# Electron excitation rate in dielectrics under an intense elliptically
polarized laser field

Prachi Venkat and Tomohito Otobe
venkat.prachi<EMAIL_ADDRESS>Kansai Photon Science
Institute Kansai Photon Science Institute National Institutes for Quantum
Science and Technology (QST) National Institutes for Quantum Science and
Technology (QST) Kizugawa Kizugawa Kyoto 619-0215 Kyoto 619-0215 Japan
Japan

###### Abstract

Electron excitation in dielectrics is studied for an elliptically polarized
laser field. As the first step, we develop an analytical formula for electron
excitation rate under elliptically polarized laser as the extension of our
previous work [T. Otobe et. al., JPSJ 88 (2019) 024709]. In the next step, we
calculate the excitation rate depending on the band structure by assuming
direction dependence of reduced mass. We find that although the ellipticity
decreases the excitation rate significantly in an isotropic system, the energy
oscillation of electrons due to the intra-band dynamics in the anisotropic
band structure increases the excitation rate with higher ellipticity. Our
results indicate that we can control the excitation rate in dielectrics by
varying the ellipticity, depending on the band anisotropy.

##  1 Introduction

Electron excitation in dielectrics by an intense laser field is the main
process in laser-matter interactions. Technical developments in femtosecond
laser processing have made it possible to produce a nano-scale laser-induced
periodic surface structure (LIPSS)[1, 2, 3, 4], and to realize non-thermal
ablation for sub-wavelength resolution. In particular, for femtosecond lasers,
electron excitation by multi-photon absorption and tunneling process is
crucial because such non-linear processes generate a controllable free-carrier
density and confine material change to the focal volume. Therefore, the
prediction of electron excitation rate using theoretical models and/or
numerical simulation is important. Non-linear carrier dynamics finds
application in various fields of laser-matter interaction such as HHG [5, 6],
transient transmission changes the carrier-envelope-phase-dependent electronic
currents [7].

Electron excitation can be carried out by various mechanisms such as multi-
photon absorption and tunneling, depending on the laser parameters. Various
models have been proposed over the years to model multi-photon processes and
tunneling of electrons numerically. The Keldysh theory describes electron
excitation in atoms and crystals [8, 9], and various mechanisms have been
developed based on the Keldysh approach [10, 11, 12, 13, 14, 15].

Incident laser intensity is a paramount factor in deciding whether the
interaction follows linear or non-linear dynamics. Similarly, the polarization
of the incident laser field significantly affects the carrier excitation
process and optical properties of a material. The time-resolved dynamic
Franz–Keldysh effect, which is responsible for the change in the dielectric
function of target material, depends significantly on the polarization of
incident field [16].

Change in the polarization of laser field also affects the dynamic phase of
electrons undergoing intra-band motion, ultimately affecting the excitation
rate during multi–photon absorption. For example, circularly polarized lasers
are also important as an ultrafast laser waveguides[17], and in controlling
laser–induced nonastructure [18]. M. Kozák et. al. have reported that the
ellipticity of the laser polarization decreases the excitation rate in diamond
[19]. On the other hand, V. V. Temnov et. al. have reported that the electron
excitation rate induced by a circularly polarized laser is twice that induced
by a linearly polarized laser at the same laser irradiance in fused silica and
sapphire [20]. We have to study the laser-matter interaction with elliptically
polarized laser in order to understand the above two conflicting experiments.

The electron excitation dynamics has been numerically studied, employing the
time-dependent density functional theory (TDDFT) [21, 22, 23] as well as
analytical models [24, 15]. Although the numerical simulation employing TDDFT
is one of the most accurate approaches, it needs a sizeable computational
resource, and analyzing the details of the physical process is sometimes
demanding. The analytical approach is not accurate like TDDFT, although we can
use it to study the physics of the interaction in detail.

In this paper, we adopt an analytical approach to obtain the electron
excitation rate for the interaction of isotropic dielectrics with an intense
elliptically polarized laser field. The formulation is based on the general
formula presented in the previous study [15]. The effect of ellipticity of
incident laser field on excitation rate is also studied. Furthermore, we study
the dominance of $n$-order photonic processes at different intensities for
elliptical polarization. We also extended the fundamental equation to the
anisotropic band by assuming the direction-dependent reduced mass. We
performed real-time simulations for carrier density after the laser pulse ends
and analyzed the effect of anisotropy of the system.

This paper is organized as follows: Section 2 describes the analytical
formulation of the excitation rate for elliptically polarized field with
isotropic and anisotropic band structures, section 3 presents results of
electron excitation in a diamond for different laser and material parameters,
and then finally, we would like to summarize the work in section 4.

##  2 Formulation

Starting with the Schrödinger equation which is assumed as,

$\epsilon^{G}_{n,\textbf{k}}u^{G}_{n,\textbf{k}}(\textbf{r})=\left(\frac{1}{2m}(\textbf{p}+\hbar\textbf{k})^{2}+V(\textbf{r})\right)u^{G}_{n,\textbf{k}}(\textbf{r})$
(1)

p being the momentum operator and k being the Bloch wave vector.
$V(\textbf{r})$ is the potential and $\epsilon^{G}_{n,\textbf{k}}$ and
$u^{G}_{n,\textbf{k}}(\textbf{r})$ are the energy and wave functions,
respectively, corresponding to the k and $n^{th}$ band. Both $V(\textbf{r})$
and $u^{G}_{n,\textbf{k}}(\textbf{r})$ are periodic in space. The time-
dependent Schrödinger equation is described as

$\left[\frac{1}{2m}\left(\textbf{p}+\hbar\textbf{k}+\frac{e}{\hbar
c}\textbf{A}(t)\right)^{2}+V(\textbf{r})\right]u_{n,\textbf{k}}(\textbf{r},t)=i\hbar\frac{\partial}{\partial
t}u_{n,\textbf{k}}(\textbf{r},t),$ (2)

where $\textbf{A}(t)$ is the vector potential for an elliptically polarized
field [25]:

$\vec{A}(t)=A_{0}(\eta\sin(\omega t),0,\cos(\omega t)).$ (3)

Here, $\eta$ is the parameter that decides the ellipticity of the field and is
$\eta=0(1)$ for linear(circular) polarization. Varying the $\eta$ between $0$
and $1$ will change the ellipticity of polarization.

The time-dependent wave function, $u_{n,\textbf{k}}(\textbf{r},t)$, can be
expressed by Houston function,

$w_{n,\textbf{k}}(\textbf{r},t)=u^{G}_{n,\textbf{k}+\frac{e}{c}\textbf{A}(t)}(\textbf{r})\exp\left[-i\int^{t}\epsilon^{G}_{n,\textbf{k}+\frac{e}{c}\textbf{A}(t^{\prime})}dt^{\prime}\right],$
(4)

as

$u_{n,\textbf{k}}(\textbf{r},t)=\sum_{n^{\prime}}C_{nn^{\prime}}^{\textbf{k}}(t)w_{n^{\prime},\textbf{k}}(\textbf{r},t).$
(5)

Here, $u^{G}_{n,\textbf{k}}(\textbf{r})$ is the wave function of ground state.
As presented in Ref.[15], we can obtain a simplified expression for the time-
evolution of the coefficient $C_{nn^{\prime}}^{\textbf{k}}(t)$ as follows:

$\begin{split}i\frac{\partial C^{\textbf{k}}_{nn^{\prime}}(t)}{\partial
t}=&i\frac{e}{mc}\frac{d\textbf{A}(t)}{dt}\sum_{n^{\prime\prime}}C^{\textbf{k}}_{nn^{\prime\prime}}(t)\left\langle
u^{G}_{n^{\prime},\textbf{k}+\frac{e}{c}\textbf{A}(t)}\,\middle|\,\frac{\partial
u^{G}_{n^{\prime\prime},\textbf{k}+\frac{e}{c}\textbf{A}(t)}}{\partial\textbf{k}}\right\rangle\\\
&\times\exp\left[-i\int^{t}dt^{\prime}\Delta\epsilon^{G}_{n"n^{\prime},\textbf{k}+\frac{e}{\hbar
c}\textbf{A}(t^{\prime})}\right].\end{split}$ (6)

The important quantity in Eq. (6) is the time integration of the relative
energy, $\Delta\epsilon^{G}_{n"n^{\prime},\textbf{k}+\frac{e}{\hbar
c}\textbf{A}(t)}=\epsilon^{G}_{n^{\prime\prime},\textbf{k}+\frac{e}{\hbar
c}\textbf{A}(t)}-\epsilon^{G}_{n^{\prime},\textbf{k}+\frac{e}{\hbar
c}\textbf{A}(t)}$, because it includes multi-photon absorption processes as
discussed in following sections.

###  2.1 Isotropic band

In the following discussion, we assume the isotropic parabolic two-band
system,

$\epsilon^{G}_{c,\textbf{k}}-\epsilon^{G}_{v,\textbf{k}}=\Delta\epsilon^{G}_{cv,\textbf{k}}=E_{g}+\frac{\hbar^{2}\textbf{k}^{2}}{2\mu},$
(7)

to simplify the equations and results. Here $E_{g}$ is the band gap, $\mu$ is
the reduced mass and the subscript $c$ ($v$) corresponds to conduction
(valence) band.

If we assume the transition dipole momentum to be isotropic, the transition
coefficient for this model is simplified to

$\begin{split}C^{\textbf{k}}_{vc}=&-\frac{ieP_{cv}A_{0}}{2\hbar
mc}\int_{-T}^{T}dt[M^{-}e^{i\omega t}+M^{+}e^{-i\omega t}]\\\
&\times\exp\left[\frac{i}{\hbar}\int^{t}\Delta\epsilon^{G}_{cv,\textbf{k}+\frac{e}{\hbar
c}\textbf{A}(t^{\prime})}dt^{\prime}\right]\end{split}$ (8)

where $M^{\pm}=1\pm i\eta$, and $P_{cv}$ is the absolute value of the
transition dipole momentum, it can be estimated as $P_{cv}\simeq
m\sqrt{E_{g}/4\mu})$ [26]. The exponential term in Eq. (8) can be expressed
as,

$\begin{split}&\exp\left[\frac{i}{\hbar}\int^{t}\Delta\epsilon_{cv,\textbf{k}+\frac{e}{\hbar
c}\textbf{A}(t^{\prime})}^{G}dt^{\prime}\right]\\\
&=\sum_{l,m,n}\exp\left[\frac{i}{\hbar}\left(E_{g}+\frac{\hbar^{2}k_{0}^{2}}{2\mu}+\frac{e^{2}A_{0}^{2}}{4\mu
c^{2}}(1+\eta^{2})\right)t\right]\\\ &\times
J_{l}(\alpha)J_{m}(\beta)J_{n}(\gamma)e^{i(2l+m+n)\omega
t}e^{im\frac{\pi}{2}}\end{split}$ (9)

where $\theta$ and $\phi$ are the angle of k from $z$\- and $x$-axis
respectively, and $J_{i}$ is the $i$-th order Bessel function with arguments:

$\alpha=(1-\eta^{2})\frac{e^{2}A_{0}^{2}}{8\hbar\mu\omega c^{2}}$ (10)
$\beta=\eta\frac{ek_{0}A_{0}\sin\theta\cos\phi}{\mu c\omega}$ (11)
$\gamma=\frac{ek_{0}A_{0}\sin\theta}{\mu c\omega}.$ (12)

The transition rate for every k is
$w_{\textbf{k}}=\lim_{T\to\infty}\frac{|C_{cv}^{\textbf{k}}|^{2}}{2T}$. The
total transition rate per unit time ($W$) is obtained by

$\displaystyle\begin{split}W=&\sum_{k}w_{\textbf{k}}=\int\frac{2k_{0}^{2}}{8\pi^{3}}dk_{0}\sin\theta
d\theta d\phi w_{\textbf{k}}\end{split}$ (13)

The final expression for the transition rate is

$\displaystyle\begin{split}W&=\frac{e^{2}P_{cv}^{2}A_{0}^{2}\mu^{3/2}}{4\sqrt{2}\pi^{2}\hbar^{4}m^{2}c^{2}}\sum_{L=l_{0}}^{\infty}\sum_{m_{1},m_{2}}\int
d\theta\sin\theta\int d\phi\\\
&(-1)^{m_{1}-m_{2}}\sqrt{\kappa_{L}}\Big{[}(1+\eta^{2})\Big{\\{}\\\
&J_{L-m_{1}-1}(\tilde{\gamma},\tilde{\alpha})J_{m_{1}}(\tilde{\beta})J_{L+m_{1}-2m_{2}-1}(\tilde{\gamma},\tilde{\alpha})J_{2m_{2}-m_{1}}(\tilde{\beta})\\\
&+J_{L-m_{1}+1}(\tilde{\gamma},\tilde{\alpha})J_{m_{1}}(\tilde{\beta})J_{L+m-2n+1}(\tilde{\gamma},\tilde{\alpha})J_{2m_{2}-m_{1}}(\tilde{\beta})\Big{\\}}\\\
+&2(1-\eta^{2})J_{L-m_{1}-1}(\tilde{\gamma},\tilde{\alpha})J_{m_{1}}(\tilde{\beta})\\\
&\times
J_{L+m_{1}-2m_{2}+1}(\tilde{\gamma},\tilde{\alpha})J_{2m_{2}-m_{1}}(\tilde{\beta})\Big{]},\end{split}$
(14)

where $J_{i}(a,b)$ is the $i$-th order of generalized Bessel function [27],
$L$ is the order of multi-photon absorption, $l_{0}$ is the lowest photon
process order to overcome the band gap with ponderomotive energy

$\kappa_{L}=L\hbar\omega-\left\\{E_{g}+(1+\eta^{2})\frac{e^{2}A_{0}^{2}}{4\mu
c^{2}}\right\\}\geq 0,$ (15)

and the new arguments for Bessel functions
($\tilde{\alpha},\tilde{\beta},\tilde{\gamma}$) are

$\tilde{\alpha}=(\eta^{2}-1)\frac{e^{2}A_{0}^{2}}{8\hbar\mu\omega c^{2}}$ (16)
$\tilde{\beta}=\eta\frac{e\sqrt{2\kappa_{L}}A_{0}\sin\theta\cos\phi}{\sqrt{\mu}c\omega}$
(17)
$\tilde{\gamma}=\frac{e\sqrt{2\kappa_{L}}A_{0}\sin\theta}{\sqrt{\mu}c\omega}.$
(18)

To derive Eq. (14), we impose symmetry in the integration about $\phi$.

####  2.1.1 linear and circular polarization limit

From the definition of the ellipticity $\eta$, we can reproduce the linear and
circular polarization cases as the limit of $\eta=0$ and 1 of Eq. (14). For
$\eta=0$, the excitation rate with linearly polarized laser is

$\displaystyle\begin{split}W=&\frac{e^{2}P_{cv}^{2}A_{0}^{2}\mu^{3/2}}{2\sqrt{2}\pi\hbar^{4}m^{2}c^{2}}\\\
\times&\sum_{L=l_{0}}^{\infty}\int
d\theta\sin\theta\sqrt{\kappa_{L}}\left(J_{L+1}(\tilde{\gamma},\tilde{\alpha})+J_{L-1}(\tilde{\gamma},\tilde{\alpha})\right)^{2}\end{split}$
(19)

with $\tilde{\alpha}=-e^{2}A_{0}^{2}/8\hbar\mu\omega c^{2}$ and
$\kappa_{L}=L\hbar\omega-\left(E_{g}+e^{2}A_{0}^{2}/4\mu c^{2}\right)$, while
$\eta=1$ gives the rate with circularly polarized laser as

$\displaystyle\begin{split}W=&\frac{e^{2}P_{cv}^{2}A_{0}^{2}\mu^{3/2}}{2\sqrt{2}\pi^{2}\hbar^{4}m^{2}c^{2}}\sum_{L=l_{0}}^{\infty}\sum_{m_{1},m_{2}}\int
d\theta\sin\theta\int d\phi(-1)^{m_{1}-m_{2}}\sqrt{\kappa_{L}}\\\
&\times\Big{\\{}J_{L-m_{1}-1}(\tilde{\gamma})J_{m_{1}}(\tilde{\beta})J_{L+m_{1}-2m_{2}-1}(\tilde{\gamma})J_{2m_{2}-m_{1}}(\tilde{\beta})\\\
&+J_{L-m_{1}+1}(\tilde{\gamma})J_{m_{1}}(\tilde{\beta})J_{L+m_{1}-2m_{2}+1}(\tilde{\gamma})J_{2m_{2}-m_{1}}(\tilde{\beta})\Big{\\}},\end{split}$
(20)

with $\tilde{\beta}=(ek_{0}A_{0}/\mu c\omega)\sin\theta\cos\phi$ and
$\kappa_{L}=L\hbar\omega-\left(E_{g}+e^{2}A_{0}^{2}/2\mu c^{2}\right)$. To
derive Eq. (20), we use the relation of generalized Bessel function
$J_{i}(a,0)=J_{i}(a)$. Although Eq. (19) is consistent with our previous
result [15], Eq. (20) is not because the definition of polarization direction
is different.

###  2.2 Band structure dependence

In general, the reduced mass $\mu$ around the band-edge has direction
dependence due to the band structure, $\mu\rightarrow\mu(\theta,\phi)$. Under
the vector potential $\textbf{A}(t)$, the momentum of electron in the Bloch
phase space is changed from
$\textbf{k}=k_{0}(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ to

$\displaystyle\begin{split}\textbf{k'}(t)&=(k_{0}\sin\theta\cos\phi+\frac{e}{\hbar
c}A_{0}\eta\sin\omega t,\\\ &k_{0}\sin\theta\sin\phi,\\\
&k_{0}\cos\theta+\frac{e}{\hbar c}A_{0}\cos\omega t)\\\
&\equiv(k^{\prime}_{0}\sin\theta^{\prime}\cos\phi^{\prime},k^{\prime}_{0}\sin\theta^{\prime}\sin\phi^{\prime},k^{\prime}_{0}\cos\theta^{\prime}).\end{split}$

We have a new time-dependent Bloch wavevector with,

$\displaystyle\begin{split}k^{\prime}_{0}(t)&=\Big{(}k_{0}^{2}+\frac{e^{2}}{\hbar^{2}c^{2}}A_{0}^{2}(\cos^{2}\omega
t+\eta^{2}\sin^{2}\omega t)\\\ &+2\frac{e}{\hbar
c}k_{0}A_{0}(\cos\theta\cos\omega t+\eta\sin\theta\cos\phi\sin\omega
t)\Big{)}^{1/2}\end{split}$ (21)
$\theta^{\prime}(t)=\arccos\left[\frac{k_{0}\cos\theta+\frac{e}{\hbar
c}A_{0}\cos\omega t}{k^{\prime}_{0}(t)}\right]$ (22)

, and

$\phi^{\prime}(t)=\arctan\left[\frac{k_{0}\sin\theta\sin\phi}{k_{0}\sin\theta\cos\phi+\frac{e}{\hbar
c}A_{0}\eta\sin\omega t}\right],$ (23)

we can calculate the energy phase including band structure as

$\begin{split}&\exp\Big{[}\frac{i}{\hbar}\int^{t}\Delta\epsilon^{G}_{cv,\textbf{k}+\frac{e}{\hbar
c}\textbf{A}(t^{\prime})}dt^{\prime}\Big{]}\\\
&=\exp\Big{[}\frac{i}{\hbar}\int^{t}\left(B_{g}+\frac{\hbar^{2}\textbf{k}^{\prime
2}_{0}}{2\mu(\theta^{\prime},\phi^{\prime})^{2}}\right)dt^{\prime}\Big{]}.\end{split}$
(24)

For the case of the simple $M$-fold symmetric system in $\theta$ and $\phi$,
it can be expressed as

$\mu(\theta,\phi)=\mu_{0}\left[1+\xi\left(\sin^{2}\frac{M}{2}\theta+\sin^{2}\frac{M}{2}\phi\right)\right].$
(25)

The relative energy and transition dipole moment have the $M$-fold symmetry
with the reduced mass in Eq. (25). By substituting the reduced mass to the
general formula of the transition coefficient Eq (6), we can get the
transition coefficient with band structure,

$C^{\textbf{k}}_{vc}(t)=-e\int^{t}dt^{\prime}\frac{\textbf{E}(t^{\prime})\cdot\textbf{P}_{\textbf{k'}(t^{\prime}),cv}}{E_{g}+\frac{\hbar\textbf{k'}(t^{\prime})^{2}}{2\mu(\theta^{\prime},\phi^{\prime})}}\exp\left[\frac{i}{\hbar}\int^{t^{\prime}}dt"\Delta\varepsilon^{G}_{cv,\textbf{k'}(t")}\right].$
(26)

##  3 Results

###  3.1 Isotropic band

Diamond is a typical dielectric in non-linear laser-matter interaction
studies, and we select it as an example to illustrate the application of the
preceding formalism. The transition probability of diamond by 800 nm (1.55 eV)
light is shown in Fig. 1. We assume a bandgap ($B_{g}$) of 7.4 eV and a
reduced mass of 0.5 $m$. The lowest order of the multi-photon absorption is
the 5-photon process.

The excitation rates with linear (solid red line) and circular polarization
(purple double-doted and dashed line) are same as the previous results[19].
The excitation rate decrease as $\eta$ increases, which is consistent with the
experimental results qualitatively.

Figure 1:  Laser irradiance dependence of the excitation rate with some
ellipticity ($\eta$).

The ratio of $W$ with respect to circular polarization is shown in Fig. 2. The
ratio of $W$ accesses to one up to $7\times 10^{13}$ W/cm2 for all $\eta$,
while the ratio becomes less than one above $7\times 10^{13}$ W/cm2. To
clarify the excitation process, we plot the Keldysh parameter ($\gamma_{k}$)
in Fig. 2 as a solid black line. In general, the tunnelling process is
dominant at $\gamma_{k}\ll 1.0$. The reversing relationship in the ratio
occurs at $\gamma_{k}<0.5$, which suggests that the ellipticity dependence is
different for the multi-photon and tunneling excitation processes.

Figure 2:  Relative excitation rate with respect to circular polarization.

Although linear polarization shows clear 5-photon absorption below $1\times
10^{12}$ W/cm-2, the irradiance dependence of elliptical polarization does
not. Figure 3 shows 5–photon absorption (5PA) and 6–photon absorption (6PA),
which is the $L=5$ and $6$ terms of Eq. (14) respectively. The 5PA channel is
closed at $2.6\times 10^{12}$ W/cm-2, while the 6PA channel is closed at
$1.4\times 10^{13}$ W/cm-2. The 5PA rate is larger than the 6PA absorption
rate with $\eta=0.0$ and $0.25$. The relationship between 5PA and 6PA is
reversed as $\eta$ increases because the 5PA rate decreases significantly
below the channel closing with higher $\eta$. We have to consider some multi-
photon processes to evaluate the excitation rate with elliptical and circular
polarization.

Figure 3:  Excitation rate via 5- and 6-photon processes. The abrupt drop of
the rate indicates the channel closing of each photonic process.

###  3.2 Anisotropic band

In the previous subsection, we studied the $\eta$ dependence with isotropic
parabolic band. However, the $\eta$ dependence is much larger than the
experimental result[19]. In this subsection, we would like to study how the
band structure affects $\eta$ dependence. Since the analytical formulation
with direction-dependent reduced mass like Eq. (25) is complicated, we solve
Eq. (26) by real-time method.

We assume the Gaussian-shape laser pulse,

$\textbf{A}(t)=A_{0}e^{-\frac{t^{2}}{\tau^{2}}}(\eta\sin\omega t,0,\cos\omega
t).$ (27)

Since diamond has 4–fold symmetry, the reduced mass can be mimicked by

$\mu(\theta,\phi)=\mu_{0}\left[1+\xi\left(\sin^{2}2\theta+\sin^{2}2\phi\right)\right],$
(28)

where $\xi$ is the parameter for anisotropy of band structure. Figure 4 shows
the time-evolution of the carrier density,

$n(t)=\int d\textbf{k}|C^{\textbf{k}}_{vc}(t)|^{2},$ (29)

with the ellipticity of 0.0, 0.5, and 1.0. The irradiance is $1\times 10^{12}$
W/cm2 with frequency of 1.55 eV. The pulse duration parameter $\tau$ is 100
fs. The band structure parameter, $\xi$, is 0.005. The isotropic band case
indicates that the excitation rate with linear polarization ($\eta=0.0$) is
about 50 times higher than that with circular polarization. On the other hand,
the anisotropic case indicates a factor of 2.5, which is close to the factor
of 5 in experiment[19].

Figure 5 shows the $\xi$ and $\eta$ dependence of the carrier density. The
carrier density increases at higher $\xi$ in all $\eta$, and the $\eta$
dependence decreases as the $\xi$ increases. In particular, in the case of
$\xi=1\times 10^{-2}$, the excitation rate has the minimum at $\eta=0.5$. The
change of $\eta$ giving minimum excitation rate indicates that the band
structure changes the $\eta$ dependence not only quantitatively but also
qualitatively.

Figure 4: Time evolution of carrier density with $\eta$ as 0, 0.5, and 1.0.
The grey shaded region is laser pulse irradiance. $\xi$ is fixed at $5\times
10^{-3}$. The solid line with gray shaded area indicates the time-evolution of
the laser irradiance. Figure 5:  Ellipticity and $\xi$ dependence of the
carrier density at the end of time evolution. The inset figure shows the
carrier density scaled at $\eta=0.0$.

To clarify why the excitation $\xi$-dependence is more intense with higher
ellipticity, we would like to revisit the phase by the time integral of the
energy. If we see the electron at $\Gamma$-point with $\eta=1.0$, the time
evolution of the relative energy is

$\Delta\epsilon^{G}_{cv,\textbf{k'}(t)}=\frac{e^{2}A_{0}^{2}}{\hbar^{2}c^{2}\mu_{0}}\frac{1}{1+\xi\frac{1-\cos
4\omega t}{2}}.$ (30)

The relative energy oscillates with a frequency of $4\omega$. On the other
hand, in the linear polarization case, the relative energy oscillates with
$2\omega$. The higher relative energy oscillation frequency induces the
increase of excitation rate. Therefore, higher $\eta$ is more sensitive to
$\xi$. Since the intra-band oscillation under elliptically polarized laser
changes the $\eta$-dependence, the material with a significant anisotropic
band structure may have the maximum excitation rate under circularly polarized
laser.

It should be noted that $\xi$ affects excitation rate more than we expected
from the experiment [19]. The sensitive behavior of our model may be due to
the naive treatment of the band structure. Diamond has almost isotropic band
structure around the $\Gamma$-point and the direction dependence of effective
mass becomes more prominent as $k_{0}$ increases. Our results indicate that
the realistic band structure is essential to reproduce the experiments
quantitatively.

##  4 Summary

We extended our previous analytical formula for the electron excitation rate
in the dielectrics[15] to elliptical polarization. In general, ellipticity
decreases the excitation rate. In particular, the ellipticity dependence is
significant in the isotropic system. However, the decrease of excitation rate
by elliptically polarized laser becomes small by assuming the anisotropic band
structure because the symmetry-dependent oscillation of energy in the intra-
band dynamics occurs with higher ellipticity. If the anisotropy of the band
structure is significant, the ellipticity increases the excitation rate.

The conflict between two experimental results reported by Temnov et. al. [20]
and Kozák et. al.[19], and our numerical results indicate that the anisotropy
of the band structures is important to understand the laser excitation
process.

##  5 Acknowledgment

This research is supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP)
under Grant No. JPMX0118067246. This research is also partially supported by
JST-CREST under Grant No. JP-MJCR16N5. The numerical calculations are carried
out using the computer facilities of the SGI8600 at Japan Atomic Energy Agency
(JAEA).

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# Optimising quantum tomography via shadow inversion

Andrea Caprotti  0000-0002-4404-2216<EMAIL_ADDRESS>University
of Vienna, Faculty of Physics, Vienna Center for Quantum Science and
Technology (VCQ), Boltzmanngasse 5, 1090 Vienna, Austria  University of
Vienna, Vienna Doctoral School in Physics, Boltzmanngasse 5, 1090 Vienna,
Austria  Joshua Morris  0000-0002-1022-7976<EMAIL_ADDRESS>University of Vienna, Faculty of Physics, Vienna Center for Quantum Science
and Technology (VCQ), Boltzmanngasse 5, 1090 Vienna, Austria  University of
Vienna, Vienna Doctoral School in Physics, Boltzmanngasse 5, 1090 Vienna,
Austria  Borivoje Dakić  0000-0001-9895-4889<EMAIL_ADDRESS>University of Vienna, Faculty of Physics, Vienna Center for Quantum Science
and Technology (VCQ), Boltzmanngasse 5, 1090 Vienna, Austria  Institute for
Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria.

###### Abstract

In quantum information theory, the accurate estimation of observables is
pivotal for quantum information processing, playing a crucial role in compute
and communication protocols. This work introduces a novel technique for
estimating such objects, leveraging an underutilised resource in the inversion
map of classical shadows that greatly refines the estimation cost of target
observables without incurring any additional overhead. A generalised framework
for computing and optimising additional degrees of freedom in the homogeneous
space of the shadow inversion is given that may be adapted to a variety of
near-term problems. In the special case of local measurement strategies we
show feasible optimisation leading to an exponential separation in sample
complexity versus the standard approach and in an exceptional case we give
non-trivial examples of optimised post-processing for local measurements,
achieving the same efficiency as the global Cliffords shadows.

## Introduction

Quantum state tomography with classical shadows Huang  _et al._ (2020),
initially motivated by the seminal work of Aaronson Aaronson (2017),
represents a significant advancement in the realm of quantum information
processing, particularly in the efficient characterisation of quantum systems.
This technique, developed as a solution to the prohibitive resource
requirements of full quantum state tomography has since seen a plethora of
modifications Hu  _et al._ (2023); Elben  _et al._ (2020); Huang  _et al._
(2021); Neven  _et al._ (2021); Crawford  _et al._ (2021); Chen  _et al._
(2021); Hadfield  _et al._ (2022); Zhao  _et al._ (2021); Seif  _et al._
(2023); Wan  _et al._ (2023); Levy  _et al._ (2024); Bu  _et al._ (2024);
Coopmans  _et al._ (2023); Bertoni  _et al._ (2023); Koh and Grewal (2022);
McGinley and Fava (2023); Kunjummen  _et al._ (2023); Iosue  _et al._ (2024);
Nakaji  _et al._ (2023); Feldman  _et al._ (2022); Wyderka  _et al._ (2023),
which all seek to provide a viable alternative for obtaining meaningful
information about quantum states with substantially reduced computational and
experimental overhead Sack  _et al._ (2022); Anshu  _et al._ (2021); Huang
_et al._ (2023); Nguyen  _et al._ (2022); Zhang  _et al._ (2021); Struchalin
_et al._ (2021); Stilck França  _et al._ (2024); Ippoliti  _et al._ (2023);
Morris  _et al._ (2022); Glos  _et al._ (2022); Stricker  _et al._ (2022);
Vermersch  _et al._ (2023). Naturally, the main focus has been the ’quantum’
element of the procedure as this is where most practical efforts are stymied,
either in gate or sample complexity. Other than machine learning Huang  _et
al._ (2023, 2022) and adaptive techniques García-Pérez  _et al._ (2021); Glos
_et al._ (2022), little thought has been put towards explicit optimisation of
the classical postprocessing with it often being treated as a mostly fixed
step.

At the core of shadow tomography lies the concept of constructing these
aforementioned shadows; efficiently representable compressed classical
descriptions of quantum states. These shadows are generated through a process
involving random measurements on copies of a state, followed by classical
processing to reconstruct a succinct representation of the state, the
’shadow‘. These classical shadows, though not providing a complete description
of the quantum state, contain enough information to estimate a wide range of
properties with high accuracy when an appropriate POVM is used. An archetypal
example is the set of measurements performed by random perturbations sampled
from the Clifford group Webb (2016); Zhu (2017) followed by a computational
basis measurement

$\mathcal{C}_{N}=\\{T\in
SU(2^{N})|TP_{N}T^{\dagger}\in\mathcal{P}_{N}\\}/U(1),$ (1)

for all $P_{N}$, the set of Pauli operators on $N$ qubits. Critically for
shadow tomography, we have that the Clifford group constitutes a $2,3$-design
Webb (2016) i.e., it can reproduce the statistical properties of the full set
of unitaries up to the second and third moment. It is this that forms the
basis upon which shadow tomography derives its impressive performance,
implementing a map whose consequent inverse lies at the core of efficient
estimation with classical shadows. If $U_{k}$ are elements of the $N$ qubit
Clifford group and $\Pi_{\ell}$ are computational measurement projectors all
acting on an input $N$ qubit quantum state then

$\mathcal{M}[\rho]=\frac{1}{|\mathcal{C}_{N}|}\sum_{k=1}^{|\mathcal{C}_{N}|}\sum_{\ell=1}^{2^{N}}T_{k\ell}\rho
T_{k\ell}=\lambda\rho+(1-\lambda)\tr(\rho)\frac{\mathbbm{1}}{2^{N}},$ (2)

for $T_{k\ell}=U_{k}^{\dagger}\Pi_{\ell}U_{k}$. The inverse operation
$\mathcal{M}^{-1}$ may be easily computed to recover the input state $\rho$
from the output $\mathcal{M}^{-1}$. Classical shadows’ great insight was to
show that this inversion procedure remains effective even when only a tiny
fraction of Clifford measurements are used in the above sum. This rapid
convergence to a known configuration means the linear map of (2) may be
readily inverted to recover the original state $\rho$. As shown in Huang  _et
al._ (2020), the inverse map of (2) for global Clifford measurements on $N$
qubits seems to easily appear as the inverse depolarising channel:
$\mathcal{M}_{N}^{-1}(X)=(2^{N}+1)X-\mbox{Tr}\left[{X}\right]\mathbbm{1}_{2^{N}}$;
in the case of tensor products of random single-qubit Pauli measurements, the
corresponding inverted quantum channel reduces to
$\mathcal{M}^{-1}_{\mathcal{P}}=\bigotimes_{i=1}^{N}\mathcal{M}_{1}^{-1}$.

Figure 1: (a) Classical shadow tomography with optimised shadow inversion.
After a collection of $M$ POVM outcomes have been acquired from a fixed
quantum state, any estimation can be improved for a dynamic choice of target
observable $A$ by computing an optimal ‘inverse’ operation for that observable
and POVM. (b) Shadow norm for different strategies. The shadow norm cofactor
when optimised for certain target projectors, using local Pauli measurements
on single qubits. The shadow norm for such an observable on $N$ qubits is
normally $(3/2)^{N}$ but when optimised over additional degrees of freedom can
be shrunk to $(1.15)^{N}$. Though still exponential in the number of qubits,
the optimised strategy is exponentially cheaper than standard classical shadow
tomography with local measurements.

In these cases, the inversion map takes a particularly simple form, and the
estimation of consumed resources (sampling complexity) critically depends on
it. Nevertheless, what is often neglected in studies is the fact that such a
map is not unique, i.e., as long as the POVM used is overcomplete (in the
operator, Hilbert-Schmidt space), an additional set of free parameters appear
in the inversion process which may be optimised to improve the post-processing
performance of the estimation procedure. Surprisingly, this has been only
recently recognised in the literature Innocenti  _et al._ (2023) by employing
the techniques of measurement dual frames. In this work, we demonstrate that
such additional degrees of freedom reside in the homogeneous space, i.e. the
general inverse map contains a free homogeneous component
$\mathcal{M}^{-1}=\mathcal{M}_{p}^{-1}+\mathcal{M}_{h}^{-1}$ which can be
further optimized. Our goal is to pursue such an optimisation by considering a
hereto before missed resource in the tomography procedure that appears most
vividly in classical shadows and the post-processing thereof. Though the
natural choice of the inverse map (such as the one given above) is the inverse
of the forward channel being applied, there is nothing that says it must be
so. It is this inversion process, normally treated as a fixed mathematical
procedure, that we aim to modify and improve upon. As long as the set of
performed measurements is tomographically over-complete, the additional
degrees of freedom may be exploited to reduce case-specific variance for a
given observable. In doing so, we will show that an exponential improvement in
estimations is possible compared to the standard choice of inverse map
presented in the literature. With this novel approach to processing acquired
quantum data, we open a new route to exploit areas where advantages may be
gained without incurring an unacceptable resource cost.

## Shadow map inversion

We begin by describing a generalised procedure Morris and Dakić (2020); Nguyen
_et al._ (2022), depicted in Figure 1(a) fulfilling the random measurement
scheme with a fixed POVM $\\{E_{k}\\}_{k=1}^{n}$ on a $d$-dimensional system.
The POVM elements sample randomly an unknown state $\rho$ with probability
$p_{k}=\mbox{Tr}\left[{\rho E_{k}}\right]$. These POVM elements span a
subspace $\mathcal{V}_{D}=\mathrm{Span}\\{E_{k}\\}_{k=1}^{n}$ in the Hilbert-
Schmidt space (HS) of the dimension $|\mathcal{V}_{D}|=D$. Typically,
$D=d^{2}$ (the entire HS space), but for the sake of generality, we shall keep
$D\leq d^{2}$. Furthermore, we assume $D<n$, thus $E_{k}$ forms an over-
complete basis in $\mathcal{V}_{D}$. The estimation task aims to approximate
the mean values of an observable $A\in\mathcal{V}_{D}$, i.e.,
$\expectationvalue{A}=\mbox{Tr}\left[{A\rho}\right]$. Note that
$\expectationvalue{A}=\mbox{Tr}\left[{A\bar{\rho}}\right]$, with $\bar{\rho}$
being the projection of a quantum state $\rho$ onto $\mathcal{V}_{D}$ which
follows directly from $A\in\mathcal{V}_{D}$. For the case of the entire HS
space, we have $\bar{\rho}=\rho$. Similarly, the measured probabilities
trivially satisfy
$p_{k}=\mbox{Tr}\left[{E_{k}\rho}\right]=\mbox{Tr}\left[{E_{k}\bar{\rho}}\right]$.
The post-processing task assigns a (matrix) estimator
$\hat{\eta}\in\\{\dots,\hat{\eta}_{k},\dots\\}$, i.e. each outcome $E_{k}$ the
corresponding matrix $\hat{\eta}_{k}$, which represents a single-shot
approximation of $\bar{\rho}$. They can be calculated via an inversion map
$\hat{\eta_{k}}=\mathcal{M}^{-1}(E_{k})$ Huang  _et al._ (2020); Nguyen  _et
al._ (2022) which we will show is not unique, i.e.
$\mathcal{M}^{-1}=\mathcal{M}_{p}^{-1}+\mathcal{M}_{h}^{-1}$ with the latter
(homogeneous) component containing auxiliary parameters that may be used as a
free resource in post-processing.

The overall classical shadow itself is obtained by averaging over the
estimators obtained from $M$ random measurements:
$M^{-1}\sum_{s}^{M}\hat{\eta}^{(s)}$. Assuming that all samples are i.i.d.,
the average value $\expectationvalue{\hat{\eta}}$ is immediately

$\expectationvalue{\hat{\eta}}=\sum_{k=1}^{n}p_{k}\hat{\eta}_{k}=\sum_{k=1}^{n}\mbox{Tr}\left[{\bar{\rho}E_{k}}\right]\hat{\eta}_{k}\overset{!}{=}\bar{\rho},~{}~{}~{}\forall\bar{\rho}\in\mathcal{V}_{D}.$
(3)

This is the requirement on the estimator $\hat{\eta}$ such that the statistics
of the shadow mimics, in expectation, that of the actual unknown state. Since
the POVM forms an over-complete basis in $\mathcal{V}_{D}$, any solution to
Eq. (3) is not unique. This multiplicity results in a set of free parameters,
which becomes an extra resource that may be used in the post-processing
component of an estimation to reduce the variance.

To achieve this, we now show a constructive method to recover families of
equivalent classical shadows in terms of the POVM elements and their free
parameters. We rely on an equivalent problem to (3), that is, to impose the
equivalence between the expectation value of an observable $A$ and the
corresponding classical estimator $a=\mbox{Tr}\left[{A\hat{\eta}}\right]$:

$\expectationvalue{\,a\,}=\mbox{Tr}\left[{A\expectationvalue{\hat{\eta}}}\right]=\sum_{k=1}^{n}p_{k}\mbox{Tr}\left[{A\,\hat{\eta}_{k}}\right]\overset{!}{=}\mbox{Tr}\left[{\bar{\rho}A}\right].$
(4)

It is instructive to express the operators as vectors in $\mathcal{V}_{D}$,
i.e., for an operator $X\in\mathcal{V}_{D}$, we shall use the ket notation
$|{X}\rangle\\!\rangle$. We also identify an orthonormal basis
$\beta_{D}=\\{|{B_{j}}\rangle\\!\rangle\\}_{j=1}^{D}$ in $\mathcal{V}_{D}$.
The expectation value from (4) can be expressed as

$\expectationvalue{\,a\,}=\sum_{k=1}^{n}\langle\\!\langle{A^{\dagger}}|{\eta_{k}}\rangle\\!\rangle\langle\\!\langle{E_{k}^{\dagger}}|{\rho}\rangle\\!\rangle\overset{!}{=}\langle\\!\langle{A^{\dagger}}|{\bar{\rho}}\rangle\\!\rangle$
(5)

where
$\langle\\!\langle{X}|{Y}\rangle\\!\rangle=\mbox{Tr}\left[{X^{\dagger}Y}\right]$.
To guarantee the estimators are unbiased, the requirement reduces to a
resolution of the identity in the subspace $\mathcal{V}_{D}$

$\sum_{k=1}^{n}|{\eta_{k}}\rangle\\!\rangle\langle\\!\langle{E_{k}^{\dagger}}|=\sum_{k=1}^{n}|{\eta_{k}}\rangle\\!\rangle\langle\\!\langle{E_{k}}|\equiv\mathbbm{1}_{D},$
(6)

which can be solved for $|{\eta_{k}}\rangle\\!\rangle$ using
$E_{k}^{\dagger}=E_{k}$. The last equation expressed in the basis $\beta_{D}$
reads

$\sum_{k=1}^{n}\langle\\!\langle{B_{s}}|{\hat{\eta}_{k}}\rangle\\!\rangle\langle\\!\langle{E_{k}}|{B_{t}}\rangle\\!\rangle=\sum_{k=1}^{n}L_{sk}R_{tk}^{*}\overset{!}{=}\delta_{st},$
(7)

with $L_{sk}=\langle\\!\langle{B_{s}}|{\hat{\eta}_{k}}\rangle\\!\rangle$ and
$R_{tk}=\langle\\!\langle{B_{t}}|{E_{k}}\rangle\\!\rangle$. This implies the
matrix equation $LR^{\dagger}=\mathbbm{1}_{D}$; thus, $L$ is a generalised
inverse of $R^{\dagger}$. Since $R$ is a $D\times n$ matrix having
$|{E_{k}}\rangle\\!\rangle$ as columns, it is a rank-$D$ matrix (recall there
are only $D$ independent $E_{k}$ matrices). We can write its singular value
decomposition (SVD), i.e., $R=U(\Sigma|\hat{0})W^{\dagger}$ with
$\Sigma=\mathrm{diag}[\sigma_{1},\dots,\sigma_{D}]$ ($\sigma_{s}>0$) and
$\hat{0}$ being the $D\times(n-D)$ zero-matrix. Having this, a simple exercise
shows $L=W(\Sigma^{-1}|H)U^{\dagger}$ with $H$ being an arbitrary
$D\times(n-D)$ matrix. Accordingly, we can split solution into two parts
$L=L^{(p)}+L^{(h)}$, with $L^{(p)}=W(\Sigma^{-1}|\hat{0})U^{\dagger}$
(particular) and $L^{(h)}=W(\hat{0}|H)U^{\dagger}$ (homogeneous). This results
in a family of equivalent estimators

$\hat{\eta_{k}}=\sum_{s=1}^{D}L_{sk}\hat{B_{s}}=\sum_{s=1}^{D}\left(L_{sk}^{(p)}+L_{sk}^{(h)}\right)\hat{B_{s}}=\hat{\eta}_{k}^{(p)}+\hat{\eta}_{k}^{(h)},$
(8)

parameterised by a homogeneous solution $\hat{\eta}_{k}^{(h)}$. As we shall
see, this multiplicity of possible inversion schemes represents an additional
resource for error optimisation in post-processing. We turn to observable
properties which may be expressed in terms of estimator coefficients
$a_{k}=\mbox{Tr}\left[{A\hat{\eta}_{k}}\right]=a_{k}^{(p)}+a_{k}^{(h)}$, with
$a_{k}^{(p/h)}=\mbox{Tr}\left[{A\hat{\eta}_{k}^{(p/h)}}\right]$ where the
homogeneous coefficients do not influence expectation values or any other
linear combination of classical shadows

$\expectationvalue{\,a\,}=\sum_{k=1}^{n}p_{k}a_{k}=\sum_{k=1}^{n}p_{k}\left(a_{k}^{(p)}+a_{k}^{(h)}\right)=\sum_{k=1}^{n}p_{k}a_{k}^{(p)}.$
(9)

By substituting $p_{k}=\mbox{Tr}\left[{\rho E_{k}}\right]$ and
$\expectationvalue{\,a\,}=\mbox{Tr}\left[{AE_{k}}\right]$ into the previous
equation we find

$\displaystyle\sum_{k}a_{k}^{(p)}E_{k}=A,$ (10)
$\displaystyle\sum_{k}a_{k}^{(h)}E_{k}=0.$ (11)

The set of solutions to the latter (homogeneous) equation defines an auxiliary
optimisation space of dimension $n-D$ independent of the target observable
(fixed by the choice of POVM). This is a free resource that may now be used in
post-processing.

Figure 2:  (a) Optimal free parameter for projectors on the Bloch sphere. The
contour plot shows the minimal shadow norm (13) of the single-qubit projector
characterised by the Bloch sphere angles $\theta,\,\phi$ (in units of $\pi$).
Optimisation results in bounds lower or equal than $3/2$, obtained via the
Clifford inversion map (15) found in Huang  _et al._ (2020) (b) Optimal free
parameter for plane projectors. The optimal value of the shadow norm of the
single-qubit projector in the equatorial plane in terms of a single angle
parameter $\phi$ (in units of $\pi$). This enhanced norm is always lower or
equal to the standard shadow norm
$\|\outerproduct{\psi}{\psi}(\phi)\|^{2}_{\text{sh}}\leq 5/4$ (see main text).
In the case of $\phi=k\frac{\pi}{2},\,k\in\mathbb{Z}$, which corresponds to
the projectors on the Pauli $X$ and $Y$ basis. The bound corresponds to the
combinations which present optimal variance
$\|\outerproduct{\psi}{\psi}(\phi)\|^{2}_{\text{sh}}\equiv 1$.

Now, to effectively compare the different choices for estimators, we refer to
the efficiency of the estimation process, that is the expected sample size of
copies required to estimate a property of the state up to a fixed accuracy. It
is customary in classical shadow tomography to refer to variance as figure of
merit Huang  _et al._ (2020). In particular, we consider the state-independent
upper bound on variance, defined as the shadow norm, which only depends on the
decomposition of the particular observable:

$\mbox{Var}\left[{a}\right]\leq\max_{\sigma:\text{state}}\mbox{Tr}\left[{\sigma\sum_{k=1}^{n}|a_{k}^{(p)}+a_{k}^{(h)}|^{2}E_{k}}\right]=\|A\|^{2}_{\text{shadow}}.$
(12)

This quantity is equivalent to the maximal eigenvalue of the variance operator
defined as
$\hat{O}_{A}\coloneq\sum_{k=1}^{n}|a_{k}^{(p)}+a_{k}^{(h)}|^{2}E_{k}$; we will
use both definitions interchangeably. This is akin to the original upper bound
on shadow norm and thence sampling complexity found in Huang  _et al._ (2020),
where the expected number of samples scales with
$M=O(\varepsilon^{-2}\log{\delta^{-1}}\|A\|^{2}_{\text{shadow}})$.
Nevertheless, this can be further optimised over the free parameters in the
homogeneous part of the estimator. By optimising over these coefficients, it
is possible to identify the configuration which guarantees the minimal sample
size for the estimation of the particular observable $A$:

$\|A\|^{2}_{\text{opt}}=\min_{a_{k}^{(h)}:H}\max_{\sigma:\text{state}}\mbox{Tr}\left[{\sigma\sum_{k=1}^{n}|a_{k}^{(p)}+a_{k}^{(h)}|^{2}E_{k}}\right],$
(13)

The existence of a minimum is guaranteed through the positive semidefiniteness
of the variance operator. We will provide examples to illustrate that
different choices of free parameters may result in a sampling complexity with
exponential separation compared to that of the naive estimator construction,
which shows, in turn, that optimisation in (13) yields exceptional gains for
relatively minuscule cost.

## Product observables and shadow optimisation

Let us now discuss the estimation of product observables $A^{(N)}=A_{1}\otimes
A_{2}\ldots\otimes A_{N}$ via local POVMs, i.e., of the type
$E_{k_{1}}\otimes\dots\otimes E_{k_{n}}$. This case is particularly
interesting for near-term applications, such as the VQE algorithm Peruzzo  _et
al._ (2014); Tilly  _et al._ (2022). In such a scenario, it is easy to show
that the shadow norm in Eq. (12) has a product form (see also Nguyen  _et al._
(2022)) and consequently Eq. (13) becomes

$\|A^{(N)}\|^{2}_{\text{opt}}=\prod_{j=1}^{N}\|A_{j}\|^{2}_{\text{opt}}$ (14)

The optimisation in Eq. (12) appears efficient, given the total number of free
parameters scales linearly with the number of subsystems $N$. We see that even
minimal deviation from the minimum in local shadow norms
$\|A_{j}\|^{2}_{\text{opt}}$ may accumulate exponentially fast with $N$. This
is illustrated in Figure 1(b) for the case of the canonical inversion map of
Eq. (15). We now present examples of such exponential improvement for qubits
for particular sets of local observables, specifically projectors, whereby by
adjusting the available free parameters, we are able to reduce the expected
bound on sample size based on assumptions for the observable itself.

## Fidelity estimation of product states

We first consider the case of estimation of fidelity with product states. The
target observables are thus tensor products of a projector on a single qubit
$\bigotimes_{j=1}^{N}\outerproduct{\psi_{j}}{\psi_{j}}$, parameterised on the
Bloch sphere by two angles $(\theta_{j},\,\phi_{j})$. The measurement scheme
is described by the POVM of projectors on the Pauli basis
$E^{\pm}_{\zeta}=\frac{1}{2}(\mathbbm{1}\pm\sigma_{\zeta})$, where
$\zeta=x,\,y,\,z$. As reference, we consider the bound
$\|\bigotimes_{j=1}^{N}\outerproduct{\psi_{j}}{\psi_{j}}\|^{2}_{\text{shadow}}\equiv(3/2)^{N}$
obtained using the estimators

$\hat{\eta}^{\pm}_{\zeta}=3\left(3E^{\pm}_{\zeta}-\mbox{Tr}\left[{E^{\pm}_{\zeta}}\right]\mathbbm{1}_{3}\right).$
(15)

found in Huang  _et al._ (2020). Now we show how to improve this bound by our
techniques. The POVMs $E^{\pm}_{\zeta}$ span the entire HS space. We have
$n=6$ and $D=4$, thus the equation (11) gives two free optimisation parameters
for each qubit See supplementary material at [URL from publisher] for
information concerning estimator explicit parameter dependence . In Fig.2(a),
we present the minimal upper bound as a function of angles $(\theta,\phi)$,
which is contained within the interval
$1.15\leq\|\outerproduct{\psi_{j}}{\psi_{j}}\|^{2}_{\text{sh}}\leq 1.5$. Since
the overall shadow norm (14) is of the product form, there are entire regions
in the $(\theta,\phi)$-space for which deviations from from the standard $3/2$
leads to the exponential improvement in $N$.

## Locally efficient fidelity estimation implies globally efficient

We now present a configuration in which the optimisation of the inversion map
not only improves the standard bound, but we are actually able to achieve
efficiency, i.e. $\|A\|^{2}_{\text{opt}}\equiv 1$. We consider the POVM
constituted by the projectors of the X and Y Paulis, which spans the ”equator”
on the Bloch sphere. The target observables are tensor products of a projector
on a single qubit, parameterised by a single angle:
$A=\outerproduct{\psi(\phi)}{\psi(\phi)}$ with
$|{\psi(\phi)}\rangle=(|{0}\rangle+e^{i\phi}|{1}\rangle)\sqrt{2}$. As before,
we follow the ansatz given in Huang  _et al._ (2020) to get the particular
solution for the inversion map and estimators

$\hat{\eta}^{\pm}_{\zeta}=4E^{\pm}_{\zeta}-\mbox{Tr}\left[{E^{\pm}_{\zeta}}\right]\mathbbm{1}_{2}.$
(16)

From here we obtain the reference bound for the shadow norm of
$\|\bigotimes_{j=1}^{N}\outerproduct{\psi_{j}}{\psi_{j}}\|^{2}_{\text{sh}}\equiv(5/4)^{N}$.
We reference to Supplementary Information See supplementary material at [URL
from publisher] for information concerning shadow norm expression for local
Clifford measurements  for more precise remarks on how we obtain this result.

Now we turn to our optimisation technique. We have the POVM with $n=4$
elements spanning a subspace of dimension $D=3$, therefore the generic
estimators obtained from our procedure depend on a single free parameter $p$.
The shadow norm of single-qubit projector depends on this free parameter and
on the angle parameter $\phi$:

$\displaystyle\|\outerproduct{\psi}{\psi}(\phi)\|^{2}_{\text{sh}}=\frac{1}{2}$
$\displaystyle(1+p_{i}^{2}+B(\phi,p_{r})+$ (17)
$\displaystyle\quad\sqrt{2}\sqrt{C(\phi,p_{r})})$ $\displaystyle
B(\phi,p_{r})=(4$ $\displaystyle p_{r}-2)\cos{\phi}+\cos{2\phi}+2$ (18)
$\displaystyle\quad+2p_{r}(p_{r}-1)$ $\displaystyle C(\phi,p_{r})=(4$
$\displaystyle p_{r}-1)\cos{\phi}+2p_{r}\cos{2\phi}$ (19)
$\displaystyle\quad+\cos{3\phi}+2+2p_{r}(p_{r}-1),$

where $p_{r}=\Re{p},\,p_{i}=\Im{p}$.

In general, the dependency on $\phi$ remains explicit, changing the expected
bound based on the projector to be estimated. In Fig.2(b) we present the
optimal value of the shadow norm for each parameter $\phi\in[0,\pi]$. We note
that for each value of $\phi$ the optimised norm is
$\|\outerproduct{\psi}{\psi}(\phi)\|^{2}_{\text{opt}}\leq 5/4$. Moreover, it
can be seen that for specific angles, the bound is exactly
$\|E_{\zeta}^{\pm}\|^{2}_{\text{opt}}\equiv 1$., i.e., for the eigenprojectors
of $X$ and $Y$ Paulis, corresponding to the POVM elements themselves. This
implies efficient estimation: given the product form of the overall shadow
norm (14), this scaling represents not only an exponential improvement
compared to $(5/4)^{N}$, as for other projectors, but also an expected
constant scaling of the sample size with the number of qubits. This is
particularly relevant considering that X and Y are non-commuting observables,
as it opens the possibility of the simultaneous estimation of up to $2^{N}$
projections simultaneously for complementary observables. Previously this
result has been found only via global measurement strategies relying on
Clifford unitaries consisting of many joint operations, acting on all $N$
qubits at the same time; we have shown instead that by introducing the initial
assumption over the space of observables, this could be achieved using only
local single-qubit measurements.

## Discussion

In the course of this letter we have observed how post-processing, an often
overlooked aspect in estimation, represents a relevant resource to be examined
in the accounting of a more efficient tomography. By introducing a generalised
framework to obtain families of equivalent parameter-dependent estimators, we
have found a clear relation between over-complete POVMs and their ability to
reduce the variance in estimation of arbitrary observables. When considering
local measurements on large composite quantum systems, even a small gain on
each estimator represents an overall improvement in sample complexity,
yielding an exponential separation between the naive sample variance and the
optimised one as seen in Figure 1(b). In the case of example two, this
optimised gain can achieve a variance that performs as well as global
classical shadows for given observables, with modest additional post
processing that remains efficient. We anticipate that by considering
overcomplete POVMs containing joint measurements on multiple qubits, further
advantage may be found at increased (but still tractable) optimisation
difficulty, either in sample complexity or the set of observables. Finally,
interleaving optimised multiqubit measurements may yield sample improvements
that approach regular classical shadows within some limit. Such questions are
natural candidates for further investigation and the identification of such a
hierarchy of optimisation will be the topic of future work.

Note \- In preparation of this manuscript, we became aware of two other works
Malmi  _et al._ (2024); Fischer  _et al._ (2024) considering a similar
problem. The authors consider a similar optimisation of the post-processing
stage in estimation tasks, albeit referring to different frameworks.

## Acknowledgements

We thank Richard Küng for his insight and excellent discussions on the subject
matter. This research was funded in whole, or in part, by the Austrian Science
Fund (FWF) [10.55776/F71] (BeyondC). For the purpose of open access, the
author(s) has applied a CC BY public copyright licence to any Author Accepted
Manuscript version arising from this submission.

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##  Appendix A Parameter dependence for estimators

The POVM of projectors on the Pauli basis
$E^{\pm}_{\zeta}=\frac{1}{2}\left(\mathbbm{1}\pm\sigma_{\zeta}\right)$, where
$\zeta=x,y,z$, admits a homogeneous combination
$\sum_{i=\pm}\sum_{\zeta}h_{i,\zeta}E^{i}_{\zeta}=0$ (see discussion around
(11) in the main text) determined by $n=6$ coefficients; given the symmetry of
the system, they reduce to $3$ since $h_{\zeta}^{+}=h_{\zeta}^{-}$:

$\\{h_{i,\zeta}\\}_{\zeta=x,y,z}=\\{h_{\zeta}\\}_{\zeta=x,y,z}=\\{p_{1},p_{2},-p_{1}-p_{2}\\}$
(20)

A expected, we find $n-D=2$ parameters $p_{1},p_{2}$ which are used to
optimise the shadow norm in (14). For the particular solution (see discussion
around (10)) of
$\sum_{i=\pm}\sum_{\zeta}q_{i,\zeta}E^{i}_{\zeta}=A=a\mathbbm{1}+\sum_{\zeta=x,y,z}\zeta\sigma_{\zeta}$
we can obtain the particular coefficients

$q_{\pm,\zeta}=a\pm 3\zeta$ (21)

where $\zeta=x,y,z$. In order to find the expression for the generic projector
$\outerproduct{\psi}{\psi}(\theta,\phi)$, one can simply rely on the
parametrisation $a\to\frac{1}{2}$, $x\to\frac{1}{2}\sin\theta\cos\phi$,
$y\to\frac{1}{2}\sin\theta\sin\phi$, $z\to\frac{1}{2}\cos\theta$. This
particular solution corresponds to the coefficients found with the estimators
obtained with the standard inversion map, as introduced in B.0.1.

##  Appendix B Shadow norm expression for local Clifford measurements

In this section, we detail the explicit expressions for the shadow norm
obtained using the standard estimators (15), as introduced in the seminal work
on Classical Shadows.

###  B.0.1 Shadow norm for local Pauli measurements

The POVM of projectors on the Pauli basis
$E^{\pm}_{\zeta}=\frac{1}{2}\left(\mathbbm{1}\pm\sigma_{\zeta}\right)$, where
$\zeta=x,y,z$. For a generic single-qubit observable
$A=a\mathbbm{1}+\sum_{\zeta=x,y,z}\zeta\sigma_{\zeta}$ the shadow norm in (12)
simply evaluates to

$\|A\|^{2}_{\text{sh}}=a^{2}+3(x^{2}+y^{2}+z^{2})+2|a|\sqrt{x^{2}+y^{2}+z^{2}}.$
(22)

Considering the particular case of projectors on the Bloch sphere, with the
parameterization $a\to\frac{1}{2}$, $x\to\frac{1}{2}\sin\theta\cos\phi$,
$y\to\frac{1}{2}\sin\theta\sin\phi$, $z\to\frac{1}{2}\cos\theta$, (22)
trivially reduces to
$\|\outerproduct{\psi}{\psi}\|^{2}_{\text{sh}}\equiv\frac{3}{2}$.

###  B.0.2 Shadow norm for local Pauli measurements on the plane

In this simplified case, the set of POVMs spend a subspace on the equatorial
plane of the Bloch sphere, i.e.,
$\mathcal{V}_{D=3}=\mathrm{span}\\{\mathbbm{1},X,Y\\}$. The generic target
observable can be thus represented as

$A=\spmqty{undef}.$ (23)

As a reference, we use the ansatz from Huang  _et al._ (2020) to find the
inversion map in the form

$\hat{\eta}_{k}=\mathcal{M}^{-1}[E_{k}]=\mathcal{N}(\lambda
E_{k}-\mu\,\mbox{Tr}\left[{E_{k}}\right]\mathbbm{1}_{2}).$ (24)

A simple inspection shows $\mathcal{N}=2$, $\lambda=2$ and $\mu=\frac{1}{2}$.
The shadow norm of the generic observable results as

$\|A\|^{2}_{\text{sh}}=a^{2}+2(x^{2}+y^{2})+2|a|\sqrt{x^{2}+y^{2}}.$ (25)

For the case of projectors we have $x\to\cos{\phi}/2$ and $y\to\sin{\phi}/2$,
thus we get
$\|\outerproduct{\psi(\phi)}{\psi(\phi)}\|^{2}_{\text{sh}}\equiv\frac{5}{4}$.