Random Bosonic States for Robust Quantum Metrology

We study how useful random states are for quantum metrology, i.e., whether they surpass the classical limits imposed on precision in the canonical phase sensing scenario. First, we prove that random pure states drawn from the Hilbert space of distinguishable particles typically do not lead to superclassical scaling of precision even when allowing for local unitary optimization. Conversely, we show that random pure states from the symmetric subspace typically achieve the optimal Heisenberg scaling without the need for local unitary optimization. Surprisingly, the Heisenberg scaling is observed for random isospectral states of arbitrarily low purity and preserved under loss of a fixed number of particles. Moreover, we prove that for pure states, a standard photon-counting interferometric measurement suffices to typically achieve resolution following the Heisenberg scaling for all values of the phase at the same time. Finally, we demonstrate that metrologically useful states can be prepared with short random optical circuits generated from three types of beam splitters and a single nonlinear (Kerr-like) transformation.

Popular Summary

Ultraprecise measurements of physical quantities lie at the heart of modern science and span fields ranging from physics to biology. In recent decades, scientists have discovered that entanglement, a peculiar form of physical correlations predicted by quantum mechanics, can be used to attain measurement precision beyond the reach of classical statistics. Researchers can accordingly sense physical quantities such as the magnitude of electric or magnetic fields or the deformation of interferometer arms by gravitational waves with unprecedented resolution. However, entanglement alone is not sufficient for quantum-enhanced precision to be observed. Here, we show that, surprisingly, usefulness for quantum-enhanced sensing is not specific to a peculiar type of quantum state but is rather a generic property of bosonic particles such as photons.

We theoretically study the usefulness of random states for quantum-enhanced precision measurements in paradigmatic interferometric scenarios. We show that typical pure states of many distinguishable particles, despite their high entanglement, do not offer an asymptotic advantage for parameter estimation compared with classical strategies (i.e., the squared error of estimation is not smaller than $1/N$). On the other hand, we show that symmetric (bosonic) states, even when chosen randomly, yield enhanced precision in parameter sensing. Moreover, we prove that this improvement can be realized in the presence of noise and finite particle losses and by using a single measurement device. Finally, we demonstrate that short random bosonic circuits can be employed to approximately generate the necessary bosonic states. Going forward, we suggest that future studies focus on other sources of noise such as dephasing and depolarization.

We expect that our findings will advance our qualitative understanding of which states offer super-resolution in quantum sensing. Our results also pose a number of open questions, such as whether similar techniques can be used in a continuous-variable setting or when simultaneously sensing many parameters.

Figure 1

Local-unitary (LU) optimized lossy quantum metrology protocol: A given state $\rho$ is adjusted with LU operations $V_1,\dots ,V_N$ to probe, as precisely as possible, small fluctuations of the parameter $\phi$, which is independently and unitarily encoded into each of the $N$ constituent particles. Finally, only $N-k$ particles are measured, reflecting the possibility of losing $k$ of them. The most general measurement is then described by a positive-operator-valued measure (POVM), i.e., a collection of positive semidefinite operators $\{{\mathrm{\Pi }}_n^{N-k}{\}}_n$ that act on the remaining $N-k$ particles and sum up to the identity. The process is repeated $\nu$ times, in order to construct the most sensitive estimate of the parameter, $\tilde{\phi }$, based on the measurement outcomes $\{n_i{\}}_{i=1}^{\nu }$.

Figure 2

Mach-Zehnder interferometer: We consider a bosonic pure state ${\psi }_N$ of $N$ photons in modes $a$ and $b$ inside the interferometer. The estimated phase $\phi$ is acquired because of the path difference in the arms. After a balanced beam splitter, the number of photons $n$ and $N-n$ are measured in arms $a$ and $b$, respectively.

Figure 3

Convergence of QFI and FI of random circuit states for increasing depth $K$. The main plot shows $F$ and $F_{\mathrm{cl}}$ of $N=100$ two-mode photons (indistinguishable qubits) for the measurement from Fig. 2 averaged over 150 realizations (sufficient to make the finite sample size irrelevant) of random circuits for different depths $K$. The starting states before the random circuit are of the form $|{\psi }_N⟩=\sum _{n=0}^N\sqrt{x_n}|n,N-n⟩$, with $x_0=1$ polarized (red line), $x_n=(\genfrac{}{}{0ex}{}{N}{n})/2^N$ balanced (blue line), or $x_0=x_N=1/2$ N00N (yellow line). Fast convergence to the values $N(N+1)/3$ and $N(N+1)/6$ (black horizontal lines) is evident in all cases. The shaded regions mark “worse than SQL” and “better than HL” precisions. The $F_{\mathrm{cl}}$ curves are plotted for $\phi =\pi /2$ (solid line) and $\phi =\pi /3$ (dotted line). The inset depicts the sufficient circuit depth $K_{\mathrm{suf}}$ as a function of $N$, such that the corresponding sample-averaged QFI (black curves) or FI (red curves) is at most 1% (top curves) or 10% (bottom curves) from its typical value. As $K_{\mathrm{suf}}$ grows, at most, mildly with $N$, for realistically achievable photon numbers [78], $K\approx 20$ may be considered sufficient.

Figure 4

Mean-squared error attained by random bosonic states generated by sufficiently deep random circuits. We depict the ultimate limit of the resolution $\nu {\mathrm{\Delta }}^2\tilde{\phi }$ attainable with random circuit states generated by applying a deep random circuit ($K=60$) onto a two-mode balanced [$x_n=(\genfrac{}{}{0ex}{}{N}{n})\frac{1}{2^N}$ in Fig. 3] state for both the interferometric measurement of Fig. 2 (red line) and the theoretically optimal one yielding the QFI (black line). The corresponding sample-averaged FI and QFI quickly concentrate around the typical values $N^2/3$ and $N^2/6$ (dashed lines), respectively. The shaded regions mark the “worse than SQL” and “better than HL” precisions.