscr H$ evolves according to the von Neumann equation \begin{align*} i\hbar\dot\rho=[H,\rho]. \end{align*} Given a solution $\rho(t)$ to the above equation, the time-evolution of density matrices associated with subsystems can be obtained by taking the trace of $\rho$. Specifically, if $S$ is a subsystem (with complement $S^C$, which can be thought of as the 'bulk' of the full system) then $\rho_S(t)=\text{tr}_{S^C} (\rho(t))$.

Under reasonable assumptions, the long-time dynamics of $\rho_S(t)$ typically involves equilibration to a thermal state. However, memory effects are apparent at intermediate times, which can be thought of as echos of the initial state of $\rho_S$ that reflect the state of the bulk. Using essentially a sort of scattering theory, it should be possible to use the dynamics of multiple instances of $\rho_S(t)$ to characterize the state of the bulk.

Question

More fundamentally, I'm interested in the extent to which the full state $\rho$ together with the time-evolution generator $H$ can be inferred from a generic single instance of $\rho_S(t)$. The most basic version of the question is this: given a physically valid trajectory of a density matrix $\rho_S(t)$, what is the 'simplest' solution $(\mathscr H, \rho(t),H)$ to the von Neumann equation that induces the dynamics of $\rho_S(t)$?

For example, if you looked at the dynamics of a single classical particle from an ensemble of hard spheres, you could theoretically enumerate the possible many-particle trajectories that would induce the dynamics of the single particle you can actually see. Of course, you can always add auxiliary particles that never produce an observable effect, so we are interested in 'simple' or irreducible auxiliary ensembles.

[You can think of the problem as 'inverting' the hypothetical quantum master equation that $\rho_S(t)$ satisfies. One challenge is that the dimension of the Hilbert space that contains the solution $\rho(t)$ to the inverse problem must be inferred from $\rho_S(t)$ as well.]

Bonus questions:

1) Is it possible to express candidate solutions $(\mathscr H, \rho(t),H)$ as functionals of $\rho_S(t')$ for $t'\leq t$?

2) How much regularity of $\rho_S(t)$ is needed to ensure that a candidate $(\mathscr H , \rho(t),H)$ valid up to time $\tau$ will correctly predict the behavior of $\rho_S(t')$ for $t'>\tau$?

The reason I ask this question is that it seems like a tentative brute-force framework for quantizing any system based only on the dynamics of local operators.

Progress

Ok, here's 'the physicist's' approach. The idea here is to find a way of constructing moments of the full density matrix (and associated Hamiltonian) as a functional of $\rho_S(t)$. The equations to be solved are $\rho_S(t)=\text{tr}_{S^C}(\rho(t))$ and $i\hbar\dot\rho(t)=[H,\rho(t)]$, for some suitable $\rho$ and $H$ and identification of $S$, where we are given $\rho_S(t)$. Hence, we can introduce formal delta functions $$ \delta^\infty(\rho_S(t)-\text{tr}_{S^C}(\rho(t))),\quad \delta^\infty(i\hbar\dot\rho-[H,\rho]) $$ and weights $$ \exp(-\beta F[H]),\quad \exp(-\beta G[\mathscr H]) $$ that penalize candidate $\mathscr H$, $H$ pairs that are too 'large' or 'contrived' in a certain sense. In the limit as $\beta\rightarrow \infty$, it seems reasonable that there is a way to frame the problem such that a unique optimal choice of $H$ and $\mathscr H$ exists.

We can then perform formal averages over solutions via \begin{align*} \langle\mathcal O[\rho]\rangle=\frac{1}{Z}\int\mathscr D \mathscr H \Bigg\{\mathscr D\rho(t)\mathscr DH\bigg[&\exp[-\beta F[H]-\beta G[\mathscr H]]\times\dots\\&\delta^\infty(\rho_S-\text{tr}_{S^C}\rho)\delta^\infty(i\hbar\dot\rho-[H,\rho])\mathcal O[\rho]\bigg]\Bigg\} \end{align*} where $Z$ is the associated partition function. Implicitly, we also have the constraints $\rho=\rho^\dagger$, $\text{tr}(\rho)=1$, and that $\rho$ must be nonnegative.

To make sense of $\text{tr}_{S^C}(\rho)$ we need some standard way of identifying the subsystem $S$ in $\mathscr H$. This could be done, for example, by summing only over Hilbert spaces obtained by sequentially adding new degrees of freedom to $\mathscr H_S$ (and summing over the 'cardinalities' of the new degrees of freedom). Once this has been done, we can use a formal version of the Fourier representation of the delta function in order to make the integrand look more like a path integral. This entails introducing a large number of auxiliary fields, depending on the size of the subsystem and the size of the 'ambient space' $\mathscr H$. When $\rho_S(t)$ is finite dimensional and has a finite discrete Fourier transform, we expect $\mathscr H$ to be finite dimensional.

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  • $\begingroup$ This question reminds me of Nielsen's work on quantum computational complexity, where he mentioned the question: Will adding ancilla system lead to a more efficient algorithm? Following his formulation on Riemannian manifold, I do believe your question is related with the curvature of the manifold (at least on how reliable we can predict the evolution). Susskind assumes in "Switchbacks and the Bridge to Nowhere" that a submanifold of the problem has a constant negative curvature, Nielsen assumes that most likely the curvature is negative. So predication is difficult. $\endgroup$ – XXDD Feb 14 '16 at 4:07
  • $\begingroup$ Your question 'given a physically valid trajectory of a density matrix ρS(t)ρS(t), what is the 'simplest' solution (H,ρ(t),H)(H,ρ(t),H) to the von Neumann equation that induces the dynamics of ρS(t)ρS(t)' is similar with the question of Nielsen on the most efficient quantum algorithm. But he only concerns the boundary condition (initial and final state of the subsytem), you set more constraints on the complete trajectory. $\endgroup$ – XXDD Feb 14 '16 at 4:11
  • $\begingroup$ If your Hamiltonian is time independent, then the system evolves as a Lie exponential, if it's time dependent, then the most efficient evolution trajectory is a geodesic on the Riemannian manifold depending on the metric. See Nielsen's paper 'The geometry of quantum computation' arXiv:quant-ph/0701004v1 $\endgroup$ – XXDD Feb 14 '16 at 4:16
  • $\begingroup$ The computation as an optimization problem is very very difficult, at least numerically. Nielsen only succeeded on 3 qubits, I used the geodesic shooting method, it can hardly work on 4 qubits case (it takes me 2 days to find one geodesic on my PC). Of course if you work with constant Hamiltonian, it might be easier, but then the trajectory may not be 'optimal'. Also considering the negative curvature, the optimization strategy will not work finally. $\endgroup$ – XXDD Feb 14 '16 at 4:23
  • $\begingroup$ It's tricky to define ' what is the 'simplest' solution'. If the trajectory of $\rho_s(t)$ is fixed, then $\rho(t)$ is just a lift of the curve $\rho_s(t)$, then the 'simplest' solution can be defined as the lifted curve $\rho(t)$ with the minimal length depending on the metric in the extended space. $\endgroup$ – XXDD Feb 14 '16 at 10:12
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The situation you mention has a long history, So you start with \begin{equation} \partial _{t}\rho =-i[H,\rho ]=-iL\rho \Rightarrow \rho (t)=\exp [-iLt]\rho (0) \tag{(1)} \end{equation} where $L$ is the Liouville operator and $\rho $ is a non-negative trace class operatot, i.e., an element of $\mathcal{B}_{1}$, with unit trace norm \begin{equation*} \parallel \rho \parallel _{1}=\mathrm{tr}\rho =1 \end{equation*} The partial trace leading to $\rho _{S}(t)$ can be formulated as the action of a projection operator \begin{equation*} \rho _{S}(t)=P\rho (t) \end{equation*} Then, with $Q=1-P$, we obtain \begin{eqnarray*} \partial _{t}P\rho (t0 &=&-iPLP\rho (t)-iPLQ\rho (t) \\ \partial _{t}Q\rho (t0 &=&-iQLQ\rho (t)-iQLP\rho (t) \end{eqnarray*} from which \begin{equation*} Q\rho (t)=\exp [-iQLQt]Q\rho (0)-i\int_{0}^{t}ds\exp [-iQLQ(t-s]QLP\rho (s) \end{equation*} Substitution into the first gives a closed equation for $P\rho (t)$ \begin{equation*} \partial _{t}P\rho (t)=-iPLP\rho (t)-iPLQ\exp [-iQLQt]Q\rho (0)-\int_{0}^{t}dsPLQ\exp [-iQLQ(t-s]QLP\rho (s) \end{equation*} This is all pretty standard stuff. The resulting equation for $P\rho (t)$ is known as the generalised master equation in statistical mechanics. In that field the initial term \begin{equation*} PLQ\exp [-iQLQt]Q\rho (0) \end{equation*} usually dies out for large $t$ in the thermodynamic limit. In other cases simply $Q\rho (0)=0$. Thus let us concentrate on \begin{equation} \partial _{t}P\rho (t)=-iPLP\rho (t)-\int_{0}^{t}dsPLQ\exp [-iQLQ(t-s]QLP\rho (s) \tag{(2)} \end{equation} We see that a time-convolution term has appeared which is a remnant of the part that is projected out. Note that (1) is still exact. It is possible

(see http://www.physicsoverflow.org/17968/how-to-handle-nonmarkovian-dynamics-in-open-quantum-system.) to recover an equation without convolution term by introducing an additional variable, say $\sigma (t)$. Then \begin{equation*} \left( \begin{array}{c} \rho (t) \\ \sigma (t) \end{array} \right) \end{equation*} obeys an equation similar to (1) but $\sigma (t)$ is not equal to $Q\rho (t) $, but acts in a subspace of $\mathcal{B}_{1}$.

Thus one does not retrieve the full original motion in this way but (2) is an exact equation for $P\rho (t)$.

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  • $\begingroup$ Thank you for your answer. I'm familiar with the quantum master equation for an open quantum system, which provides a closed equation of motion for the subsystem with additional 'initial data' given by $Q\rho(0)$. I'm not sure that this addresses the question I was hoping to resolve, however. The goal is to recover $\rho(t)$ from $P\rho(t)=\rho_S(t)$. In particular, this requires 'guessing' how the Hilbert space of $\rho_S(t)$ must be augmented in order for $\rho_S(t)$ to equal $P\rho(t)$ for some $\rho$ that satisfies the von Neumann equation (for a time-independent $H$). $\endgroup$ – TotallyRhombus Feb 15 '16 at 16:09

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