How to detect when one state can not be reached from another autonomously?

Joint work with Chau Nguyen, Yuri Minoguchi, and Tomasz Andrzejewski.

In the spirit of a previous note, here I investigate the question of reachability of quantum states. Consider the initial states $\ket\psi$, and a linearly parameterized family of Hamiltonians $\sum_{k=1}^n \lambda_k H_k$; the question is when can we determine when a given target state $\ket\phi$ can – or can not – be reached as an autonomous, unitary evolution of $\ket\psi$.

Naturally, if parameters $(\lambda_1, \ldots, \lambda_n)$ can be found such that $\ket\phi = \exp(-i\sum_{k=1}^n \lambda_k H_k) \ket\psi$, the question is decided. But generically, finding such parameters is nontrivial and one has to resort to numerical methods to find them. Naturally, numerical search can not provide a negative answer, that is: the proof that a state $\ket\phi$ is unreachable from $\ket\psi$.

Some time ago I found a method which can in some cases serve as an unreachability proof. It does not work always – quite frequently it is inconclusive, but when it does work, one can be sure that no autonomous evolution generated by a given family of Hamiltonians can lead from the initial to the final state.

The method relies on comparison of expectation values of the Hamiltonian moments. Naturally, if a nontrivial Hamiltonian $H:=\sum_{k=1}^n \lambda_k H_k$ exists such that $\ket\phi=\exp(-iH)\ket\psi$, the expectation value of $H$ must coincide on the two states: $\langle H\rangle_\psi = \langle H\rangle_\phi$. But variance of $H$, and any other moment have to be equal too, leading to $\langle H^m\rangle_\psi = \langle H^m\rangle_\phi$.

In mathematical terms, we are looking for the parameters $\lambda_1, \ldots, \lambda_n$ such that the difference of moments is zero. When written down, the differences for $m=1, 2$ can be recognized as linear and quadratic expressions in the parameters:

\begin{equation}\begin{aligned} \langle H\rangle_\psi -\langle H\rangle_\phi &= \sum_{k=1}^n \lambda_k (\langle H_k\rangle_\psi -\langle H_k\rangle_\phi),\\ \langle H^2\rangle_\psi -\langle H^2\rangle_\phi &= \sum_{k,l=1}^n \lambda_k \lambda_l (\langle H_k H_l\rangle_\psi -\langle H_k H_l\rangle_\phi). \end{aligned} \end{equation}

The quadratic part is always real for real $(\lambda_1, \ldots, \lambda_n)$, and hence can be rewritten with a real symmetric matrix: $\langle H^2\rangle_\psi -\langle H^2\rangle_\phi = \sum_{k,l=1}^n \lambda_k Q_{k,l} \lambda_l$. Let us name the linear part by $L$, and the quadratic by $Q$, and consider the following expression:

\begin{equation}\begin{aligned} R(C)=\overbrace{Q}^{\langle H^2\rangle_\psi -\langle H^2\rangle_\phi}\!\!\!\!\!\!\!\! + C\!\!\!\!\!\!\!\! \underbrace{L^2}_{(\langle H\rangle_\psi -\langle H\rangle_\phi)^2}. \end{aligned} \end{equation}

For any real number $C$, the above expression is a quadratic expression in the parameters $(\lambda_1, \ldots, \lambda_n)$. If a number $C$ can be found such that the real symmetric matrix defining $R$ is positive definite – it has strictly positive eigenvalues – then it means that for no choice of the parameters $R_C$ can be zero (apart from $\lambda_1=\ldots=0$, of course – but this corresponds to no dynamics at all, and has no physical relevance). And if $R_C$ can not be zero, $Q$ can not be zero when $L$ vanishes! In other words, the two first moments of $H$ can not be simultaneously equal for $\ket\psi$ and $\ket\phi$.

As a simple example, consider the system of $N$ bosons with one- and two-body interaction 1, spanned by

\begin{equation}\begin{aligned} (H_1, \ldots, H_8) = (&a_1^\dagger a_1, a_2^\dagger a_2, a_1^\dagger a_2 + \text{h.c.}, (i a_1^\dagger a_2)+ \text{h.c.}, a_1^\dagger a_1^\dagger a_1 a_1,\\& a_1^\dagger a_2^\dagger a_1 a_2, a_1^\dagger a_1^\dagger a_2 a_2 + \text{h.c.}, (i a_1^\dagger a_1^\dagger a_2 a_2)+ \text{h.c.} ) \end{aligned} \end{equation}

Starting from $N=9$ bosons, the state $\frac1{\sqrt2}\left(\ket{N-1,1}+\ket{1,N-1}\right)$ is unreachable from $\ket{N,0}$, and this can be proven by the method sketched above.

  1. This can be considered also as a rotor system, with the Hamiltonian family spanned by the spin matrices $S_x, S_y, S_z$ and their linearly independent anticommutators $S_z^2$, $S_x S_y + \text{h.c.}$, $S_x S_z + \text{h.c.}$, $S_y^2$, $S_y S_z + \text{h.c.} $ ↩