# Transmission Electron Microscopy at the Quantum Limit

Stewart A. Koppell  Physics Department, Stanford University, Stanford,
California 94305, USA  Department of Electrical Engineering and Computer
Science, Massachusetts Institute of Technology  Yonatan Israel  Adam J. Bowman
Brannon B. Klopfer  Physics Department, Stanford University, Stanford,
California 94305, USA  Mark A. Kasevich  Authors to whom correspondence should
be addressed<EMAIL_ADDRESS>Physics Department, Stanford University,
Stanford, California 94305, USA

###### Abstract

A number of visions for a new generation of dose-efficient electron
microscopes have been advanced. These proposals, while inspired by quantum
principles, make little contact with the broader field of quantum metrology.
We discuss a framework calculating the amount of information carried by each
electron. This makes it possible to evaluate the potential effectiveness of
any particular microscope architecture relative to the quantum limit for
information per dose. In the case of phase imaging, we argue this limit is at
least an order of magnitude beyond what is possible with aberration-free
Zernike phase contrast.

††preprint: APS/123-QED

##  I Introduction

Transmission electron microscopes (TEMs) are versatile tools, essential to
many fields of science, from material science to structural biology. While
modern TEMs are capable of atomic resolution, a fundamental limitation
remains: many samples are degraded by high energy electrons. For example,
cryo-electron microscopy (cryo-EM) makes it possible to image biological
molecules in their (ostensibly) native configurations[1, 2, 3], but these
molecules are destroyed long before a high resolution image can be formed. It
is possible to form a near-atomic-resolution reconstruction of the molecule
using a technique called single particle analysis (SPA), whereby a large
number of low resolution images are collected and then combined using a
sophisticated algorithm. However this laborious procedure can only be
performed on large ensembles of identical particles heavier than about $40$
kDa[4]. Imaging of single protein instances can provide valuable insights into
structural biology beyond SPA [5, 6], but current techniques are constrained
by the dose limit of TEM. A typical dose for cryo-EM is tens of electrons per
square angstrom, and is often divided into multiple exposures. With such
limited measurement resources, it is important to extract as much information
as possible from each electron.

Quantum metrology is dedicated to pursuing the ultimate sensitivity of
measurements allowed by quantum mechanics. This is made possible by developing
formalism to calculate such limits and developing experimental means to
achieve them in practice. In this perspective we comment on research that lies
in the intersection of the two fields, whereby tools of quantum metrology are
used to achieve the ultimate limits allowed by quantum mechanics in electron
microscopy.

##  II Visions of a Quantum Electron Microscope

In the context of quantum metrology, a ‘quantum’ enhanced measurement is
generally one which uses a non-classical probe state to achieve lower error
per measurement resource. Arguably the most important result of quantum
metrology is that by using a probe of $n$ particles prepared in an entangled
state, it is possible to reduce measurement variance by a factor scaling like
$n^{2}$ (the Heisenberg Limit) rather than like $n$ (the Standard Quantum
Limit)[7, 8]. However, Heisenberg-limited measurements are only possible for
lossless system[9]. In damage-limited TEM, where loss is intrinsic, what is
the benefit of a quantum treatment? We will argue that there is a quantum
limit for information per damage which an ideal, traditional TEM architecture
cannot access. We will call any microscope which approaches this limit a
Quantum Electron Microscope (QEM).

Figure 1: (A) Elitzur-Vaidman bomb test with transparent (top) or opaque
(bottom) sample $S$. (B) Elitzur-Vaidman bomb test with weak, repeated
interrogations. This configuration is known as interaction-free measurement
(IFM). (C) Multi-pass (MP) interferometry. (D) MP microscopy. For simplicity,
the schematics in (B), (C), and (D) show multiple copies of the sample. In
practice these measurements can be performed with a single copy by forming the
optical axis into a loop. (E) Interferometer scheme compatible with
entanglement-enhanced fermionic probes. The Pauli-exclusion principle is
circumvented by using inter-mode correlations.

In optical microscopy, entanglement-enhanced measurements have been shown to
increase imaging sensitivity and reduce damage [10, 11, 12]. In principle,
fermionic probes can also be enhanced using entanglement (and indeed, achieve
the Heisenberg limit) by partitioning states with $n$ electrons into $k$ modes
(where $k>n$)[13]. There has been no experimental demonstration of
entanglement between free electrons, but free electrons have been entangled
with other systems[14, 15]. One proposal for entanglement-enhanced TEM
suggests using a flux qubit to coherently sum the phase shifts measured by a
train of single electron pulses[16].

There have also been proposals which invoke quantum principles other than
entanglement, such as interaction-free measurement[17] (IFM). The simplest
kind of IFM can be performed using a balanced interferometer as depicted in
Fig. 1(A). The goal is to determine whether an opaque object ($S$) has been
placed in one of the interferometer arms. When $S$ is absent, the
interferometer is configured to have complete constructive interference at one
output port ($D_{1}$) and destructive interference at the other ($D_{2}$).
When $S$ is present, there is a 50% chance that it will absorb each photon (a
failed IFM) and a 25% chance that the photon arrives at $D_{2}$. A single
photon detection at $D_{2}$ unequivocally establishes the presence of $S$
while also precluding interaction[18] with $S$. The success probability can be
increased arbitrarily close to unity using an asymmetric interferometer to
repeatedly (weakly) interrogate the sample[19] as shown in Fig 1(B). IFM can
be thought of in terms of the quantum Zeno effect[20, 21, 19], where the
obstructing object acts as a measuring device to freeze the coherent evolution
of the probe.

The first proposal for IFM in electron microscopy called for a scanning probe
of $100\text{\,}\mathrm{keV}$ electrons confined in a hybrid Paul-Penning
trap[22]. The scheme relied on sample staining (with heavy atoms) to produce
the high contrast necessary for efficient IFM, and predicted a reduction in
sample damage by two orders of magnitude. In light of subsequent analysis,
this seems overly optimistic[23]. Furthermore, the high-contrast requirement
excludes the most important applications for a QEM (e.g., imaging weak phase
objects like proteins in vitreous ice). A later proposal was based on a more
traditional TEM architecture with the addition of a coherent beam-splitter and
‘resonator’ (a re-imaging cavity with fast in/out-coupling optics) to allow
multiple passes through the sample[24]. It was also suggested that IFM could
be used to measure a phase shift, but the proposal did not include a
calculation of the efficiency of this technique relative to standard TEM.
Recently, IFM was demonstrated for electrons with an efficiency of 14% using
diffraction grating based beam-splitters [Fig. 1(A)] in a balanced
interferometer [25].

A well-known method for enhancing sensitivity in measurements of a static
phase shifts involves simply passing the probe through the sample $m$ times
before detection [see Fig. 1(C)]. Such sequential or multi-pass (MP)
measurements have been shown to be quantum-optimal[26, 27] and proposed as a
substitution for entanglement in interferometric measurements[28, 29]. The
idea can also be applied to full-field imaging, where it is known as MP
microscopy[30, 31]. In this technique [see Fig. 1(D)], a probe is re-imaged
$m$ times onto the specimen, thereby enhancing the phase shift accrued by a
factor of $m$. This results in an increase of the signal-to-noise ratio (SNR)
by a factor of $m$, or by a factor $\sqrt{m}$ at constant damage. Unlike IFM,
a MP microscope does not require a beam-splitter. Nevertheless, for weakly
scattering samples, there is a strong physical analogy between full-field MP
and interaction-free measurements. The specimen in a MP microscope
heuristically acts like a beam-splitter, scattering and combining the beam
with each pass. The result is an increase in the probability of detecting the
presence of a scatterer (through an elastic interaction) relative to the
probability of causing damage (through an inelastic interaction).

There have also been proposed solutions to a more specialized measurement
scenario where the goal is to identify the sample from a finite list of
structural hypotheses. The corresponding problem in quantum metrology is
called quantum state discrimination (QSD). In the simplest case, where the
position and orientation of the sample are known and losses are negligible, it
is possible to distinguish between two phase objects in a single shot using an
entangled probe[32] or a MP measurement with a single probe particle[33]. The
key to applying QSD to TEM is inventing optics for projecting into an optimal
measurement basis. When the position and orientation are known, a single phase
mask may be sufficient[34]. When the orientation is unknown, efficient
discrimination is still possible by using a sequence of diffractive elements
to measure in an orbital angular momentum basis [35]. An efficient strategy
for QSD when both the position and orientation are unknown has yet to be
proposed.

A number of projects are underway to turn some of the conceptual innovations
described in this section into proof-of-principle devices. How can we know
whether any of these schemes have the potential to saturate the quantum limit
for dose-limited information? In the next section, we map the problem of phase
imaging in TEM onto phase estimation in quantum metrology.

##  III Electron Microscopy as a Quantum Metrology Protocol

The paradigmatic approach to quantum metrology begins by describing the
unknown degrees of freedom in the sample using a parameter vector
$\Theta=[\theta_{0},\ldots,\theta_{n}]$. If the purpose of the measurement is
to image a completely general phase object, the parameter values could
represent the projected atomic potential $V(\mathbf{r})=\int
dzV(z,\mathbf{r})$ at each transverse coordinate $\mathbf{r}$ or the amplitude
of the Fourier coefficients of $V$ for each spatial frequency $\mathbf{k}$. It
is often possible to reduce the dimension of $\Theta$ using prior information
about the sample. For single particle analysis, $\Theta$ may describe the
species and coordinates of each atom in a model particle and the positions and
orientations of each (hypothetically identical) particle in the field of view.
When measuring summary material properties like a lattice constant, average
grain size, or vacancy populations, $\Theta$ may be scalar.

A metrological protocol for estimating $\Theta$ consists of four steps.

  1. 1.

Create a probe in (mixed) state $\rho$.

  2. 2.

Interact the probe with the sample represented by transmission function
$S_{\Theta}$ to produce output state $\rho_{\Theta}=S_{\Theta}(\rho)$.

  3. 3.

Project the output state onto a set of $d$ detectors using transfer function
$T$ to produce intensities $I_{j}=\left\langle
j\right|T(\rho_{\Theta})\left|j\right\rangle$.

  4. 4.

Analyze $I$ using a set of estimating functions (estimators) $\hat{{\Theta}}$.

The standard way to formally optimize this protocol is by minimizing a cost
function

$C=\mathbf{w}\cdot\text{Var}(\hat{{\Theta}})=\sum_{i}w_{i}\text{Var}(\hat{\theta}_{i})\
,$ (1)

where the weights $w_{i}$ define the relative value of information about each
parameter. We may use a non-uniform weighting, for example, to prioritize the
high spatial-frequency information which tends to degrade more quickly as a
result of radiation damage.

The information $\mathcal{I}_{i}$ contained in $I$ about $\theta_{i}$ is
related to the minimum measurement variance achieved using an optimal (and
unbiased) estimator. Similarly, the information $\mathcal{J}_{i}$ available in
$\rho_{\Theta}$ about $\theta_{i}$ is related to the maximum $\mathcal{I}_{i}$
achieved by using an optimal transfer function. The quantities
$\mathcal{I}_{i}$ and $\mathcal{J}_{i}$ are called the Fisher Information (FI)
and Quantum Fisher Information (QFI)[36, 37, 38] respectively. They are
related to the measurement variance by the Cramér-Rao Bound (CRB) and the
Quantum CRB (QCRB)

$\text{Var}(\hat{\theta}_{i})\underset{\text{CRB}}{\geq}\frac{1}{N\mathcal{I}_{i}(\Theta,\rho,T)}\underset{\text{QCRB}}{\geq}\frac{1}{N\mathcal{J}_{i}(\Theta,\rho)}\
,$ (2)

where $N$ is the number of independent measurements. It is worth noting that
the FI and QFI have units of inverse variance rather than bits. However, like
the more traditional notion of information (i.e., entropy) and unlike figures
of merit such as contrast and resolution, the FI and QFI from independent
measurements are additive. The utility of this formalism comes from explicit
formulae for the FI and QFI, which establish optimization benchmarks for
$\hat{\Theta}$ and $T$, respectively.

Given a model for $I$ as a function of $\Theta$, the FI is

$\mathcal{I}_{i}=\sum_{j=1}^{d}\frac{1}{I_{j}}\left(\frac{\partial}{\partial\theta_{i}}I_{j}\right)^{2}$
(3)

and in the special case that the probe is a pure state $\rho_{\Theta}=\psi$,
the QFI is

$\mathcal{J}_{i}=4\left\langle\left.\frac{\partial}{\partial\theta_{i}}\psi\right|\frac{\partial}{\partial\theta_{i}}\psi\right\rangle-4\left|\left\langle\left.\frac{\partial}{\partial\theta_{i}}\psi\right|\psi\right\rangle\right|^{2}\
.$ (4)

As an example, consider a sample with transmission function
$S_{\Theta}(\rho)=\exp\left(i\phi(\mathbf{r})\right)\left|\mathbf{r}\right\rangle\left\langle\mathbf{r}\right|\rho$
where $\phi(\mathbf{r})=aV(\mathbf{r})$, $a$ is an interaction constant
depending on the beam energy (derivable from the WKB approximation), and
$V(\mathbf{r})=\sum_{\mathbf{k}}\theta_{\mathbf{k}}\cos(\mathbf{r}\cdot\mathbf{k})$.
Suppose we use a pure, plane wave probe and the projection optics are
described by transfer function
$T(\rho)=\mathcal{F}^{-1}\left\\{\exp\left(i\chi(|\mathbf{k}|)\right)\mathcal{F}\\{\rho\\}\right\\}$
which implements a radially-symmetric phase mask in the back focal plane of
the objective lens. In the weak phase object approximation (WPOA,
$\phi(\mathbf{r})\ll 1$)

$\mathcal{I}_{i}=4a^{2}\sin^{2}(\chi(|\mathbf{k}_{i}|)-\chi(0))=4a^{2}C(|\mathbf{k}_{i}|)^{2}\
,$ (5)

where $C(|\mathbf{k}|)$ is the phase contrast transfer function (CTF) as it is
typically defined for TEM[39]. The QFI can be calculated using equation 4 or
by optimizing $\chi$ by inspection. Regardless of the weighting $\mathbf{w}$,
the optimal transfer function has $\chi(|\mathbf{k}|)-\chi(0)=\pi/2$ for all
$k$, corresponding to Zernike phase contrast (ZPC) and
$\mathcal{J}_{i}=4a^{2}$.

When the WPOA cannot be applied, each spatial frequency component of $V$ may
generate contrast at many spatial frequencies at the detector (i.e., a strong
phase grating scatters to many angles). In this case, to account for possible
covariance between measurements of $\theta_{i}$ and $\theta_{j}$, the FI and
QFI can be formulated as matrices $\mathcal{I}_{i,j}$ and $\mathcal{J}_{i,j}$.
Then CRB and QCRB provide lower bounds for the covariance matrix
$\Sigma(\hat{\Theta})$. In general, the optimal transfer function depends on
$\mathbf{w}$ and the particular value of $\Theta$. For example, a Zernike
phase plate produces negligible contrast when the majority of the probe is
scattered.

While ZPC produces an apparently ideal CTF in the WPOA, more information is
attainable using more general measurements. If the probe is passed through the
sample $m$ times or includes $n$ maximally entangled particles, the QFI can
improve by a factor as large as $m^{2}$ or $n^{2}$ (or both can be done at
once)[27]. In principle, the information per probe particle is unbounded,
however the initial uncertainty in the sample phase is also amplified,
eventually causing $2\pi$ ambiguities. In practice, $m$ and $n$ should be
initially chosen to avoid this, and can be increased as the variance of
intermediate estimates of the phase decreases.

So far our model has not included loss and damage. As we will show, the total
loss will generally affect the optimal measurement parameters (e.g., $m$ or
$n$). Damage, which is associated with inelastic scattering in the sample,
will limit the total number of probes $N$. Some inelastic interactions
preserve partial spatial coherence in the probe (e.g., plasmon scattering)[39,
40, 41, 42] so they retain some value for phase estimation. However, for
highly localized interactions (e.g., chemical bond breaking), the sample
presumably receives enough which-way information to destroy all interference
at the detector. For simplicity, we will assume that inelastically scattered
electrons contain no information about the sample phase and can be filtered
from the imaging beam before they arrive at the detector or return to the
sample (in a MP measurement). With this assumption, inelastic scattering in
electron microscopy is similar to absorption in photon microscopy.

For measurements of a single phase shift with transmissivity $\eta$, MP and
NOON state measurements are optimal (producing the maximum QFI) at fixed
$n\times m$ when $\eta$ is high[43]. However, with damage as the measurement
resource, MP and NOON states are sub-optimal. It has been shown that for the
measurement of a lossy phase shift using photons, the QFI per absorbed photon
is bounded by[44]

$\xi\leq\xi_{\text{QL}}=\frac{4\eta}{(1-\eta)^{2}}\ .$ (6)

The calculation of $\xi_{\text{QL}}$ does not rely on bosonic particle
statistics[9], so it also bounds measurements performed with electrons (note
the bound applying to an estimate of $V=\phi/a^{2}$ is scaled by $a^{2}$). The
bound can be saturated by squeezed states[44] (which can be formed with
fermionic probes using inter-mode entanglement[45]). Due to the great
technical challenge of forming such states using free electrons, it is worth
considering whether other measurements can approach this bound.

MP and NOON state measurements produce the same amount of information when
$n=m$, however they inflict a different dose on the sample. If all loss is due
to inelastic scattering in the sample, the dose is $1-\eta^{m}$ for a MP
measurement and $n(1-\eta)$ for a NOON state measurement. When $\eta\sim 1$,
$\xi_{\text{NOON}}\lesssim\xi_{\text{QL}}/e=0.38\xi_{\text{QL}}$ for optimal
$n$ and $\xi_{\text{MP}}\lesssim 0.65\xi_{\text{QL}}$ for optimal $m$. An IFM
architecture can be be used to perform a still more efficient measurement.
While the standard scheme uses a beam-splitter with transmissivity
$\tau=\sin(\pi/2m)^{2}$, we have found the dose efficiency is improved by
setting $\tau\ll(\pi/2m)^{2}$, in which case
$\xi_{\text{IFM}}\gtrsim.9\xi_{\text{QL}}$. By changing $\tau$ each pass
according to

$\tau_{k}=\epsilon\eta^{m-k}$ (7)

with $\epsilon\ll 1$, IFM can saturate the quantum limit. Fig. 2 shows
$\xi/\xi_{\text{QL}}$ for each of these measurements as a function of $\eta$.

Figure 2: Fisher information per dose, $\xi$, relative to the quantum limit $\xi_{\text{QL}}$ for estimation of a single lossy phase shift. At high transmissivity $\eta$, a single-pass measurement with a classical probe (SP) attains only a small fraction of the information available from more general measurements. Multi-pass (MP) and NOON state measurements offer a significant improvement when the number of passes ($m$) or probe particles ($n$) is optimized. The quantum limit can be saturated using a squeezed gaussian probe (SQZ) or using a sequence of weak interactions with a single-particle probe (IFM). | Single Pass | NOON | Multi-Pass | Squeezed
---|---|---|---|---
$\mathcal{J}$ | $4\eta a^{2}$ | $4\eta n^{2}a^{2}$ | $4\eta m^{2}a^{2}$ | $\dfrac{4\eta\alpha^{2}a^{2}}{1-\eta+\eta 10^{-S/10}}$
dose | $1-\eta$ | $n(1-\eta)$ | $1-\eta^{m}$ | $\alpha^{2}(1-\eta)$
$\xi(\eta=0.9)$ | 36 | 3.9$\xi_{\text{SP}}$ | 6.4$\xi_{\text{SP}}$ | $\leq 10\xi_{\text{SP}}$
$\xi(\eta=0.95)$ | 76 | 7.5$\xi_{\text{SP}}$ | 13$\xi_{\text{SP}}$ | $\leq 20\xi_{\text{SP}}$
$r(\eta=0.9)$ | $20\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ | $10\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ | $7.9\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ |  $\geq$$6.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
$r(\eta=0.95)$ | $20\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ | $7.7\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ | $5.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ |  $\geq$$4.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
Table 1: Formulae for the QFI, $\mathcal{J}$, and dose for various
measurements of a single phase shift. The NOON state measurement uses $n$
particles, the multi-pass measurement uses $m$ passes, and the squeezed state
has squeezing $S$ (in dB) and expects $a^{2}$ particles to pass through the
sample. Numerical values for the QFI per dose, $\xi$, and a typical[46] dose-
limited resolution $r$ are provided for $\eta=0.9$ and $\eta=0.95$.

In order to ground this discussion in quantities more familiar to electron
microscopy, we can write $\xi$ in terms of elastic and inelastic scattering
cross sections. We will focus on MP measurements, which would likely be the
simplest to implement. Using the WPOA, the probability of elastic scattering
is $tn_{s}\sigma_{e}^{(m)}=m^{2}a^{2}\left\langle\Delta V^{2}\right\rangle$
where $\left\langle\Delta V^{2}\right\rangle$ is the spatial variance in
atomic potential, $t$ is the sample thickness, and $\sigma_{e}^{(m)}$ is the
effective $m$-pass elastic scattering cross section for scatterers with number
density $n_{s}$. The probability of inelastic scattering is given by
$tn_{s}\sigma_{i}^{(m)}=1-\eta^{m}$, so we have

$\xi_{\text{MP}}=\frac{4}{\left\langle\Delta V^{2}\right\rangle}\zeta^{(m)}\
,$ (8)

where

$\zeta^{(m)}=\eta^{m}\frac{\sigma_{e}^{(m)}}{\sigma_{i}^{(m)}}$ (9)

is the MP generalization of the information coefficient introduced by Peet et
al [47]. It is a convenient figure of merit because it depends only on
material properties rather than the particular configuration of the sample. It
also has a straightforward interpretation: when $\zeta^{(m)}>1$, each
measurement has a greater chance of detecting a scatterer (probability
$\eta^{m}\sigma_{e}^{(m)}$) than damaging it (probability $\sigma_{i}^{(m)}$).
If we imagine the scatterer acts like an Elitzer-Vaidman bomb[48] and is
destroyed by a single inelastic event, then, assuming post-selection is
possible (by registering inelastically scattered electrons), the probability
of a counterfactual (guaranteed damage-free) detection is

$P_{\text{CF}}=\frac{\zeta^{(m)}}{1+\zeta^{(m)}}\ .$ (10)

In Fig. 3, we plot various quantities associated with the multi-pass
information coefficient. Fig. 3 (A) shows the effective elastic cross section
$\sigma_{e}^{(m)}$ and effective inelastic cross section $\sigma_{i}^{(m)}$
vs. beam energy for amorphous carbon. We calculate $\sigma_{e}^{(1)}$ using a
theoretical relationship proposed by Lamgmore and Smith[49]. Values for
$\sigma_{i}^{(1)}$ are taken from the Estar database[50]. In (B), we calculate
$\zeta^{(m)}$ vs. beam energy for a sample $10\text{\,}\mathrm{nm}$ thick. In
(C) we show the energy which maximizes $\zeta^{(m)}$. The optimal energy
$E\sim$830\text{\,}\mathrm{keV}$$ seems to be mostly independent of $t$.
Finally, (D) shows the probability of a successful counterfactual detection of
a scatterer. To obtain $P_{\text{CF}}>$0.75(0.9) using a MP measurement, the
sample thickness must be $t<$$22(7)\text{\,}\mathrm{nm}$. Using a quantum-
limited measurement, the requirement is $t<$$33(12)\text{\,}\mathrm{nm}$.

Figure 3: (A) Elastic (solid) and inelastic (dashed) scattering cross sections
vs. beam energy for $m$ passes through a thin ($\eta\sim 1$) sample. Notice
$\sigma_{e}^{(m)}\gtrsim\sigma_{i}^{(m)}$ for $m\gtrsim 4$. (B) Information
coefficient, $\zeta^{(m)}$, vs. beam energy for various $m$ assuming a sample
thickness of $t=$10\text{\,}\mathrm{nm}$$. The optimal electron energy is
higher for larger $m$. (C) Beam energy which maximizes the information
coefficient as a function of sample thickness. Regardless of the sample
thickness, the optimal energy (when jointly optimized with $m$) is near
$830\text{\,}\mathrm{keV}$. (D) Probability of a successful counterfactual
detection of a scatterer at the optimal beam energy for various sample
thicknesses. The dotted line represents the quantum limit.

We can also use a MP measurement to increase the dose efficiency of phase
object discrimination. An optimal single pass measurement for discriminating
between output wavefunctions $\psi_{A}$ and $\psi_{B}$ has success probability
equal to the Helstrom Bound[51]

$P_{s}=\frac{1}{2}+\frac{1}{2}\sqrt{1-4P_{A}P_{B}\left|\left\langle\left.\psi_{A}\right|\psi_{B}\right\rangle\right|^{2}}\
,$ (11)

where $P_{A}$ and $P_{B}$ are the a priori probabilities of $A$ and $B$.
Taking into account the finite transmissivity of the sample, the MP
generalization Helstrom Bound is

$\displaystyle P_{s}^{(m)}$
$\displaystyle=\sum_{N^{\prime}=0}^{N}\eta^{mN^{\prime}}(1-\eta^{m})^{N-N^{\prime}}{N\choose
N^{\prime}}P_{s}^{(m)}(N^{\prime})$ $\displaystyle P_{s}^{(m)}(N^{\prime})$
$\displaystyle=\frac{1}{2}+\frac{1}{2}\sqrt{1-4P_{A}P_{B}\left|\left\langle\left.\psi_{A}^{(m)}{}^{\otimes
N^{\prime}}\right|\psi_{B}^{(m)}{}^{\otimes
N^{\prime}}\right\rangle\right|^{2}}$
$\displaystyle=\frac{1}{2}+\frac{1}{2}\sqrt{1-4P_{A}P_{B}\cos^{2N^{\prime}}(\beta^{(m)})}\
.$

As an example, suppose we are using a Mach-Zehnder interferometer to
categorize an atom as being either species $A$ or $B$ (with prior
probabilities $P_{A}=P_{B}=0.5$). We’ll assume the difference in the phase
shifts caused by the two species is $\Delta\phi=0.05$ and also that
$\eta=0.98$. Then we can achieve $P_{\text{success}}\gtrsim 0.9$ using either
$N=418$ single pass measurements or $N=3$ electrons passed through the sample
$m=14$ times each. The single pass method causes, on average, 8.4 inelastic
interactions while the MP method causes 0.7 inelastic interactions.

##  IV How good can QEM become?

In his 2016 commentary[52] ‘How good can cryo-EM become?’, Glaeser discusses
improvements in sample preparation, cameras, and phase contrast techniques
that could bring cryo-EM performance to the estimated theoretical limit for a
traditional microscope column. He also wondered whether the various proposals
for QEMs, which at the time made vague promises about their potential
advantage, could save “true single-particle electron microscopy”. Based on our
analysis, quantum-limited cryo-EM would yield 10-20 times more information per
dose, or 3-4 times better resolution. This would be a significant advancement,
but still insufficient for damage free imaging of a single molecule at atomic
resolution. However we have made a number of assumptions which could hide the
potential for further improvement.

We have not yet emphasized the potentially significant differences between
measuring a single phase shift and measuring many simultaneously. In a MP
measurement, a transformation could be applied between each pass, modulating
the intensity of the beam in a way which could depend on previous interactions
with the sample. This ‘incremental phase contrast’ could be especially
important in an application like cryo-EM, where the allowable dose can vary
throughout the field of view. The structure of the vitreous ice background,
for example, need not be preserved. Additionally, the intensity pattern of the
initial probe could also be adaptively adjusted between each measurement. This
would be similar to adaptively adjusting the dwell time in scanning TEM, which
is estimated[53] to reduce the required dose by a factor of 10.

Another assumption worth examining is that inelastically-scattered electrons
carry no useful information. As discussed in the previous section, post-
selection by detection of inelastically-scattered electrons could reveal when
and where the sample has been damaged. This would make it possible to expose a
sample region of size $R$ precisely until it has been damaged. Taking
advantage of this information would require a detailed model of the possible
damage processes to determine an appropriate size for $R$.

As an example, suppose it takes 10 counter-factual detections to identify a
region. Using the data from Fig 3, a $20\text{\,}\mathrm{nm}$-thick sample has
a $P_{\text{CF}}=0.72$(0.8) for optimal MP(IFM) and $P_{\text{CF}}=0.23$ for a
single pass. Then the chance of a non-destructive MP(IFM) identification is
$(P_{\text{CF}})^{10}=4\%(10\%)$ compared to $4\times 10^{-5}\%$ for a single
pass.

In addition to acting as heralds of damage, inelastically-scattered electrons
could provide extra image contrast. While they may be unable to directly
contribute to the phase contrast image, it is possible they could be filtered
out and measured with a different protocol. For example, they could contribute
to a darkfield image, perhaps after being deflected to a second detector.
Assuming mean free paths
$\lambda_{\text{ice}}\approx$300\text{\,}\mathrm{nm}$$ and
$\lambda_{\text{protein}}\approx$150\text{\,}\mathrm{nm}$$ at beam energy
$300\text{\,}\mathrm{keV}$[54], the scattering contrast for a
$10\text{\,}\mathrm{nm}$ thick protein is less than 2% for 1 pass and 26% for
16 passes. In addition, when the electron source is subject to Poisson noise
(contrary to our assumption in section III), detection of inelastic electrons
could mitigate the extra error[55].

Many of these more speculative techniques would require advances in both
hardware and theory. Perhaps in the distant future electron microscopes will
use deterministic sources of arbitrary multi-electron quantum states. However,
as we have discussed, we can mostly saturate the quantum limit with a
classical probe. In order to do that, we would need fast-switching electron
optics in order to build a MP TEM column. A number of designs have been
proposed for electron resonators using gated electron mirrors [56, 57, 58].
Another critical electron optic is a spatial phase modulator, variations of
which are currently being demonstrated[59, 60].

Compared to photon microscopy, which is subject to an endless proliferation of
design, innovation in electron microscopy is slow. However, the perspective
afforded by quantum metrology could motivate more ambitious microscope
schemes. There may also be a reciprocal effect on the field of quantum
metrology, which could discover a rich new space of applications where damage
(rather than probe size) is the limiting resource.

##  V Acknowledgments

This work was done as part of the Quantum Electron Microscope collaboration
funded by the Gordon and Betty Moore foundation. A.J.B. acknowledges support
from the Stanford Graduate Fellowship and from the National Science Foundation
Graduate Research Fellowship Program under Grant No. 1656518.

## References

  * [1] Jacques Dubochet, Marc Adrian, Jiin-Ju Chang, Jean-Claude Homo, Jean Lepault, Alasdair W McDowall, and Patrick Schultz.  Cryo-electron microscopy of vitrified specimens.  Quarterly Reviews of Biophysics, 21(2):129–228, 1988.
  * [2] Joachim Frank.  Three-dimensional electron microscopy of macromolecular assemblies: visualization of biological molecules in their native state.  Oxford university press, 2006.
  * [3] Xiao-Chen Bai, Greg McMullan, and Sjors HW Scheres.  How cryo-EM is revolutionizing structural biology.  Trends in Biochemical Sciences, 40(1):49–57, 2015.
  * [4] Richard Henderson.  The potential and limitations of neutrons, electrons and x-rays for atomic resolution microscopy of unstained biological molecules.  Quarterly Reviews of Biophysics, 28(2):171–193, 1995.
  * [5] J. P. Rickgauer, N. Grigorieff, and W. Denk.  Single-protein detection in crowded molecular environments in cryo-EM images.  eLife, 6:e25648, 2017.
  * [6] J. P. Rickgauer, H. Choi, J. Lippincott-Schwartz, and W. Denk.  Label-free single-instance protein detection in vitrified cells.  BioRxiv, 2020.
  * [7] Hwang Lee, Pieter Kok, and Jonathan P Dowling.  A quantum rosetta stone for interferometry.  Journal of Modern Optics, 49(14-15):2325–2338, 2002.
  * [8] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone.  Quantum-enhanced measurements: beating the standard quantum limit.  Science, 306(5700):1330–1336, 2004.
  * [9] B. Escher, R. Filho, and Liema Davidovich.  General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology.  Nature Physics, 7, 2012.
  * [10] Peter C Humphreys, Marco Barbieri, Animesh Datta, and Ian A Walmsley.  Quantum enhanced multiple phase estimation.  Physical Review Letters, 111(7):070403, 2013.
  * [11] Yonatan Israel, Shamir Rosen, and Yaron Silberberg.  Supersensitive polarization microscopy using NOON states of light.  Physical Review Letters, 112(10):103604, 2014.
  * [12] Catxere A. Casacio, L. Madsen, Alex Terrasson, M. Waleed, Kai Barnscheidt, B. Hage, Michael Taylor, and W. Bowen.  Quantum-enhanced nonlinear microscopy.  Nature, 594 7862:201–206, 2021.
  * [13] Fabio Benatti, Roberto Floreanini, and Ugo Marzolino.  Entanglement in fermion systems and quantum metrology.  Physical Review A, 89(3):032326, 2014.
  * [14] Zhexin Zhao, Xiao-Qi Sun, and Shanhui Fan.  Quantum entanglement and modulation enhancement of free-electron–bound-electron interaction.  Physical Review Letters, 126:233402, 2021.
  * [15] A. Hayun, O. Reinhardt, Jonathan Nemirovsky, Aviv Karnieli, N. Rivera, and I. Kaminer.  Shaping quantum photonic states using free electrons.  Science Advances, 7:eabe4270, 2021.
  * [16] Hiroshi Okamoto and Yukinori Nagatani.  Entanglement-assisted electron microscopy based on a flux qubit.  Applied Physics Letters, 104, 2013.
  * [17] Avshalom C Elitzur and Lev Vaidman.  Quantum mechanical interaction-free measurements.  Foundations of Physics, 23(7):987–997, 1993.
  * [18] Lev Vaidman.  The meaning of the interaction-free measurements.  Foundations of Physics, 33, 2001.
  * [19] Paul Kwiat, Harald Weinfurter, Thomas Herzog, Anton Zeilinger, and Mark A Kasevich.  Interaction-free measurement.  Physical Review Letters, 74(24):4763, 1995.
  * [20] Baidyanath Misra and EC George Sudarshan.  The Zeno’s paradox in quantum theory.  Journal of Mathematical Physics, 18(4):756–763, 1977.
  * [21] Wayne M Itano, Daniel J Heinzen, John J Bollinger, and David J Wineland.  Quantum Zeno effect.  Physical Review A, 41(5):2295, 1990.
  * [22] William P. Putnam and Mehmet Fatih Yanik.  Noninvasive electron microscopy with interaction-free quantum measurements.  Physical Review A, 80:040902, 2009.
  * [23] Sebastian Thomas, C. Kohstall, P. Kruit, and P. Hommelhoff.  Semitransparency in interaction-free measurements.  Physical Review A, 90:053840, 2014.
  * [24] P. Kruit, R. G. Hobbs, C-S. Kim, Y. Yang, V. R. Manfrinato, J. Hammer, S. Thomas, P. Weber, B. Klopfer, C. Kohstall, T. Juffmann, M. A. Kasevich, P. Hommelhoff, and K. K. Berggren.  Designs for a quantum electron microscope.  Ultramicroscopy, 164:31–45, 2016.
  * [25] Amy E. Turner, Cameron W. Johnson, Pieter Kruit, and Benjamin J. McMorran.  Interaction-free measurement with electrons.  Physical Review Letters, 127:110401, 2021.
  * [26] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone.  Quantum metrology.  Physical Review Letters, 96:010401, 2006.
  * [27] V. Giovannetti, S. Lloyd, and L. Maccone.  Advances in quantum metrology.  Nature Photonics, 5:222–229, 2011.
  * [28] Alfredo Luis.  Phase-shift amplification for precision measurements without nonclassical states.  Physical Review A, 65:025802, 2002.
  * [29] B Higgins, D Berry, S Bartlett, Howard Wiseman, and G Pryde.  Entanglement-free heisenberg-limited phase estimation.  Nature, 450:393–6, 2007.
  * [30] Thomas Juffmann, Brannon B Klopfer, Timmo LI Frankort, Philipp Haslinger, and Mark A Kasevich.  Multi-pass microscopy.  Nature Communications, (1):1–5, 2016.
  * [31] Thomas Juffmann, Stewart A Koppell, Brannon B Klopfer, Colin Ophus, Robert M Glaeser, and Mark A Kasevich.  Multi-pass transmission electron microscopy.  Scientific Reports, 7(1):1–7, 2017.
  * [32] A. Acín.  Statistical distinguishability between unitary operations.  Physical Review Letters, 87:177901, 2001.
  * [33] R. Duan, Y. Feng, and M. Ying.  Entanglement is not necessary for perfect discrimination between unitary operations.  Physical Review Letters, 98 10:100503, 2007.
  * [34] Hiroshi Okamoto, Tatiana Latychevskaia, and Hans-Werner Fink.  A quantum mechanical scheme to reduce radiation damage in electron microscopy.  Applied Physics Letters, 88(16):164103, 2006.
  * [35] F. Troiani, E. Rotunno, S. Frabboni, R. B. G. Ravelli, P. J. Peters, E. Karimi, and V. Grillo.  Efficient molecule discrimination in electron microscopy through an optimized orbital angular momentum sorter.  Physical Review A, 102:043510, 2020.
  * [36] Holevo A S.  Probabilistic and Statistical Aspects of Quantum Theory.  Amsterdam: North- Holland, 1982.
  * [37] C W Helstrom.  Minimum mean-squared error of estimates in quantum statistics.  Physics Letters A, 25(2):101–102, 1967.
  * [38] Samuel L. Braunstein and Carlton M. Caves.  Statistical distance and the geometry of quantum states.  Physical Review Letters, 72:3439–3443, 1994.
  * [39] L. Reimer and H. Kohl.  Transmission Electron Microscopy.  Springer Science, 2008.
  * [40] Peter Schattschneider and Wolfgang SM Werner.  Coherence in electron energy loss spectrometry.  Journal of electron spectroscopy and related phenomena, 143(2-3):81–95, 2005.
  * [41] RA Herring.  Energy-filtered electron-diffracted beam holography.  Ultramicroscopy, 104(3-4):261–270, 2005.
  * [42] RF Egerton.  Scattering delocalization and radiation damage in STEM-EELS.  Ultramicroscopy, 180:115–124, 2017.
  * [43] Rafał Demkowicz-Dobrzański, Uwe Dorner, B. J. Smith, Jeff S Lundeen, Wojciech Wasilewski, Konrad Banaszek, and Ian A. Walmsley.  Quantum phase estimation with lossy interferometers.  Physical Review A, 80:013825, 2009.
  * [44] Patrick M. Birchall, Jeremy L. O’Brien, Jonathan C. F. Matthews, and Hugo Cable.  Quantum-classical boundary for precision optical phase estimation.  Physical Review A, 96:062109, 2017.
  * [45] K. Svozil.  Squeezed fermion states.  Physical Review Letters, 65:3341–3343, 1990.
  * [46] Robert Glaeser and Richard Hall.  Reaching the information limit in cryo-EM of biological macromolecules: Experimental aspects.  Biophysical Journal, 100:2331–7, 2011.
  * [47] M. Peet, R. Henderson, and C. J. Russo.  The energy dependence of contrast and damage in electron cryomicroscopy of biological molecules.  Ultramicroscopy, 203:125 – 131, 2019.
  * [48] A.C. Elitzur and L. Vaidman.  Quantum mechanical interaction-free measurements.  Foundations of Physics, 23:987–997, 1993.
  * [49] John P Langmore and Michael F Smith.  Quantitative energy-filtered electron microscopy of biological molecules in ice.  Ultramicroscopy, 46(1-4):349–373, 1992.
  * [50] M.J. Berger, J.S. Coursey, M.A. Zucker, and J. Chang.  Stopping-power and range tables for electrons, protons, and helium ions, July 2017.
  * [51] Carl W. Helstrom.  Quantum detection and estimation theory.  Academic Press New York, 1976.
  * [52] Robert Glaeser.  How good can cryo-EM become?  Nature methods, 13:28–32, 2015.
  * [53] Tim Dahmen, Michael Engstler, Christoph Pauly, Patrick Trampert, Niels Jonge, Frank Mücklich, and Philipp Slusallek.  Feature adaptive sampling for scanning electron microscopy.  Scientific Reports, 6:25350, 2016.
  * [54] Miloš Vulović, Raimond B.G. Ravelli, Lucas J. van Vliet, Abraham J. Koster, Ivan Lazić, Uwe Lücken, Hans Rullgård, Ozan Öktem, and Bernd Rieger.  Image formation modeling in cryo-electron microscopy.  Journal of Structural Biology, 183(1):19–32, 2013.
  * [55] Akshay Agarwal, Karl K. Berggren, Yuri J. van Staaden, and Vivek K. Goyal.  Reduced damage in electron microscopy by using interaction-free measurement and conditional reillumination.  Physical Review A, 99:063809, 2019.
  * [56] Stewart A Koppell, Marian Mankos, Adam J Bowman, Yonatan Israel, Thomas Juffmann, Brannon B Klopfer, and Mark A Kasevich.  Design for a 10 kev multi-pass transmission electron microscope.  Ultramicroscopy, 207:112834, 2019.
  * [57] Marco Turchetti, Chung-Soo Kim, Richard Hobbs, Yujia Yang, Pieter Kruit, and Karl K. Berggren.  Design and simulation of a linear electron cavity for quantum electron microscopy.  Ultramicroscopy, 199:50–61, 2019.
  * [58] Brannon B Klopfer, Stewart A Koppell, Adam J Bowman, Yonatan Israel, and Mark A Kasevich.  Fast pulse shaping for a novel gated electron mirror.  Review of Scientific Instruments, 92(4):043705, 2021.
  * [59] Jo Verbeeck, Armand Béché, Knut Müller-Caspary, Giulio Guzzinati, Minh Luong, and Martien Hertog.  Demonstration of a 2 × 2 programmable phase plate for electrons.  Ultramicroscopy, 190, 2017.
  * [60] Osip Schwartz, Jeremy Axelrod, Sara Campbell, Carter Turnbaugh, Robert Glaeser, and Holger Müller.  Laser phase plate for transmission electron microscopy.  Nature Methods, 16:1016–1020, 2019.