# Matters Arising: Entanglement-enhanced matter-wave interferometry in a high-
finesse cavity

Liam P. McGuinness  Laser Physics Centre, Research School of Physics,
Australian National University, Acton, Australian Capital Territory 2601,
Australia
Email<EMAIL_ADDRESS>
###### Abstract

In their paper “Entanglement-enhanced matter-wave interferometry in a high-
finesse cavity” Nature (2022) [1], Greve et. al. claim to use entanglement in
a matter-wave interferometer to achieve a sensitivity beyond that achievable
with the same number of independent particles – a limit known as the standard
quantum limit (SQL). In particular, using squeezed momentum states of 700
atoms, the authors claim to directly observe a sensitivity 3.4 dB (a factor of
1.5) below the SQL. This claim is incorrect. The authors do not measure
anything beyond the SQL, nor do they achieve a sensitivity beyond what one
could obtain with a single atom. The achieved sensitivity is at least a factor
of 39 worse than the claimed value.

In “Entanglement-enhanced matter-wave interferometry in a high-finesse cavity”
Nature (2022) [1], Greve et. al. describe measuring the phase $\phi$ between
two quantum states $\left|a\right>,\left|b\right>$, given by
$\frac{1}{\sqrt{2}}\left(\left|a\right>+e^{{i\mkern
1.0mu}\phi}\left|b\right>\right)$. With no prior information on $\phi$, i.e.
$0<\phi\leq 2\pi$, one cannot estimate this phase with a single measurement to
an angular uncertainty better than $\Delta\phi=1$ radians. Allowing for $N$
trials, which can be implemented by encoding $\phi$ into the state of $N$
independent atoms, the uncertainty must be greater than
$\Delta\phi_{\mathrm{SQL}}=1/\sqrt{N}$ rad, a limit known as the standard
quantum limit (SQL). As Greve et. al. note, this is a fundamental limit which
cannot be improved upon, even with entanglement. In fact, if one entangles $N$
atoms to obtain the state
$\frac{1}{\sqrt{2}}\left(\left|a_{N}\right>+e^{{i\mkern
1.0mu}\phi}\left|b_{N}\right>\right)$, where
$\left|a_{N}\right>,\left|b_{N}\right>$ are states in an $N$-dimensonal
Hilbert space, measurement of this phase has an uncertainty is restricted to
$\Delta\phi>1$ rad, much worse than the SQL. The reason being that we have
lost the ability to perform $N$ independent trials and are thus limited to the
uncertainty of a single measurement. This is clear to see, since the entangled
state is identical to the single atom state up to a relabelling.

While the above is well-known for estimating an unknown quantum phase, it is
widely accepted that despite this entanglement can lead to improved estimation
of some other phase $\theta$. Why is this? The argument is that entangled
states accumulate a bigger quantum phase in response to some physical
Hamiltonian to be measured ($N$-fold greater than a single atom). With
particular reference to Mach-Zehnder interferometry, discussed by Greve et.
al., this is the phase that particles in one arm of the interferometer
accumulate with respect to particles in the other arm. If we also allow
somewhat sneakily that we now have more prior information on $\phi$ – it is
known to within a much narrow range, then it is expected that, for the same
measurement time, entangled states achieve greater sensitivity to small phase
shifts than unentangled states [2]. As a result, although the uncertainty in
estimating $\phi$ is $\sqrt{N}$ worse with entanglement than with independent
atoms, the value of $\phi$ in an entangled state is $N$-fold greater than any
of the independent atomic states. Again, it is important to be clear here.
With entanglement we have not improved the uncertainty in estimating $\phi$,
it has gotten worse. Whenever measurement of a quantum phase is performed as
described above, one can immediately rule out surpassing the SQL, it is only
when the quantum phase is used to infer some other parameter $\theta$ that the
uncertainty in $\theta$ can be reduced.

Let’s assume an element with phase $\theta$ is in one arm of the
interferometer, and passing a single atom through the interferometer performs
a one-to-one mapping of $\theta$ to the atomic phase $\phi$. Then with no
entanglement we have $\phi=\theta$, whereas with entanglement $\phi=N\theta$.
In the former case (assuming no additional errors) $\Delta\theta=\Delta\phi$
and estimation of $\theta$ with $N$ atoms is limited by the SQL to:
$\Delta\phi>1/\sqrt{N}\Longrightarrow\Delta\theta>1/\sqrt{N}$. With
entanglement $\Delta\theta=\Delta\left(\phi/N\right)$ again assuming no
additional errors, so we obtain $\Delta\phi>1\Longrightarrow\Delta\theta>1/N$.
If the second inequality can be saturated with no overheads, then entanglement
outperforms unentangled sensors. Contrary to what is often stated, this
superiority of entanglement in sensing is not proven and only holds under
strong experimental assumptions. For that reason, experimental evidence with
correct analysis and complete details is critical in validating the theory
[3].

So what is the phase shown in Fig. 1b that Greve et. al. measure with a
precision beyond the SQL? Described in section Entangled matter-wave
interferometry: as “A relative phase accumulates between the wave packets
during a free evolution time $T_{\mathrm{evol}}$”, at first reading one might
be surprised to find that the cause of the relative phase shift is not
explicitly defined in the main text. One reason for this oversight could be
that Fig. 1b and the description of the matter-wave interferometer is somewhat
misleading. Greve et. al. do not really perform Mach-Zehnder interferometry
because they do not measure a phase shift between different arms of the
interferometer. So what are the authors measuring? Greve et. al. allow the
atoms to fall through a gravitational potential and measure the energy shift
of the atoms – manifesting as a Doppler shift of atomic resonance with respect
to the Raman laser (see Methods – Raman transitions and velocity selection).
Notably this shift is the same for both arms of the interferometer and can be
measured without a Mach-Zehnder interferometer. Most importantly, entangled
atoms do not experience a greater phase shift (see Fig. 4c), since their
velocity is the same as unentangled atoms.

Put simply, the authors measure the difference between the atomic phase $\phi$
and the laser phase $\theta$ at the end of the experiment. As entanglement
produces no enhanced phase shift, the uncertainty in measuring either phase
when using entanglement is strictly worse than the SQL, and we have shown that
Greve et. al.’s claim in beating the SQL is incorrect. In fact, assuming the
ensemble is fully entangled, the obtained sensitivity must be worse than the
single atom precision limit. Even assuming the ensemble is not fully
entangled, there are many experimental imperfections that prevent Greve et.
al. reaching the single atom limit. With $N=660$ atoms, this means that the
achieved sensitivity is at least a factor of 31 worse than the claimed 1.7 dB
enhancement. Similar analysis can show that all other claims made by the
authors are similarly incorrect. So how can Greve et. al. claim to have done
otherwise? Maybe it is better to reframe this question with a focus on the
audience. These are the questions one should ask anybody claiming to beat the
SQL.

  1. 1.

Precisely what parameter do you measure?

  2. 2.

What sensitivity/uncertainty for this parameter do you explicitly achieve in
your experiment? This should have the correct units, including the total
measurement time and the number of particles used.

  3. 3.

Is it _really_ impossible to measure this parameter with better sensitivity
using the same measurement time and the same number of independent particles?
How about just a single particle. Is it impossible to measure this parameter
with better sensitivity using the same measurement time and a single particle?

If 1) and 2) are properly defined, then the answer for all experiments to
date, including the work by Greve et. al. is a resounding – ‘No!’.

It is important that the scientific community is made aware of the current
state of the art in quantum metrology. I am sure that many people would be
extremely surprised to learn that entanglement has never been used to improve
any experiment beyond what one could achieve without entanglement. Even more
surprising is that fully entangled ensembles have never demonstrated a
precision beyond the single particle limit. The community should be made aware
of this for a variety of reasons. First, if the experimental data conflicts
with the message being presented then we should demand better scientific
rigour in published papers. In quantum metrology broadly, the standards have a
long way to improve. Secondly, misrepresentation of data is hindering progress
since people are currently unaware of a massive discrepancy between quantum
mechanics as interpreted and experiment; even going so far as to prevent
plausible explanations from being investigated. Thirdly, there is currently
huge investment in technologies dependent on quantum entanglement (in both
quantum sensing and quantum computing), if entangled ensembles cannot provide
fundamentally more information than a single atom then these technologies will
never reach their goals.

## References

  * [1] Greve, G. P., Luo, C., Wu, B. & Thompson, J. K.  Entanglement-enhanced matter-wave interferometry in a high-finesse cavity.  _Nature_ 610, 472–477 (2022).  URL https://doi.org/10.1038/s41586-022-05197-9.
  * [2] I am presenting the view of the general quantum metrology community, however I must make my stance clear. I do not share the same views. Indeed I expect that entanglement cannot improve the amount of information that can be obtained from the same number of atoms in the same time. Specifically the information gain from fully entangled states cannot exceed that obtained with a single spin in the same time.
  * [3] One specific overhead is the total measurement time $T$. For example by increasing the interaction time of a single atom with the phase element we can also achieve a mapping $\phi=N\theta$ with a single atom. This is why $\Delta\theta>1/\sqrt{N}$ is not the SQL for estimating $\theta$. The SQL for estimating $\theta$ is $\Delta\theta>1/(\Omega T\sqrt{N})$, where $\Omega$ is the interaction strength between the atom and the phase element.