Unbounded separation between definite and indefinite causal order in finite-dimensional quantum metrology

Abstract

Indefinite causal order is known to offer enhancements in quantum metrology, notably including an unbounded advantage in the measurement of a geometric phase of a harmonic oscillator. This advantage, however, is specific to the infinite dimensional setting, and its finite dimensional analogue remains elusive, with recent findings suggesting that advantages in finite dimensions may be fundamentally limited to bounded constant factors. Here we show that, in fact, arbitrarily large advantages arise for finite dimensional systems in the finite sample regime. Specifically, we establish an unbounded separation between definite and indefinite causal order in the estimation of a geometric phase associated to two sets of $N$ displacements generated by discrete position and momentum operators on a $d$-dimensional quantum system: for any given constant $R$, there exist values of $N$ and $d=Ω(N^2)$ such that a strategy with indefinite order uses an initial probe with $R$ times less energy than the probe required by every strategy with definite order achieving the same mean squared error, whenever the number of measurement shots $ν$ is bounded as $ν=\mathcal{O}(\exp(πd/16)/\mathrm{poly}(d))$. In other words, indefinite order offers an energy saving that grows arbitrarily large with the parameters of the problem. To prove this result, we establish an approximate Weyl relation for discrete Gaussian wavepackets, which is of independent technical interest.