# Semidefinite programming lower bounds
on the squashed entanglement

Hamza Fawzi  DAMTP, University of Cambridge, United Kingdom Omar Fawzi  Univ
Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France

###### Abstract

The squashed entanglement is a widely used entanglement measure that has many
desirable properties. However, as it is based on an optimization over
extensions of arbitrary dimension, one drawback of this measure is the lack of
good algorithms to compute it. Here, we introduce a hierarchy of semidefinite
programming lower bounds on the squashed entanglement.

##  1 Introduction

The squashed entanglement is an entanglement measure introduced by Christandl
and Winter [CW04]. It satisfies many desirable properties of an entanglement
measure such as monotonicity under local operations and classical
communication (LOCC), additivity under tensor products, a monogamy relation
[KW04] and faithfulness [BCY11]. In fact, it is the only known entanglement
measure satisfying these properties. These mathematical properties also have
operational applications: the squashed entanglement provides an upper bound on
the distallable entanglement and the distillable key [Chr06, CEH+07, CSW12,
Wil16]. For a bipartite state $\rho_{AB}$ on $A\otimes B$, the squashed
entanglement is defined as

$\displaystyle
E_{\mathrm{sq}}(A:B)_{\rho}=\frac{1}{2}\inf_{\rho_{ABE}}I(A:B|E)_{\rho}\ ,$
(1)

where the infimum is taken over all finite-dimensional Hilbert spaces $E$ and
all extensions $\rho_{ABE}$ of the state $\rho_{AB}$, i.e.,
$\mathrm{tr}_{E}\rho_{ABE}=\rho_{AB}$. The conditional mutual information is
defined in equation (5) below. Note that as shown in [Shi16], considering
infinite dimensional separable Hilbert spaces $E$ leads to the same quantity.
We also note that a similar quantity was defined in [Tuc99, Tuc02].

To define the conditional mutual information it is useful to start with the
quantum relative entropy. For density operators $\rho,\sigma$ on a finite-
dimensional Hilbert space $\mathcal{H}$, the quantum relative entropy is
defined by

$\displaystyle
D(\rho\|\sigma)=\left\\{\begin{array}[]{ll}\mathrm{tr}(\rho(\log\rho-\log\sigma))&\text{if
}\mathrm{supp}(\rho)\subseteq\mathrm{supp}(\sigma)\\\
+\infty&\text{otherwise.}\end{array}\right.$ (4)

For a density operator $\rho_{AE}$ on $A\otimes E$, we define the conditional
von Neumann entropy $H(A|E)_{\rho}=-D(\rho_{AE}\|I_{A}\otimes\rho_{E})$ where
$\rho_{E}=\mathrm{tr}_{A}\rho_{AE}$ and $I_{A}$ denotes the identity operator
on $A$. The conditional mutual information is then defined by

$I(A:B|E)_{\rho}=H(A|E)_{\rho}-H(A|BE)_{\rho}\ .$ (5)

Note that by considering a purification $\rho_{ABDE}$ of the state
$\rho_{ABE}$ and using the fact that $H(A|BE)_{\rho}=-H(A|D)_{\rho}$, we can
write the squashed entanglement as follows:

$\displaystyle
E_{\mathrm{sq}}(A:B)_{\rho}=\frac{1}{2}\inf_{\rho_{ABDE}}H(A|D)_{\rho}+H(A|E)_{\rho}\
,$ (6)

where the infimum is over arbitrary finite-dimensional Hilbert spaces $D$ and
$E$ and extensions $\rho_{ABDE}$ of the state $\rho_{AB}$.

##  2 Squashed entanglement and $f$-divergences

To obtain our hierarchy of semidefinite programs, it is useful to generalize
the squashed entanglement (1) to standard $f$-divergences (also known as
quasi-relative entropies). Let $f:(0,\infty)\to\mathbb{R}$ be a continuous
function such that $\lim_{x\to\infty}\frac{f(x)}{x}=0$ and let
$f(0^{+}):=\lim_{x\to 0}f(x)$ (which could be infinite).

######  Definition 2.1 (Standard $f$-divergences [Pet86, HM17]).

Let $\rho,\sigma$ be positive semidefinite operators on $\mathcal{H}$ with
spectral decompositions $\rho=\sum_{j}\lambda_{j}P_{j}$ and
$\sigma=\sum_{k}\mu_{k}Q_{k}$. We define

$\displaystyle
D_{f}(\rho\|\sigma):=\sum_{j:\lambda_{j}>0}\sum_{k:\mu_{k}>0}\lambda_{j}f(\mu_{k}\lambda_{j}^{-1})\mathrm{tr}(P_{j}Q_{k})+f(0^{+})\mathrm{tr}\left(\rho(I-\sigma^{0})\right)\
,$ (7)

where $\sigma^{0}$ denotes the projector onto the support of $\sigma$.111Note
that in this definition, the roles of $\rho$ and $\sigma$ are sometimes
inverted.

If $f$ is operator convex, then $D_{f}$ is jointly convex and satisfies the
data processing inequality, i.e., for any quantum channel $\mathcal{N}$, we
have $D_{f}(\mathcal{N}(\rho)\|\mathcal{N}(\sigma))\leq D_{f}(\rho\|\sigma)$.

Note that $D_{f}(\rho\|\sigma)$ is linear and monotone in $f$ for fixed
positive semidefinite operators $\rho,\sigma$, i.e., $D_{f}(\rho\|\sigma)\leq
D_{g}(\rho\|\sigma)$ if $f(x)\leq g(x)$ for all $x\in(0,\infty)$. When
$f(x)=-\log x$, we obtain the quantum relative entropy. As done in [FSP18,
BFF21], we use an approximation of the function $-\log$ using rational
functions based on a Gauss-Radau quadrature (see e.g., [DR84, page 103]). In
order to do that, we introduce the functions $f_{t}:(0,+\infty)\to\mathbb{R}$
for $t\in[0,1]$ as

$\displaystyle f_{t}(x):=\frac{x-1}{t(x-1)+1}\ .$ (8)

It is simple to see that $-f_{t}$ is operator convex for all $t\in[0,1]$. In
addition, the condition $\lim_{x\to\infty}\frac{-f_{t}(x)}{x}=0$ is satisfied
for all $t\in(0,1]$, but not when $t=0$. For $t=0$, we take the convention
that $D_{-f_{0}}$ is still defined via (7), i.e.,
$D_{-f_{0}}(\rho\|\sigma)=1-\mathrm{tr}(\rho^{0}\sigma)$ (which is closely
related to the so-called min-relative entropy
$D_{\min}(\rho\|\sigma)=-\log(\rho^{0}\sigma)$), and not as in [HM17] which
would simply be $1-\mathrm{tr}(\sigma)$. With our convention
$D_{-f_{0}}(\rho\|\sigma)=\lim_{t\to 0}D_{-f_{t}}(\rho\|\sigma)$. In summary,
for all $t\in[0,1]$, $D_{-f_{t}}$ is jointly convex and is thus a valid
divergence satisfying the data-processing inequality.

The following proposition shows that one can choose $t_{1},\dots,t_{m}$ and
positive weights $w_{1},\dots,w_{m}$ such that taking the linear combination
of these divergences gives a good approximation of the quantum relative
entropy.

######  Proposition 2.2 (Gauss-Radau quadrature).

For any positive integer $m\geq 1$, there exists a choice of nodes
$t_{1},\dots,t_{m}\in(0,1]$ and weights $w_{1},\dots,w_{m}>0$ such that for
all $x\in(0,+\infty)$

$\displaystyle 0\leq\left(-\frac{1}{\ln 2}r_{m}(x)\right)-\left(-\log
x\right)\leq\frac{1}{m^{2}\ln 2}\left(x+\frac{1}{x}-2\right)\quad\text{ where
}\quad r_{m}(x):=\sum_{i=1}^{m}w_{i}f_{t_{i}}(x)\ .$ (9)

###### Proof.

As in [BFF21], we fix the node $t_{m}=1$ and the Gauss-Radau quadrature gives
the existence of nodes $t_{1},\dots,t_{m-1}\in(0,1)$ and weights
$w_{1},\dots,w_{m}>0$ such that $\ln x-r_{m}(x)\geq 0$ for all
$x\in(0,+\infty)$. This establishes the first inequality
$0\leq\left(-\frac{1}{\ln 2}r_{m}(x)\right)-\left(-\log x\right)$.

To prove the second inequality, we will establish the statement

$\displaystyle
r_{m}(x)+r_{m}(1/x)\geq\frac{-1}{m^{2}}\left(x+\frac{1}{x}-2\right)\ .$ (10)

Note using the fact that $\ln x-r_{m}(x)\geq 0$, the inequality (10) implies
that

$\displaystyle r_{m}(x)-\ln(x)\geq r_{m}(x)-\ln
x-(\ln(1/x)-r_{m}(1/x))\geq\frac{-1}{m^{2}}\left(x+\frac{1}{x}-2\right)\ ,$

which establishes the desired upper bound in (9).

We now focus on (10). Using the Gauss-Radau quadrature property, one can show
that we have $\ln(x)-r_{m}(x)=O((x-1)^{2m})$ in the neighborhood of $x=1$ and
$w_{m}=\frac{1}{m^{2}}$ (see [BFF21] for more details). Recalling that

$\displaystyle r_{m}(x)$
$\displaystyle=\sum_{i=1}^{m-1}w_{i}\frac{x-1}{t_{i}(x-1)+1}+\frac{1}{m^{2}}\frac{x-1}{x}$
$\displaystyle=\sum_{i=1}^{m-1}\frac{w_{i}}{1-t_{i}}\frac{x-1}{\frac{t_{i}}{1-t_{i}}x+1}+\frac{1}{m^{2}}\frac{x-1}{x}$

and putting all the fractions to the same denominator, we can write
$r_{m}(x)=\frac{-1}{m^{2}}\frac{F_{m}(x)}{xG_{m-1}(x)}$ where

$\displaystyle G_{m-1}(x)$
$\displaystyle=\prod_{i=1}^{m-1}\left(\frac{t_{i}}{1-t_{i}}x+1\right)$
$\displaystyle-F_{m}(x)$
$\displaystyle=\sum_{i=1}^{m-1}\frac{w_{i}}{1-t_{i}}(x-1)\cdot
x\cdot\prod_{j\in\\{1,\dots,m-1\\},j\neq
i}\left(\frac{t_{j}}{1-t_{j}}x+1\right)+(x-1)\cdot\prod_{j\in\\{1,\dots,m-1\\}}\left(\frac{t_{j}}{1-t_{j}}x+1\right)\
.$

Note that $\deg F_{m}=m$, $\deg G_{m-1}=m-1$ and they are normalized so that
$F_{m}(0)=G_{m-1}(0)=1$. Then we have:

$r_{m}(1/x)=-\frac{1}{m^{2}}\frac{xF_{m}(1/x)}{G_{m-1}(1/x)}=-\frac{1}{m^{2}}\frac{\hat{F}_{m}(x)}{\hat{G}_{m-1}(x)}$

where we let $\hat{F}_{m}(x)=x^{m}F_{m}(1/x)$ and
$\hat{G}_{m-1}=x^{m-1}G_{m-1}(1/x)$. Note that $\hat{F}_{m}$ and
$\hat{G}_{m-1}$ are both monic polynomials (i.e., the leading coefficient is
$1$) since $F_{m}(0)=G_{m-1}(0)=1$.

Summing we get

$r_{m}(x)+r_{m}(1/x)=-\frac{1}{m^{2}}\frac{F_{m}(x)\hat{G}_{m-1}(x)+x\hat{F}_{m}(x)G_{m-1}(x)}{xG_{m-1}(x)\hat{G}_{m-1}(x)}.$
(11)

We know that around $x=1$, $r_{m}(x)-\ln(x)=O((x-1)^{2m})$, which implies
$r_{m}(1/x)+\ln(x)=O((x-1)^{2m})$ and so by summing we get
$r_{m}(x)+r_{m}(1/x)=O((x-1)^{2m})$. Since the numerator of (11) has degree
$2m$ this means that it has to be a multiple of $(x-1)^{2m}$, i.e., we must
have

$r_{m}(x)+r_{m}(1/x)=-\frac{1}{m^{2}}\frac{c_{0}(x-1)^{2m}}{xG_{m-1}(x)\hat{G}_{m-1}(x)}$

where $c_{0}$ is the leading term of the numerator, i.e., it is the leading
term of $G_{m-1}$ so $c_{0}=\prod_{i=1}^{m-1}\frac{t_{i}}{1-t_{i}}$. The main
observation now is to note that $G_{m-1}(x)\hat{G}_{m-1}(x)\geq(x+1)^{2m-2}$.
Indeed, we have

$\displaystyle G_{m-1}(x)\hat{G}_{m-1}(x)$
$\displaystyle=\prod_{i=1}^{m-1}\left(\frac{t_{i}}{1-t_{i}}x+1\right)\left(\frac{t_{i}}{1-t_{i}}+x\right)$
$\displaystyle=c_{0}\prod_{i=1}^{m-1}\left(x+\frac{1-t_{i}}{t_{i}}\right)\left(\frac{t_{i}}{1-t_{i}}+x\right)$
$\displaystyle\geq c_{0}\prod_{i=1}^{m-1}(x+1)^{2}=c_{0}(x+1)^{2(m-1)}$

where we used the fact that for $x,a>0$, $(x+a)(x+1/a)\geq(x+1)^{2}$. This
implies that

$\displaystyle r_{m}(x)+r_{m}(1/x)$
$\displaystyle\geq-\frac{1}{m^{2}}\frac{(x-1)^{2}}{x}\left(\frac{x-1}{x+1}\right)^{2m-2}$
$\displaystyle\geq-\frac{1}{m^{2}}\frac{(x-1)^{2}}{x}$

as desired. ∎

In analogy with the expression (6), we can define a variant of the squashed
entanglement for $f$-divergences.

######  Definition 2.3 ($f$-squashed entanglement).

Let $\rho_{AB}$ be a bipartite state acting on $A\otimes B$. We define

$\displaystyle
E_{\mathrm{sq}}^{(f)}(A:B)_{\rho}:=\frac{1}{2}\inf_{\rho_{ABDE}}-D_{f}(\rho_{AD}\|I_{A}\otimes\rho_{D})-D_{f}(\rho_{AE}\|I_{A}\otimes\rho_{E})\
,$

where the infimum is over all extensions $\rho_{ABDE}$ of $\rho_{AB}$. In
particular, we define

$\displaystyle
E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}:=E_{\mathrm{sq}}^{(-\frac{r_{m}}{\ln
2})}(A:B)_{\rho}\quad\text{ where $r_{m}$ is defined in\leavevmode\nobreak\
\eqref{eq:log_rational}}.$

Note that we have

$\displaystyle
E_{\mathrm{sq}}(A:B)_{\rho}=E_{\mathrm{sq}}^{(-\log)}(A:B)_{\rho}\geq
E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}\qquad\text{ for any }m\geq 1\ .$

We emphasize that in general the $f$-squashed entanglement could be negative
and is not an entanglement measure. It will however be a useful tool in the
rest of the paper. The main property of the $f$-divergences for
$f=-\frac{r_{m}}{\ln 2}$ that we will use to construct semidefinite programs
is a variational expression of $D_{f}$ that we introduce in the next section.
Before that we establish basic properties of
$E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}$.

######  Proposition 2.4.

For any state $\rho_{AB}$ and positive integer $m$, we have

$\displaystyle E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}\leq
E_{\mathrm{sq}}(A:B)_{\rho}\leq
E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}+\frac{2d_{A}-2}{m^{2}\ln 2}\ .$

In addition, if $\rho_{AB}$ is pure, we have

$\displaystyle E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}$ $\displaystyle=\frac{1}{\ln
2}\mathrm{tr}\big{(}\rho_{A}r_{m}(\rho_{A}^{-1})\big{)}\ ,$

where $\rho_{A}^{-1}$ denotes the generalized inverse of
$\rho_{A}=\mathrm{tr}_{B}\rho_{AB}$.

###### Proof.

We simply use Definition 2.1 together with inequality (9). This gives

$\displaystyle E_{\mathrm{sq}}(A:B)_{\rho}$
$\displaystyle=\frac{1}{2}\inf_{\rho_{ABDE}}-D(\rho_{AD}\|I_{A}\otimes\rho_{D})-D(\rho_{AE}\|I_{A}\otimes\rho_{E})$
$\displaystyle\leq\frac{1}{2}\inf_{\rho_{ABDE}}-D_{g}(\rho_{AD}\|I_{A}\otimes\rho_{D})-D_{g}(\rho_{AE}\|I_{A}\otimes\rho_{E})\
,$

where $g(x)=-\frac{1}{\ln 2}r_{m}(x)-\frac{1}{m^{2}\ln
2}\left(x+\frac{1}{x}-2\right)$. Note that for the function $f_{0}(x)=x-1$ and
$f_{1}(x)=(x-1)/x$, we have

$\displaystyle
D_{f_{0}}(\rho_{AD}\|I_{A}\otimes\rho_{D})=\mathrm{tr}(\rho_{AD}^{0}I_{A}\otimes\rho_{D})-1\leq
d_{A}-1$ $\displaystyle
D_{f_{1}}(\rho_{AD}\|I_{A}\otimes\rho_{D})=1-\mathrm{tr}(\rho_{AD}^{2}I_{A}\otimes\rho_{D}^{-1})\geq
1-d_{A}\ .$

As a result,

$\displaystyle-D_{g}(\rho_{AD}\|I_{A}\otimes\rho_{D})\leq-D_{-\frac{r_{m}}{\ln
2}}(\rho_{AD}\|I_{A}\otimes\rho_{D})+\frac{1}{m^{2}\ln 2}2(d_{A}-1)\ ,$

which proves the desired bound.

If $\rho_{AB}$ is pure, then any purification has the form
$\rho_{ABDE}=\rho_{AB}\otimes\rho_{DE}$ and as a result,

$\displaystyle D_{f}(\rho_{AD}\|I_{A}\otimes\rho_{D})$
$\displaystyle=D_{f}(\rho_{A}\|I_{A})$ $\displaystyle=\mathrm{tr}\left(\rho
f(\rho^{-1})\right)\ ,$

using the defining formula (7) and using the notation $\rho^{-1}$ for the
generalized inverse of $\rho$. ∎

##  3 Approximating the squashed entanglement with noncommutative polynomial
optimization

We now use the variational expression for $D_{-f_{t}}$ that was established in
[BFF21].

######  Theorem 3.1 ([BFF21]).

Let $t\in[0,1]$ and recall that $f_{t}(x)=\frac{x-1}{t(x-1)+1}$. Let $\rho$
and $\sigma$ be positive semidefinite operators on a finite-dimensional
Hilbert space $\mathcal{H}$. Then

$D_{-f_{t}}(\rho\|\sigma)=-\inf_{Z}\frac{1}{t}\left\\{\mathrm{tr}(\rho)+\mathrm{tr}(\rho(Z+Z^{*}))+(1-t)\mathrm{tr}(\rho
Z^{*}Z)+t\mathrm{tr}(\sigma ZZ^{*})\right\\}\ ,$ (12)

where the infimum is over bounded operators $Z$ on $\mathcal{H}$.

We fix an orthonormal basis for $A$ that we denote
$\\{|a\rangle\\}_{a\in[d_{A}]}$ and one for $B$ that we denote
$\\{|b\rangle\\}_{b\in[d_{B}]}$. For a Hilbert space $\mathcal{H}$, we say
that the density operator $\rho_{AB\mathcal{H}}$ acting on $A\otimes
B\otimes\mathcal{H}$ is an extension of $\rho_{AB}$ if
$\mathrm{tr}_{\mathcal{H}}\rho_{AB\mathcal{H}}=\rho_{AB}$. We write
$\rho_{AB\mathcal{H}}=\sum_{a_{1}a_{2}b_{1}b_{2}}|a_{1}\rangle\\!\langle
a_{2}|\otimes|b_{1}\rangle\\!\langle b_{2}|\otimes\rho[a_{1}b_{1},a_{2}b_{2}]$
where $\rho[a_{1}b_{1},a_{2}b_{2}]$ is a bounded linear operator on
$\mathcal{H}$. With this notation, an extension of $\rho_{AB}$ is defined by a
family of operator $\rho[a_{1}b_{1},a_{2}b_{2}]$ satisfying
$\mathrm{tr}\rho[a_{1}b_{1},a_{2}b_{2}]=\mathrm{tr}(|a_{1}\rangle\\!\langle
a_{2}|\otimes|b_{1}\rangle\\!\langle b_{2}|\rho_{AB})$.

######  Theorem 3.2.

Let $m\geq 1$ and let $t_{1},\dots,t_{m},w_{1},\dots,w_{m}$ be the nodes and
weights from Proposition 9. We define the matrix-valued polynomial ${\bf
P}^{(m)}$ in noncommuting variables ${\bf
Z}=\\{Z_{i}[a_{1},a_{2}],Z_{i}[a_{1},a_{2}]^{*}\\}_{i,a_{1},a_{2}}$ by

$\displaystyle{\bf P}^{(m)}({\bf
Z}):=\left(\sum_{i=1}^{m}P^{i}_{a_{1},a_{2}}({\bf
Z})\right)_{a_{1},a_{2}\in[d_{A}]}$ (13)

where

$\displaystyle P^{i}_{a_{1},a_{2}}({\bf Z})$
$\displaystyle=\frac{w_{i}}{t_{i}\ln
2}\Bigg{(}Z_{i}[a_{1},a_{2}]+Z_{i}[a_{2},a_{1}]^{*}+(1-t_{i})\sum_{a_{3}}Z_{i}[a_{3},a_{1}]^{*}Z_{i}[a_{3},a_{2}]$
$\displaystyle\qquad+\delta_{a_{1}=a_{2}}\left(1+t_{i}\sum_{a_{3},a_{4}}Z_{i}[a_{3},a_{4}]Z_{i}[a_{3},a_{4}]^{*}\right)\Bigg{)}\
,$ (14)

where $\delta_{a_{1}=a_{2}}=0$ when $a_{1}\neq a_{2}$ and
$\delta_{a_{1}=a_{2}}=1$ otherwise. Note that when the variables
$Z_{i}[a_{1},a_{2}]$ are replaced with operators on $\mathcal{H}$, then ${\bf
P}^{(m)}({\bf Z})$ is a Hermitian operator of $A\otimes\mathcal{H}$. Then for
any density operator $\rho_{AB}$ on $A\otimes B$, we have

$E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}=\inf\mathrm{tr}\left(\rho_{A\mathcal{H}}\frac{{\bf
P}^{(m)}({\bf Z})+{\bf P}^{(m)}({\bf Y})}{2}\right)$ (15)

where the infimum is over finite-dimensional Hilbert spaces $\mathcal{H}$,
extensions $\rho_{AB\mathcal{H}}$ of the state $\rho_{AB}$ and two mutually
commuting families of operators ${\bf Z}$ and ${\bf Y}$ on $\mathcal{H}$.

###### Proof.

We start from the expression in Definition (2.3) with $f=-\frac{r_{m}}{\ln 2}$
and consider a fixed extension $\rho_{ABDE}$.

We have

$\displaystyle D_{-\frac{r_{m}}{\ln 2}}(\rho_{AD}\|I_{A}\otimes\rho_{D})$
$\displaystyle=\frac{1}{\ln
2}\sum_{i=1}^{m}w_{i}D_{-f_{t_{i}}}(\rho_{AD}\|I_{A}\otimes\rho_{D})$
$\displaystyle=-\sum_{i=1}^{m}\frac{w_{i}}{t_{i}\ln
2}\inf_{Z_{i}}\mathrm{tr}(\rho_{AD}(I+Z_{i}+Z_{i}^{*}))+\mathrm{tr}\left(\rho_{AD}\left((1-t_{i})Z_{i}^{*}Z_{i}+t_{i}I_{A}\otimes\mathrm{tr}_{A}Z_{i}Z_{i}^{*}\right)\right)\
,$

where $Z_{i}$ is a bounded operator on $A\otimes D$ and we used Theorem 3.1.
We can write such operators in the form
$Z_{i}=\sum_{a_{1},a_{2}}|a_{1}\rangle\\!\langle a_{2}|\otimes
Z_{i}[a_{1},a_{2}]$. With this notation, we have

$\displaystyle Z_{i}^{*}$
$\displaystyle=\sum_{a_{1},a_{2}}|a_{1}\rangle\\!\langle a_{2}|\otimes
Z_{i}[a_{2},a_{1}]^{*}$ $\displaystyle Z_{i}^{*}Z_{i}$
$\displaystyle=\left(\sum_{a_{1},a_{2}}|a_{1}\rangle\\!\langle a_{2}|\otimes
Z_{i}[a_{2},a_{1}]^{*}\right)\left(\sum_{a_{3},a_{4}}|a_{3}\rangle\\!\langle
a_{4}|\otimes
Z_{i}[a_{3},a_{4}]\right)=\sum_{a_{1},a_{4}}|a_{1}\rangle\\!\langle
a_{4}|\otimes\sum_{a_{2}}Z_{i}[a_{2},a_{1}]^{*}Z_{i}[a_{2},a_{4}]$
$\displaystyle Z_{i}Z_{i}^{*}$
$\displaystyle=\sum_{a_{1},a_{4}}|a_{1}\rangle\\!\langle
a_{4}|\otimes\sum_{a_{2}}Z_{i}[a_{1},a_{2}]Z_{i}[a_{4},a_{2}]^{*}\ .$

Writing $\rho_{AD}=\sum_{a_{1},a_{2}}|a_{1}\rangle\\!\langle
a_{2}|\otimes\rho[a_{1},a_{2}]$ we get

$\displaystyle-D_{-\frac{r_{m}}{\ln 2}}(\rho_{AD}\|I_{A}\otimes\rho_{D})$
$\displaystyle=\inf_{\bf
Z}\sum_{a_{1},a_{2}}\mathrm{tr}\left(\rho[a_{1},a_{2}]\sum_{i=1}^{m}P^{i}_{a_{1},a_{2}}({\bf
Z})\right)$ $\displaystyle=\inf_{\bf Z}\mathrm{tr}\left(\rho_{AD}{\bf
P}^{(m)}({\bf Z})\right)\ ,$

where the infimum is over all operators ${\bf
Z}=\\{Z_{i}[a_{1},a_{2}],Z_{i}[a_{1},a_{2}]^{*}\\}_{i,a_{1},a_{2}}$ acting on
$D$ and the noncommutative polynomials $P^{i}_{a_{1},a_{2}}({\bf Z})$ are
defined by (14).

Proceeding in the same way for the term $-D_{-\frac{r_{m}}{\ln
2}}(\rho_{AE}\|I_{A}\otimes\rho_{E})$, we can write for a given extension
$\rho_{ABDE}$:

$\displaystyle-D_{-\frac{r_{m}}{\ln
2}}(\rho_{AD}\|I_{A}\otimes\rho_{D})-D_{-\frac{r_{m}}{\ln
2}}(\rho_{AE}\|I_{A}\otimes\rho_{E})$ $\displaystyle=\inf_{{\bf Z},{\bf
Y}}\mathrm{tr}(\rho_{ADE}({\bf P}^{(m)}({\bf Z})\otimes I_{E}+{\bf
P}^{(m)}({\bf Y})\otimes I_{D}))\ ,$

where ${\bf Z}$ and ${\bf Y}$ are families of operators acting on $D$ and $E$
respectively. Taking the infimum over all extensions $\rho_{ABDE}$, we obtain

$\displaystyle
E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}=\inf\mathrm{tr}\left(\rho_{ADE}\frac{{\bf
P}^{(m)}({\bf Z})\otimes I_{E}+{\bf P}^{(m)}({\bf Y})\otimes I_{D}}{2}\right)\
,$

where the infimum is over all choices of finite-dimensional spaces $D$ and $E$
and operators ${\bf Z}$ and ${\bf Y}$ on $D$ and $E$ respectively. Note that
the operators $Z_{i}[a_{1},a_{2}]\otimes I_{E}$ and $I_{D}\otimes
Y_{j}[a_{3},a_{4}]$ act on the tensor product space $\mathcal{H}=D\otimes E$
and commute. Thus, we clearly obtain a lower bound by taking the infimum over
finite-dimensional Hilbert spaces $\mathcal{H}$ and now both collections ${\bf
Z}$ and ${\bf Y}$ act on $\mathcal{H}$ and they mutually commute. This shows
the inequality $\geq$ of equation (15). Equality then follows from the now
well-known fact that two commuting families of operators on a finite-
dimensional Hilbert space can be mapped isomorphically to operators acting on
different tensor factors of a tensor product of two finite-dimensional Hilbert
spaces, see [SW08] for a proof. ∎

###  3.1 Semidefinite relaxation

The expression in (15) is a noncommutative polynomial optimization problem and
thus one can use semidefinite programming hierarchies for it [PNA10]. We now
explain how this can be done. Let $\mathcal{R}$ be the ring of noncommutative
polynomials in the $4md_{A}^{2}$ variables

$\\{{\bf Y},{\bf
Z}\\}=\\{Y_{i}[a_{1},a_{2}],Y_{i}[a_{1},a_{2}]^{*},Z_{i}[a_{1},a_{2}],Z_{i}[a_{1},a_{2}]^{*},\;\;i\in[m],\;\;a_{1},a_{2}\in[d_{A}]\\}\,,$

modulo the commutation relations

$\displaystyle\Bigl{[}Y_{i}[a_{1},a_{2}]\,,\,Z_{i^{\prime}}[a^{\prime}_{1},a^{\prime}_{2}]\Bigr{]}=\Bigl{[}Y_{i}[a_{1},a_{2}]\,,\,Z_{i^{\prime}}[a^{\prime}_{1},a^{\prime}_{2}]^{*}\Bigr{]}=0\qquad\forall
i,i^{\prime}\in[m],\forall
a_{1},a_{2},a^{\prime}_{1},a^{\prime}_{2}\in[d_{A}].$ (16)

For any integer $\nu$, we let $\mathcal{R}^{\nu\times\nu}$ be the ring of
$\nu\times\nu$ matrices where each entry is an element of $\mathcal{R}$, i.e.,
$\mathcal{R}^{\nu\times\nu}$ is the ring of $\nu\times\nu$ matrix-valued
polynomials in the variables $\\{{\bf Y},{\bf Z}\\}$. In the notations
established earlier, note that $\mathbf{P}^{(m)}({\bf Z})$ and
$\mathbf{P}^{(m)}({\bf Y})$ are both elements of $\mathcal{R}^{d_{A}\times
d_{A}}$.

We can express the optimization problem (15) in the following form:

$\begin{array}[]{cl}\displaystyle\inf_{L:\mathcal{R}^{d_{A}d_{B}\times
d_{A}d_{B}}\to\mathbb{C}}&\frac{1}{2}L\left(\mathbf{P}^{(m)}({\bf Z})\otimes
I_{B}+\mathbf{P}^{(m)}({\bf Y})\otimes I_{B}\right)\\\
\text{s.t.}&L(|a_{1}b_{1}\rangle\\!\langle a_{2}b_{2}|\otimes 1)=\langle
a_{1}b_{1}|\rho_{AB}|a_{2}b_{2}\rangle\;\;\forall
a_{1},a_{2}\in[d_{A}],b_{1},b_{2}\in[d_{B}]\end{array}$ (17)

where the optimization is over all linear forms
$L:\mathcal{R}^{d_{A}d_{B}\times d_{A}d_{B}}\to\mathbb{C}$ that can be written
as

$\begin{aligned}
L(\mathbf{P})&=\mathrm{tr}{\;\sigma_{AB\mathcal{H}}\;\mathbf{P}(\underline{{\bf
Z}},\underline{{\bf
Y}})}\end{aligned}\qquad\forall\mathbf{P}\in\mathcal{R}^{d_{A}d_{B}\times
d_{A}d_{B}}$ (18)

where $\mathcal{H}$ is a finite-dimensional Hilbert space,
$\sigma_{AB\mathcal{H}}$ is any state on $AB\mathcal{H}$, and $\underline{{\bf
Z}},\underline{{\bf Y}}$ are operators on $\mathcal{H}$. The linear
constraints on $L$ in (17) ensure that the state $\sigma_{AB\mathcal{H}}$ is
an extension of $\rho_{AB}$ (indeed, the notation
$|a_{1}b_{1}\rangle\\!\langle a_{2}b_{2}|\otimes 1$ is the element of
$\mathcal{R}^{d_{A}d_{B}\times d_{A}d_{B}}$ that has an identity in position
$(a_{1}b_{1},a_{2}b_{2})$ and zeros elsewhere).

It is hard in general to require that $L$ has the form (18). The $k$’th
semidefinite relaxation of (17) is obtained by imposing instead the necessary
condition

$L(\boldsymbol{p}\boldsymbol{p}^{*})\geq
0\;\;\forall\boldsymbol{p}\in\mathcal{R}_{k}^{d_{A}d_{B}\times 1}$ (19)

where $\mathcal{R}_{k}$ is the space of degree $k$ polynomials in
$\mathcal{R}$. The constraint (19) can be encoded as a positive semidefinite
constraint on a matrix of size $\dim\mathcal{R}_{k}^{d_{A}d_{B}}$.

###  3.2 Numerical implementation

Figure 1 shows the result of using the first level ($k=1$) of the semidefinite
relaxation on the $2\times 2$ and $3\times 3$ Werner states. Recall that
Werner states [Wer89] in dimension $d=d_{A}=d_{B}$ are defined as
$\rho_{AB}=p\frac{\Pi_{\mathrm{sym}}}{\mathrm{tr}(\Pi_{\mathrm{sym}})}+(1-p)\frac{\Pi_{\mathrm{asym}}}{\mathrm{tr}(\Pi_{\mathrm{asym}})}$,
where $\Pi_{\mathrm{sym}}$ and $\Pi_{\mathrm{asym}}$ are the projectors onto
the symmetric and antisymmetric subspace of
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ respectively. Note that when
$p\geq\frac{1}{2}$, $\rho_{AB}$ is separable, so in particular the squashed
entanglement is $0$.

$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0$$0.2$$0.4$$0.6$$0.8$$1$$p$$E_{\mathrm{sq}}$$d_{A}=d_{B}=2$

$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0$$0.2$$0.4$$0.6$$0.8$$1$$p$$E_{\mathrm{sq}}$$d_{A}=d_{B}=3$

1

Figure 1:  Lower bound obtained using the first level ($k=1$) of the
semidefinite programming relaxation. The upper bound is obtained by a
heuristic optimization for computing the infimum in (6). For $d=2$, the upper
and lower bounds approximately match. However, for $d=3$, there is still a
large gap. We note that for the antisymmetric state (i.e., $p=0$), explicit
upper bounds on the squashed entanglement are given in [CSW12]: it gives
$\frac{1}{2}\log 3\approx 0.792$ for $d=3$.

##  4 Discussion

The semidefinite programs (SDPs) that we constructed for the squashed
entanglement raise a number of questions. First, does this family of SDPs
converge to the squashed entanglement? We know from Proposition 2.4 that as
$m\to\infty$, we have $E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}$ approaches
$E_{\mathrm{sq}}(A:B)_{\rho}$, but what remains unclear is whether the SDP
relaxations for $E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}$ converge for a fixed $m$.
The difficulty comes from the fact that the general convergence results in
[PNA10] give an infinite-dimensional Hilbert space but we could only prove
(15) for finite-dimensional Hilbert spaces. In other words, similar to
Tsirelson’s problem, the challenge here comes from comparing the commuting
model to the tensor product model for the specific problem defined in (15).
Another question is to use the specific structure of the polynomials in
Theorem 3.2 in order to obtain finite convergence bounds for this family of
SDPs, where one can hope to use techniques that were used to establish
noncommutative Positivestellensatz [Hel02, HKM12]. Such a convergence could
also lead to bounds on the dimension of the extension system sufficient to
approximate the squashed entanglement. Note that the NP-hardness of the
squashed entanglement [Hua14] rules out a convergence that is too fast. For
example, if $E_{\mathrm{sq}}^{(m)}(A:B)_{\rho}$ was always equal to the SDP
relaxation at level $k=1$, we could take $m$ polynomial in the dimension and
get a polyonmial-time algorithm to approximate the squashed entanglement with
additive error that is inverse polynomial in the dimension, which is ruled out
by [Hua14]. Finally, a limitation of the method we propose is that the size of
the SDPs grow quickly in practice and this makes it difficult to run except in
very small dimension. It would be interesting to find more efficient methods.

## Acknowledgements

We thank Peter Brown, Tim Netzer, Mizanur Rahaman and William Slofstra for
useful discussions. OF acknowledges funding by the European Research Council
(ERC Grant Agreement No. 851716).

## References

  * [BCY11] F. G. Brandao, M. Christandl, and J. Yard.  Faithful squashed entanglement.  Comm. Math. Phys., 306(3):805, 2011.
  * [BFF21] P. Brown, H. Fawzi, and O. Fawzi.  Device-independent lower bounds on the conditional von neumann entropy.  2021\.  arXiv:2106.13692.
  * [CEH+07] M. Christandl, A. Ekert, M. Horodecki, P. Horodecki, J. Oppenheim, and R. Renner.  Unifying classical and quantum key distillation.  In Theory of Cryptography Conference, pages 456–478. Springer, 2007\.
  * [Chr06] M. Christandl.  The Structure of Bipartite Quantum States — Insights from Group Theory and Cryptography.  PhD thesis, University of Cambridge, 2006.  arXiv:quant-ph/0604183.
  * [CSW12] M. Christandl, N. Schuch, and A. Winter.  Entanglement of the antisymmetric state.  Comm. Math. Phys., 311(2):397–422, 2012.  arXiv:0910.4151.
  * [CW04] M. Christandl and A. Winter.  “Squashed entanglement”: an additive entanglement measure.  J. Math. Phys., 45(3):829–840, 2004.
  * [DR84] P. J. Davis and P. Rabinowitz.  Methods of numerical integration.  Academic Press, 2nd edition, 1984.
  * [FSP18] H. Fawzi, J. Saunderson, and P. A. Parrilo.  Semidefinite approximations of the matrix logarithm.  Found. Comput. Math., Mar 2018.  arXiv:1705.00812.
  * [Hel02] J. W. Helton.  “Positive” noncommutative polynomials are sums of squares.  Annals of Mathematics, pages 675–694, 2002.
  * [HKM12] J. W. Helton, I. Klep, and S. McCullough.  The convex positivstellensatz in a free algebra.  Advances in Mathematics, 231(1):516–534, 2012.
  * [HM17] F. Hiai and M. Mosonyi.  Different quantum f-divergences and the reversibility of quantum operations.  Reviews in Mathematical Physics, 29(07):1750023, 2017.
  * [Hua14] Y. Huang.  Computing quantum discord is NP-complete.  New J. Phys., 16(3):033027, 2014.
  * [KW04] M. Koashi and A. Winter.  Monogamy of quantum entanglement and other correlations.  Phys. Rev. A, 69(2):022309, Feb 2004.  arXiv:quant-ph/0310037.
  * [Pet86] D. Petz.  Quasi-entropies for finite quantum systems.  Reports on mathematical physics, 23(1):57–65, 1986.
  * [PNA10] S. Pironio, M. Navascués, and A. Acín.  Convergent relaxations of polynomial optimization problems with noncommuting variables.  SIAM J. Optim., 20(5):2157–2180, 2010.
  * [Shi16] M. E. Shirokov.  Squashed entanglement in infinite dimensions.  J. Math. Phys., 57(3):032203, 2016.
  * [SW08] V. B. Scholz and R. F. Werner.  Tsirelson’s problem.  2008\.  arXiv:0812.4305.
  * [Tuc99] R. R. Tucci.  Quantum entanglement and conditional information transmission.  1999\.  arXiv:quant-ph/9909041.
  * [Tuc02] R. R. Tucci.  Entanglement of distillation and conditional mutual information.  2002\.  arXiv:quant-ph/0202144.
  * [Wer89] R. F. Werner.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model.  Phys. Rev. A, 40(8):4277, 1989.
  * [Wil16] M. M. Wilde.  Squashed entanglement and approximate private states.  Quantum Information Processing, 15(11):4563–4580, 2016.


* [49] Head, J.W., Fassett, C.I., Kadish, S.J., Smith, D.E., Zuber, M.T., Neumann, G.A., Mazarico, E.: Global distribution of large lunar craters: Implications for resurfacing and impactor populations.  Science 329(5998), 1504–1507 (2010).  DOI 10.1126/science.1195050
  * [50] Heisterkamp, D.R., Bhattacharya, P.: Invariants of families of coplanar conics and their applications to object recognition.  Journal of Mathematical Imaging and Vision 7, 253–267 (1997).  DOI 10.1023/A:1008230528693
  * [51] Herrick, R.R., Schenk, P.M., Robbins, S.J.: Surveys of elliptical crater populations on the saturnian satellites, mercury, and mars.  Icarus 220(2), 297–304 (2012).  DOI 10.1016/j.icarus.2012.05.027
  * [52] Hiesinger, H.: Ages of mare basalts on the lunar nearside.  J. Geophys. Res.: Planets 105, 29239–29275 (2000).  DOI 10.1029/2000JE001244
  * [53] Hilbert, D.: Ueber die Theorie der algebraischen Formen.  Math. Ann. 36(4), 473–534 (1890).  DOI 10.1007/BF01208503
  * [54] Hilbert, D.: Ueber die vollen Invariantensysteme.  Math. Ann. 42(3), 313–373 (1893).  DOI 10.1007/BF01444162
  * [55] HIllier, J.K., Buratti, B.J., Hill, K.: Multispectral photometry of the moon and absolute calibration of the clementine uv/vis camera.  Icarus 141, 205–225 (1999).  DOI 10.1006/icar.1999.6184
  * [56] Hosseinzadeh, M., Latif, Y., Pham, T., Suenderhauf, N., Reid, I.: Structure aware slam using quadrics and planes.  In: Asian Conference on Computer Vision (2018).  DOI 10.1007/978-3-030-20893-6˙26
  * [57] Hughes, G.B., Chraibi, M.: Calculating ellipse overlap areas.  Computing and Visualization in Science volume 15, 291–301 (2012).  DOI 10.1007/s00791-013-0214-3
  * [58] Humphreys, J.E.: Linear algebraic groups.  Springer-Verlag, New York-Heidelberg (1975).  Graduate Texts in Mathematics, No. 21
  * [59] Hungerford, T.W.: Algebra.  Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London (1974)
  * [60] Jacobs, D.W.: What makes viewpoint-invariant properties perceptually salient?  Journal of the Optical Society of America A 20(7), 1304–1320 (2003).  DOI 10.1364/JOSAA.20.001304
  * [61] Johnson, A., Villaume, N., Umsted, C., Kourchians, A., Sternberg, D., Trawny, N., Cheng, Y., Geipel, E., Montgomery, J.: The Mars 2020 Lander Vision System Field Test.  In: AAS Guidance, Navigation and Control Conference, AAS 20-105 (2020)
  * [62] Johnson, R.A.: The conic as a space element.  Transactions of the American Mathematical Society 15, 335–368 (1914).  DOI 10.2307/1988792
  * [63] Kanatani, K., Rangarajan, P.: Hyper least squares fitting of circles and ellipses.  Computational Statistics and Data Analysis 55, 2197–2208 (2011).  DOI 10.1016/j.csda.2010.12.012
  * [64] Kannala, J., Salo, M., Heikkilä, J.: Algorithms for computing a planar homography from conics in correspondence.  In: Proceedings of the British Machine Vision Conference (BVMC) (2006).  DOI 10.5244/C.20.9
  * [65] Kariya, K., Ishida, T., Sawai, S., Fukuda, S.: Position and displacement estimation using crater-based line segments for pinpoint lunar landing.  In: International Symposium on Space Flight Dynamics (2017)
  * [66] Klear, M.: Pycda: An open-source library for automated crater detection.  In: 9th Planetary Crater Consortium, 1806 (2018)
  * [67] Kordas, J.F., Lewis, I.T., Priest, R.E., White, W.T., Nielsen, D.P., Park, H., Wilson, B.A., Shannon, M.J., Ledebuhr, A.G., Pleasance, L.D.: Uv/visible camera for the clementine mission.  Proc. SPIE 2478 (1995).  DOI 10.1117/12.210943
  * [68] Lang, D., Hogg, D., Mierle, K., Blanton, M., Roweis, S.: Astrometry.net: Blind astrometric calibration of arbitrary astronomical images.  The Astronomical Journal 139, 1782–1800 (2010).  DOI 10.1088/0004-6256/139/5/1782
  * [69] Lang, S.: Algebra, _Graduate Texts in Mathematics_ , vol. 211, third edn.  Springer-Verlag, New York (2002).  DOI 10.1007/978-1-4613-0041-0
  * [70] Lawrence, S.J., et al.: The apollo digital image archive: new research and data products.  In: Proceedings of the NLSI Lunar Science Conference (2008)
  * [71] Leake, C., Arnas, D., Mortari, D.: Non-dimensional star-identification.  Sensors 20, 2697 (2020).  DOI 10.3390/s20092697
  * [72] Lenz, R., Meer, P.: Point configuration invariants under simultaneous projective and permutation transformations.  Pattern Recognition 27(11), 1523–1532 (1994).  DOI 10.1016/0031-3203(94)90130-9
  * [73] Lepetit, V., Moreno-Noguer, F., Fua, P.: EPnP: An Accurate O(n) Solution to the PnP Problem.  International Journal of Computer Vision 81, 155–166 (2009).  DOI 10.1007/s11263-008-0152-6
  * [74] Leutenegger, S., Chli, M., Siegwart, R.: Brisk: Binary robust invariant scalable keypoints.  In: IEEE International Conference on Computer Vision (ICCV) (2011).  DOI 10.1109/ICCV.2011.6126542
  * [75] Levandowski, M., Winter, D.: Distance between sets.  Nature 234, 34–35 (1971).  DOI 10.1038/234034a0
  * [76] Levine, G.: A method of orbital navigation using optical sightings to unknown landmarks.  AIAA Journal 4(11), 1928–1931 (1966).  DOI 10.2514/3.3820
  * [77] Liebe, C.C.: Star trackers for attitude determination.  IEEE Aerospace and Electronic Systems Magazine 10(6), 10–16 (1995).  DOI 10.1109/62.387971
  * [78] Liebe, C.C.: Accuracy performance of star trackers — a tutorial.  IEEE Transactions on Aerospace and Electronic Systems 38(2), 587–599 (2002).  DOI 10.1109/TAES.2002.1008988
  * [79] Lowe, D.: Distinctive image features from scale-invariant keypoints.  International Journal of Computer Vision 60(2), 91–110 (2004).  DOI 10.1023/B:VISI.0000029664.99615.94
  * [80] Lu, T., andC. Liu, W.H., Yang, D.: Relative pose estimation of a lander using crater detection and matching.  Optics Engineering 55(2), 023102 (2016).  DOI 10.1117/1.OE.55.2.023102
  * [81] Ma, S.D.: Conics-based stereo, motion estimation, and pose determination.  International Journal of Computer Vision 10(1), 7–25 (1993).  DOI 10.1007/BF01440844
  * [82] Maass, B., Woicke, S., Oliveria, W.M., Razgus, B., Krüger, H.: Crater navigation system for autonomous precision landing on the moon.  Journal of Guidance, Control, and Dynamics (2020).  DOI 10.2514/1.G004850
  * [83] Maybank, S.J.: Geometric Invariance in Computer Vision, chap. The projection of two non-coplanar conics, pp. 105–119.  MIT Press, Cambridge, MA (1992)
  * [84] McCabe, J., DeMars, K.: Anonymous feature-based terrain relative navigation.  Journal of Guidance, Control, and Dynamics 43(3), 410–421 (2020).  DOI 10.2514/1.G004423
  * [85] Melosh, H.J.: Impact Cratering: A Geologic Process, pp. 1–253.  Oxford University Press, New York, NY (1989)
  * [86] Melosh, H.J.: Planetary Surface Processes, chap. The shapes of planets and moons, pp. 25–48.  Cambridge University Press, Cambridge, UK (2011).  DOI 10.1017/CBO9780511977848.003
  * [87] Michikami, T., Hagermann, A., Morota, T., Haruyama, J., Hasegawa, S.: Oblique impact cratering experiments in brittle targets: Implications for elliptical craters on the moon.  Planetary and Space Science 135, 27–36 (2017).  DOI 10.1016/j.pss.2016.11.004
  * [88] Morel, J., Yu, G.: ASIFT: A new framework for fully affine invariant image comparison.  SIAM Journal on Imaging Science 2(2), 438–469 (2009).  DOI 10.1137/080732730
  * [89] Mortari, D., Neta, B.: k-vector range searching techniques.  In: AAS/AIAA Spaceflight Mechanics Meeting, AAS 00-128 (2000)
  * [90] Mortari, D., Rogers, J.: A k-vector approach to sampling, interpolation, and approximation.  Journal of the Astronautical Sciences 60, 686–706 (2013).  DOI 10.1007/s40295-015-0065-x
  * [91] Mortari, D., Samaan, M., Bruccoleri, C., Junkins, J.: The pyramid star identification technique.  Navigation: Journal of The Institute of Navigation 51(3), 171–183 (2004).  DOI 10.1002/j.2161-4296.2004.tb00349.x
  * [92] Mundy, J.L., Kapur, D., Maybank, S.J., Gros, P., Quan, L.: Geometric Invariance in Computer Vision, chap. Geometric Interpretation of Joint Conic Invariants, pp. 77–86.  MIT Press, Cambridge, MA (1992)
  * [93] Mundy, J.L., Zisserman, A.: Geometric Invariance in Computer Vision, chap. Introduction–towards a new framework for vision, pp. 1–39.  MIT Press, Cambridge, MA (1992)
  * [94] Nagata, M.: On the $14$-th problem of Hilbert.  Amer. J. Math. 81, 766–772 (1959).  DOI 10.2307/2372927
  * [95] Nagata, M.: Invariants of a group in an affine ring.  J. Math. Kyoto Univ. 3, 369–377 (1963/64).  DOI 10.1215/kjm/1250524787
  * [96] Naval Research Laboratory: CLEM1-LUN/EAR/SKY-ASTAR/BSTAR/UVVIS/HRES/LWIR/NIR-2-EDR-V1.0.  NASA Planetary Data System (1995)
  * [97] Neukum, G.: Meteoritenbombardement and datierung planetarer oberflächen.  Habilitation Dissertation for Faculty Membership, Univ. of Munich (1983)
  * [98] Neukum, G.: Cratering records in the inner solar system in relation to the lunar reference system.  Space Sci. Rev. 96, 55–86 (2001).  DOI 10.1023/A:1011989004263
  * [99] Neusel, M.D.: Invariant theory, _Student Mathematical Library_ , vol. 36.  American Mathematical Society, Providence, RI (2007)
  * [100] Nozette, S., et al.: The clementine mission to the moon: Scientific overview.  Science 266(5192), 1835–1839 (1994).  DOI 10.1126/science.266.5192.1835
  * [101] Olds, R., May, A., Mario, C., Hamilton, R., Debrunner, C., Anderson, K.: The application of optical based feature tracking to OSIRIS-REx asteroid sample collection.  In: AAS Guidance, Navigation, & Control Conference, AAS 15-124 (2015)
  * [102] Olver, P.J.: Classical invariant theory, _London Mathematical Society Student Texts_ , vol. 44.  Cambridge University Press, Cambridge (1999).  DOI 10.1017/CBO9780511623660
  * [103] Owen, W.: Methods of Optical Navigation.  In: AAS/AIAA Space Flight Mechanics Meeting, AAS 11-215 (2011)
  * [104] Owen, W., Duxbury, T., Acton, C., Synnott, S., Riedel, J., Bhaskaran, S.: A Brief History of Optical Navigation at JPL.  In: AAS Guidance and Control Conference, AAS 08-053. Breckenridge, CO (2008)
  * [105] Park, W., Jung, Y., Bang, H., Ahn, J.: Robust crater triangle matching algorithm for planetary landing navigation.  Journal of Guidance, Control, and Dynamics 42(2), 402–410 (2019).  DOI 10.2514/1.G003400
  * [106] Parkes, S., Dunstan, M., Martin, I., McCrum, M., Dubois-Matra, O.: Testing advanced navigation systems for planetary landers and rovers.  In: International Astronautical Congress, IAC-09-A3.1.6 (2009)
  * [107] Povilaitis, R.Z., RObinson, M.S., van der Bogert, C.H., Hiesinger, H., Meyer, H.M., Ostrach, L.R.: Crater density differences: Exploring regional resurfacing, secondary crater populations, and crater saturation equilibrium on the moon.  Planetary and Space Science 162, 41–51 (2018).  DOI 10.1016/j.pss.2017.05.006
  * [108] Prasad, D.K., Leung, M.K.H.: Clustering of ellipses based on their distinctiveness: An aid to ellipse detection algorithms.  In: 3rd International Conference on Computer Science and Information Technology (2010).  DOI 10.1109/ICCSIT.2010.5564932
  * [109] Quan, L.: Algebraic and geometric invariant of a pair of non-coplanar conics in space.  Journal of Mathematical Imaging and Vision 5, 263–267 (1995).  DOI 10.1007/BF01248375
  * [110] Quan, L., Gros, P., Mohr, R.: Invariants of a pair of conics revisited.  Image and Vision Computing 10(5), 319–323 (1992).  DOI 10.1016/0262-8856(92)90049-9
  * [111] Quan, L., Veillon, F.: Joint invariants of a triplet of coplanar conics: Stability and discriminating power for object recognition.  Computer Vision and Image Understanding 70(1), 111–119 (1998).  DOI 10.1006/cviu.1998.0617
  * [112] Richter-Gebert, J.: Perspective on Projective Geometry: A Guided Tour Through Real and Complex Geometry.  Springer-Verlag, Berlin (2011)
  * [113] Rijlaarsdam, D., Yous, H., Byrne, J., Oddenino, D., Furano, G., Moloney, D.: A survey of lost-in-space star identification algorithms since 2009.  Sensors 20, 2579 (2020).  DOI 10.3390/s20092579
  * [114] Robbins, S.: A new global database of lunar impact craters ¿1–2 km: 1. crater locations and sizes, comparisons with published databases, and global analysis.  Journal of Geophysical Research: Planets 124, 871–892 (2018).  DOI 10.1029/2018JE005592
  * [115] Robbins, S.J., Riggs, J.D., Weaver, B.P., Bierhaus, E.B., Chapman, C.R., Kirchoff, M.R., Singer, K.N., Gaddis, L.R.: Revised recommended methods for analyzing crater size-frequency distributions.  Meteoritics & Planetary Science 53(4), 891–931 (2018).  DOI 10.1111/maps.12990
  * [116] Robinson, M.S., et al.: The apollo digital image archive.  In: Proceedings of the Lunar and Planetary Science Conference (LPSC) XXXIX (2008)
  * [117] Rosenlicht, M.: Some basic theorems on algebraic groups.  Amer. J. Math. 78, 401–443 (1956).  DOI 10.2307/2372523
  * [118] Rothwell, C.A., Zisserman, A., Forsyth, D.A., Mundy, J.L.: Planar object recognition using projective shape representation.  International Journal of Computer Vision 16, 57–99 (1995).  DOI 10.1007/BF01428193
  * [119] Rublee, E., Rabaud, V., Konolige, K., Bradski, G.: Orb: An efficient alternative to sift or surf.  In: IEEE International Conference on Computer Vision (ICCV) (2011).  DOI 10.1109/ICCV.2011.6126544
  * [120] Salamunićcar, G., Lončarić, S.: Open framework for objective evaluation of crater detection algorithms with first test-field subsystem based on mola data.  Advances in Space Research 42(1), 6–19 (2008).  DOI 10.1016/j.asr.2007.04.028
  * [121] Salamunićcar, G., Lončarić, S., Mazarico, E.: Lu60645gt and ma132843gt catalogues of lunar and martian impact craters developed using a crater shape-based interpolation crater detection algorithm for topography data.  Planetary and Space Science 60, 236–247 (2012).  DOI 10.1016/j.pss.2011.09.003
  * [122] Samaan, M.A., Mortari, D., Junkins, J.L.: Nondimensional star identification for uncalibrated star cameras.  The Journal of the Astronautical Sciences 54(1), 95–111 (2006).  DOI 10.1007/BF03256478
  * [123] Semple, J.G., Kneebone, G.T.: Algebraic Projective Geometry, pp. 191–196.  Oxford University Press, Oxford, UK (1952)
  * [124] Shiu, Y.C., Ahmad, S.: 3d location of circular and spherical features by monocular model-based vision.  In: IEEE International Conference on Systems, Man and Cybernetics (1989).  DOI 10.1109/ICSMC.1989.71362
  * [125] Shoemaker, E.M.: Physics and Astronomy of the Moon, chap. Interpretation of Lunar Craters, pp. 283–360.  Academic Press, New York, NY (1962)
  * [126] Silburt, A., Alo-Dib, M., Zhu, C., Jackson, A., Valencia, D., Kissin, Y., Tamayo, D., Menou, K.: Lunar crater identification via deep learning.  Icarus 317, 27–38 (2019).  DOI 10.1016/j.icarus.2018.06.022
  * [127] Solarna, D., Gotelli, A., Le Moigne, J., Moser, G., Serpico, S.: Crater detection and registration of planetary images through marked point processes, multiscale decomposition, and region-based analysis.  IEEE Geoscience and Remote Sensing Letters (2020).  DOI 10.1109/TGRS.2020.2970908
  * [128] Speyerer, E.: Quantifying crater production and regolith overturn on the moon with temporal imaging.  Nature 538, 215–218 (2016).  DOI 10.1038/nature19829
  * [129] Spratling, B., Moratri, D.: A survey on star identification algorithms.  Algorithms 2, 93–107 (2009).  DOI 10.3390/a2010093
  * [130] Sugimoto, A.: A linear algorithm for computing the homography from conics in correspondence.  Journal of Mathematical Imaging and Vision 13, 115–130 (2000).  DOI 10.1023/A:1026571913893
  * [131] Świrski, L., Bulling, A., Dodgson, N.: Robust real-time pupil tracking in highly off-axis images.  In: Symposium on Eye Tracking Research and Applications (2012).  DOI 10.1145/2168556.2168585
  * [132] Toda, F., Schlee, F.: Autonomous orbital navigation by optical tracking of unknown landmarks.  Journal of Spacecraft and Rockets 4(12), 1644–1648 (1966).  DOI 10.2514/3.29147
  * [133] Troglio, G., Le Moigne, J., Atli Benediktsson, J., Moser, G., Serpico, S.: Automatic extraction of ellipsoidal features for planetary image registration.  IEEE Geoscience and Remote Sensing Letters 9(1), 95–99 (2012).  DOI 10.1109/LGRS.2011.2161263
  * [134] Wang, H., Jiang, J., Zhang, G.: Crateridnet: An end-to-end fully convolutional neural network for crater detection and identification in remotely sensed planetary images.  Remote Sensing 10, 1067 (2018).  DOI 10.3390/rs10071067
  * [135] Wang, J., Cheng, W., Zhou, C.: A chang’e-1 global catalog of lunar impact craters.  Planetary and Space Science 112, 42–45 (2015).  DOI 10.1016/j.pss.2015.04.012
  * [136] Wang, W., Zhang, X., Qi, Y., Cheng, W.: An algorithm of pose estimation based on conic correspondences.  In: IEEE International Conference on Robotics and Biomimetics (2017).  DOI 10.1109/ROBIO.2017.8324436
  * [137] Woicke, S., Moreno Gonzalez, A.S., El-Hajj, I., Mes, J.W.F., Henkel, M., Autar, R.S.D., Klavers, R.A.: Comparison of crater-detection algorithms for terrain-relative navigation.  In: AIAA Guidance, Navigation, and Control Conference (2020).  DOI 10.2514/6.2018-1601
  * [138] Wu, S., Moore, H.: Experimental photogrammetry of lunar images.  In: Geological Survey Professional Paper 1046-D. U.S. Government Printing Office (1980).  DOI 10.3133/pp1046D
  * [139] Yang, S., Baum, M., Granström, K.: Metrics for performance evaluation of elliptic extended object tracking methods.  In: IEEE lnternational Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI) (2016).  DOI 10.1109/MFI.2016.7849541
  * [140] Yu, M., Cui, H., Tian, Y.: A new approach based on crater detection and matching for visual navigation in planetary landing.  Advances in Space Research 53, 1810–1821 (2014).  DOI 10.1016/j.asr.2013.04.011
  * [141] Zanetti, R.: Rotations, transformations, left quaternions, right quaternions?  The Journal of the Astronautical Sciences 66, 361–381 (2019).  DOI 10.1007/s40295-018-00151-2
  * [142] Zisserman, A., Forsyth, D., Mundy, J., Rothwell, C., Liu, J., Pillow, N.: 3d object recognition using invariance.  Artificial Intelligence 78, 239–288 (1995).  DOI 10.1016/0004-3702(95)00023-2


#  Coexistence of insulating phases in confined
fermionic chains with a Wannier-Stark potential

N. Aucar Boidi  Centro Atómico Bariloche, Instituto Balseiro, 8400 Bariloche,
Argentina<EMAIL_ADDRESS>K. Hallberg  Centro Atómico Bariloche,
Instituto Balseiro, 8400 Bariloche, Argentina  Instituto de Nanociencia y
Nanotecnología CNEA-CONICET, 8400 Bariloche, Argentina
<EMAIL_ADDRESS>Amnon Aharony  School of Physics and Astronomy, Tel
Aviv University, Tel Aviv 6997801, Israel<EMAIL_ADDRESS>Ora Entin-
Wohlman  School of Physics and Astronomy, Tel Aviv University, Tel Aviv
6997801, Israel<EMAIL_ADDRESS>Centro Atómico Bariloche, Instituto
Balseiro, 8400 Bariloche, Argentina<EMAIL_ADDRESS>Centro Atómico
Bariloche, Instituto Balseiro, 8400 Bariloche, Argentina  Instituto de
Nanociencia y Nanotecnología CNEA-CONICET, 8400 Bariloche, Argentina
<EMAIL_ADDRESS>School of Physics and Astronomy, Tel Aviv University,
Tel Aviv 6997801, Israel<EMAIL_ADDRESS>School of Physics and
Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel<EMAIL_ADDRESS>
###### Abstract

We study fermions on a finite chain, interacting repulsively when residing on
the same and on nearest-neighbor sites, and subjected to a Wannier-Stark
linearly-varying potential. Using the density matrix renormalization-group
numerical technique to solve this generalized extended Hubbard model, the
ground state exhibits a staircase of (quasi) plateaus in the average local
site density along the chain, decreasing from being doubly-filled to empty as
the potential increases. These ‘plateaus’ represent locked-in commensurate
phases of charge density waves together with band and Mott insulators. These
phases are separated by incompressible regions with incommensurate fillings.
It is suggested that experimental variations of the slope of the potential and
of the range of the repulsive interactions will produce such a coexistence of
phases which have been individually expected theoretically and observed
experimentally for uniform systems.

Introduction.— The complexity of quantum many-body systems originates from the
interplay of strong interactions, quantum statistics, and the large number of
quantum-mechanical degrees of freedom. This interplay generates a multitude of
phases, e.g., insulating commensurate charge (CDW) and spin (SDW) density
waves and compressible (metallic) phases. This complexity already shows up in
one dimension, in which one can use (and test) a variety of theoretical and
experimental tools for their study. The simplest picture for interacting
particles in one dimension (1D) is given by the Hubbard Hamiltonian, which
includes interactions, $U$, only between particles residing on the same
lattice site [1]. This interaction competes with the kinetic [nearest-neighbor
(nn) tunneling] energy, $t$, resulting for instance, in antiferromagnetic
structures [2].

However, this simple Hamiltonian cannot reproduce certain phases, like charge
density-waves. Those are generated e.g. by the extended Hubbard model, which
also includes nn interactions, $V$. Its one-dimensional version reveals a rich
phase diagram, which includes the band and Mott insulating phase [3], SDW and
CDW and metallic phases [4, 6, 7, 5]. It has also been used to describe data
collected in experiments performed on chains of cold atoms [8, 9]. In higher
dimensions, it has been used to describe bulk and edge states in electronic
insulators [10].

An exact analytic solution of the extended Hubbard Hamiltonian, in particular
on a finite chain (the system amenable to cold-atom experiments), has not yet
been found. It has been studied by a variety of numerical and approximate
methods (e.g., Refs. [11, 12, 13, 14, 15, 16]), emphasizing the half-filled
case, where one finds (for fermions in 1D), the insulating Mott
antiferromagnetic phase [3] and CDW phases.

Experiments on cold-atom arrays naturally involve finite samples. Numerical
calculations performed on such systems used various boundary conditions: hard
walls, periodic and open boundaries, or potentials representing confining
harmonic traps [9, 17, 18]. These works concentrate mostly on the region
around the ‘center’ of the confined structure, whose details are usually not
sensitive to the particular form of the boundaries, and so its possible
structures are determined by $U,~{}V$ and particle density $n$. Remarkably,
experiments (e.g., on cold atoms) have observed some of the theoretically
predicted phases [19, 20]. Less attention has been paid to the structures near
the ‘edges’ of the samples and to their dependence on the details of the
boundary conditions, in particular when the confinement is achieved by varying
site energies. Such a confining scheme has been recently considered, using the
self-consistent Hartree-Fock approximation, for the two-dimensional extended
Hubbard Hamiltonian, and found coexistence of various structures (phases) near
the free ends of the samples [10].

In this Letter we generalize the extended Hubbard Hamiltonian to a 1D
fermionic chain, confined by a linear potential, which mimics either edge
configurations in bulk systems or cold-atom arrays placed in an electric
field. Such a potential can be produced by a longitudinal electric field, as
in the Wannier-Stark model [21]. HERE

Given the complex nature of the many-body problem associated with our system,
we resort to one of the most accurate numerical methods for correlated
systems, the density matrix renormalization-group (DMRG) [22, 23, 24, 25, 26,
27], which uses quantum information to keep the most relevant states. As we
show, the linear potential generates in the ground state the simultaneous
existence of segments in which different phases coexist, each of which having
been observed separately before, on long uniform chains. Our results are
presented by plots of the local quantum-averaged density on the sites $i$ on
the chain, $\langle n_{i}\rangle$, the nn density-density correlations
$\langle n_{i}n_{i+1}\rangle$ and the nn spin-spin correlations $\langle
s^{z}_{i}s^{z}_{i+1}\rangle$, (e.g., Fig. 2). Instead of a smooth decrease,
the local average of $\langle n_{i}\rangle$ shows flat steps, corresponding to
locked-in Mott or CDW structures (e.g., $212121\dots$, $101010\dots$, [28]).
These locked-in steps are similar to those observed for commensurate wave
vectors in the devil’s staircase [29, 30]. Between these steps, $\langle
n_{i}\rangle$ decreases more smoothly, representing incommensurate regions,
which can be thought of as ‘domain walls’ with varying lengths [31]. As shown
below, the local density of states on these intermediate sites exhibits small
energy gaps, which imply that they are incompressible (insulating), in spite
of having incommensurate fillings. We will refer to them hereafter as
incompressible incommensurate-filling phases ‘IIF’. The specific sequence of
phases, and their sizes, can be modified experimentally, e.g., by changing the
slope of the potential. Neighboring structures in a sequence are often also
neighboring in the phase diagrams found for uniform systems (which are not
subjected to the linear potential).

Model.— We study the generalized 1D extended Hubbard Hamiltonian

$\displaystyle{\cal H}=$
$\displaystyle-t\sum_{i,\sigma}\big{(}c^{\dagger}_{i,\sigma}c_{i+1,\sigma}+{\rm
h.c.}\big{)}+\sum_{i}(\mu_{i}-\mu)n_{i}$
$\displaystyle+U\sum_{i}\big{(}n_{i,\uparrow}-1/2\big{)}\big{(}n_{i,\downarrow}-1/2\big{)}$
$\displaystyle+V\sum_{i}\big{(}n_{i}-1\big{)}\big{(}n_{i+1}-1\big{)}\ ,$ (1)

where $i$ is the site index, $i=0,\dots,L-1$ (we consider an odd number of
sites without loss of generality). Here, $\mu$ is the fixed external chemical
potential, $c^{\dagger}_{i,\sigma}$ creates an electron with spin
$\sigma(=\uparrow,\downarrow)$ at site $i$,
$n_{i,\sigma}=c^{\dagger}_{i,\sigma}c_{i,\sigma}$,
$n_{i}=n_{i,\uparrow}+n_{i,\downarrow}$, while $U$ and $V$ are the repulsive
interactions between electrons on the same and nn sites, respectively (see
Fig. 1). The site-dependent local energy (the Wannier-Stark potential)
$\mu_{i}$ describes a linear external potential,

$\displaystyle\mu_{i}=\mu_{0}[i/i_{c}-1]\ .$ (2)

The site $i_{c}=(L-1)/2$ represents the center of the ‘edge’, where
$\mu_{i_{c}}=0$. The particular form of ${\cal H}$ was chosen so that at
$\mu=0$ (up to a constant energy) it is particle-hole symmetric when
$i\rightarrow L-1-i$ and $n_{i}\rightarrow 2-n_{i}$. In that case we always
have $n_{i_{c}}=1$.

For an infinite chain, $\mu_{i}$ is large and negative at large and negative
$i$, and therefore we expect all the sites there to be filled,
i.e.,$n_{i}=n_{i,\uparrow}+n_{i,\downarrow}=2$. Similarly, $\mu_{i}$ is large
and positive at large and positive $i$, and therefore we expect all the sites
there to be empty, i.e. $n_{i}=0$, as drawn in Fig. 1. For a finite chain, as
we use here, this is still expected for a large slope, $\mu_{0}\gg 1$, when
the whole ‘edge’ between the fully-occupied and empty ‘phases’ is confined
within the chain. Indeed, this is confirmed by our calculations. However, the
‘end’ trivial phases disappear for small slopes, for which the observed
structures depend on the open boundaries.

Figure 1: Schematic representation of the system considered, for $L=9$ sites.

Results.— Unless otherwise stated, we use $U/t\rightarrow U=10$, $\mu=0$ and
$L=41$. All energies are measured in units of $t$. The Hamiltonian is
diagonalized exploiting the DMRG technique, with around $m=500$ states and 4
to 6 finite-size sweeps, which leads to a precision of around $10^{-10}$ in
the energy. For a very steep potential ($\mu_{0}\rightarrow\infty$) we obtain
only two coexisting ‘phases’: a completely filled band ($n_{i}=2$) up to the
center point $i_{c}$, and completely empty sites ($n_{i}=0$) above that point,
as expected. Both regions are incompressible and insulating. As the slope
$\mu_{0}$ decreases (but remains large), these two ‘phases’ remain near the
two ends of the system, but new structures (‘phases’) appear between them, in
which $\langle n_{i}\rangle$ decreases gradually from $2$ to $0$. Figure 2
presents typical results, for three values of $V$. Note the electron-hole
symmetry between the two sides of Figs. 2(a-c), which follows directly from
Eq. (1) at $\mu=0$.

Figure 2: (color online) (a)-(c): The local density $\langle n_{i}\rangle$;
(d)-(f): the nn density-density correlations $\langle n_{i}n_{i+1}\rangle$;
(g)-(i): the nn spin-spin correlations $\langle s^{z}_{i}s^{z}_{i+1}\rangle$,
for $V=0,3,6$ and different values of $\mu_{0}$. The black diamonds in (c)
indicate the mean value between neighboring sites.

For $V=0$ (i.e., the simplest Hubbard Hamiltonian, left column in Fig. 2), the
system shows the following phases: for large (but finite) values of $\mu_{0}$
it is a band insulator at both extremes, completely filled on the left and
completely empty on the right. In the region located symmetrically around the
center point $i_{c}$, we find a Mott-insulating state (one particle per site,
$\langle n_{i}\rangle=1$), and an antiferromagnetic spin-spin correlation
function, Fig. 2(g). As seen in this figure, the spin correlation function,
$\langle s^{z}_{i}s^{z}_{i+1}\rangle\simeq-0.14$ (note:
$s_{i}^{z}\equiv(n_{i,\uparrow}-n_{i,\downarrow})/2$, the $z-$direction is
arbitrarily chosen), agrees with its value of the infinite Mott phase [23].
The three insulating commensurate phases are separated by IIF regions with
very small but finite gaps, see Fig. 3. These regions differ from the
compressible regions found in Ref. [10], possibly because Ref. [10] explores
2D systems using the mean-field approximation. As $\mu_{0}$ decreases, the
band insulating phases on both ends disappear and the Mott region grows, as
estimated below. These results are also consistent with the behavior of the
density-density correlations, which vary between $4$ on the left, via $1$ in
the Mott phase, to $0$ on the right, Fig. 2(d).

For $V=3$ (middle column in Fig. 2) the above three insulating ‘phases’ are
supplemented by two regions with an incipient (doped) CDW order on the two
sides of the Mott ‘phase’, with local mean fillings ‘quasi-plateaus’ around
$\overline{\langle n_{i}\rangle}\simeq 1.5$ and $\overline{\langle
n_{i}\rangle}\simeq 0.5$ (quarter filling of holes and of electrons,
respectively). The bar indicates a local average over a few sites. Unlike the
uniform case $\mu_{i}=0$, the local average fillings in these regions are not
exactly $1.5$ and $0.5$. Rather, they can be fitted by $\langle
n_{i}\rangle=A-Bi+C\cos(i\pi)$ (note that $i$ is the site number!). The
oscillating term corresponds to a CDW, with a wave vector $q=\pi$ (our lattice
constant is $1$) and structures $212121\dots$ or $101010\dots$ [28]. However,
the term $-Bi$ represents a linear decrease of the actual average, presumably
in response to the linear potential. Without this linear ‘background’, such a
CDW is consistent with the results of the density-density and spin-spin
correlations and with previous results for the doped (non-half-filled) 1D
extended Hubbard model [32] in a uniform potential, $\mu_{i}=0$, for which
there is a transition from a Tomonaga-Luttinger liquid to a CDW phase for
intermediate values of $2t\leq V<U/2$ and large values of $U$ ($U\gg t$). In
those cases this CDW phase is insulating and incompressible. As we discuss
below, we also find that, in spite of the varying average local densities, the
local density of states has a (small) gap at the Fermi energy, which is
consistent with an incompressible state. As before, when $\mu_{0}$ decreases,
the Mott region grows, the incipient CDW regions move towards the boundaries
and the band-insulating regions disappear.

For $V=6$ (right column in Fig. 2) the Mott region disappears and is replaced
by a half-filled CDW, $202020\dots$. For large $\mu_{0}$’s this phase exists
in the center and coexists with doped CDW’s at both sides, with fillings
$\overline{\langle n_{i}\rangle}\simeq 1.5$ and $\overline{\langle
n_{i}\rangle}\simeq 0.5$ respectively (black diamonds in Fig. 2(c)). This
coexistence of two different CDW’s has not been seen before and constitutes a
situation which could be observed in cold-atom experiments. As before, the
doped CDW’s are accompanied by a very small gradual decrease of the local
average occupation -‘quasi-plateaus’, presumably due to the slope in the
potential. When $\mu_{0}$ is lowered, the half-filled CDW occupies the whole
chain. This is expected, since it is well known that when $V>U/2$ and for a
half-filled system, the uniform chain undergoes a transition from a Mott phase
to a CDW [7, 32]. The results are consistent with the behavior of the density-
density and spin-spin correlations. It is interesting to see a finite value of
the spin-spin correlations at the phase boundaries between the half-filled and
doped CDW’s. It is also interesting to see that for $V=3$ the average
occupation $\overline{\langle n_{i}\rangle}$, and the amplitude of the
incipient CDW decrease gradually towards the central Mott or CDW region, but
this decrease becomes abrupt for $V=6$. The width of the IIF region (domain
wall) between the two CDW phases seems to shrink to zero above some ‘critical’
value of $V$.

The above results exhibited ‘plateaus’ only for $1/2$, $1/4$ and $3/4$
fillings. We expect similar ‘plateaus’, corresponding to other simple
fraction, e.g., $1/8$. However, to see these one would need a much larger
number of sites, and this is not possible with our present computer
capabilities. Note, though, that calculations with a smaller number of sites
do still show similar steps for these commensurate fillings.

Figure 3: (color online) Top: Local density profile $\langle n_{i}\rangle$
showing the sites where the local density of states ( LDOS) has been
calculated, for $V=0$ ($\mu_{0}=10$), $V=3$ ($\mu_{0}=16$) and $V=6$
($\mu_{0}=20$). Bottom: LDOS showing gaps at the Fermi energy (at
$\omega=E_{F}=0$) for all cases, using $\eta=0.01$ (Eq. S1 in [33]) .

Local Density of States.— To further explore the different phases, we have
calculated the local, site dependent, density of states (LDOS), using the
lesser and greater Green’s functions, see details in Ref. [33]. In Fig. 3 we
show the LDOS for particular sites of the chain for different parameters. We
observe that there is always a gap at $E_{F}=0$, even for the partially filled
sites (we have added the filling profile for comparison). The gaps
corresponding to these sites are smaller than the corresponding gaps of the
fully formed CDW (see the $V=6$ case) and much smaller than those of the Mott
region (see Fig. 4). These gaps indicate that these regions are incompressible
(non-metallic). This is not a finite size effect (since we would have a finite
LDOS at $E_{F}$ for fractional densities), but a consequence of the linear
potential. We also observe that the LDOS consists of a series of peaks
separated by minigaps, a possible indication of Stark discretization [21].

Figure 4 shows a heatplot of the local density of states along the chain for
$V=0$, $\mu=0$ and $\mu_{0}=10$. The Fermi energy is marked by a white
(dashed) line at $\omega=0$. As the Hamiltonian is particle-hole symmetric
around the middle of the chain, the density of states for the right half of
the chain ($20\leq i\leq 40$, not shown) is inverted as a function of $\omega$
(details see in [33]). As mentioned above (Fig. 3), we always find a gap at
$E_{F}$, indicating an incompressible state. This gap is more than an order of
magnitude smaller than the Mott gap. We also see a structure in the Hubbard
bands in the form of three main substructures which evolve along the chain
sites. Each substructure extends to around three neighboring sites, also an
indication of Stark localization which requires future study [21].

Figure 4: Heatplot of the local density of states at different sites ($6\leq
i\leq 20$) for $\mu_{0}=10$ and $V=0$. The Fermi energy is marked by a white
(dashed) line at $\omega=0$

An interesting result for $V=0$ is the existence of a (negative) high-energy
localized state in the IIF region (clearly seen in the density of states plots
at the left of the chain, Fig. S2 in Ref. [33]). We can see a small and narrow
peak at energies around $\omega\sim-14$ for the first sites of this region,
which evolves to higher energies (following the increase of $\mu$), while we
approach the Mott region, increasing its width. This state is reminiscent of
the lower Hubbard band for the left regions. A similar state is seen for the
right half of the chain which is reminiscent of the upper Hubbard band (not
shown). More results for the density of states, together with some
calculations in the atomic limit, are presented in Ref. [33].

Size of the Mott region.— At the electron-hole symmetric case (and $V=0$) the
upper and lower Hubbard bands are centered at $\pm\frac{U}{2}$ respectively,
each with a total width of 4. For $\mu=0$, the size of the Mott region can be
estimated recalling that the Mott insulating state requires that the local
$\mu_{i}$ lies within the Mott gap i.e., $-U/2+2<\mu_{i}<U/2-2$. At the lower
limit $\mu_{\rm min}=-\frac{U}{2}+2$, yielding by Eq. (2) that $i_{\rm
min}\mu_{0}=i_{c}(\mu_{0}-\frac{U}{2}+2)$, while at $\mu_{\rm
max}=\frac{U}{2}-2$ one finds $i_{\rm
max}\mu_{0}=i_{c}(\mu_{0}+\frac{U}{2}-2)$. Consequently, assuming that the
width of the Hubbard bands is not modified by the presence of the confining
potential, the size of the Mott region is:

$\displaystyle L_{\rm Mott}=i_{\rm max}-i_{\rm
min}=\left(U/2-2\right)(L-1)/\mu_{0}\ .$ (3)

As the confining potential slightly increases the width of the Hubbard bands
(not shown), the gap in-between them and $L_{\rm Mott}$ are slightly
overestimated.

To compare Eq. (S1) with our numerical results, we have estimated the size of
the Mott region by defining its boundaries at the points where the linear fits
of the numerical derivative of the local occupation intercept $0$ for each
value of $\mu_{0}$, using the results shown in Fig. 2(a). This procedure
reveals that indeed the size of the Mott region is proportional to $1/\mu_{0}$
(see Fig. S1 in Ref. [33]), and it shrinks to zero for very steep potentials.

Changing the global chemical potential.— The coexisting phases are robust
against changes in the global chemical potential $\mu$. In Fig. 5 we show our
results for two cases, $V=0$ (with coexisting band and Mott insulators,
separated by intermediate IIF regions), and $V=4$ (with CDW’s and Mott
insulators). The different phases shift towards the right or the left with
respect to their position for $\mu=0$ but otherwise are not changed, except
for the regions close to the boundaries where they are affected by the open
boundaries.

Figure 5: (color online) The local density at $\mu_{0}=10$, for $V=0$ (a), and
$V=4$ (b), for different values of the global chemical potential $\mu$.

Discussion.— In this paper we study the one-dimensional extended Hubbard
model, subject to a linearly-varying Wannier-Stark potential on a finite
chain, applying the density-matrix renormalization group. We find an
interesting sequence of several insulating electronic phases in the ground
state, in which regions with commensurate charge density waves coexist with
band and Mott insulating phases. These regions are separated by incompressible
domain walls with incommensurate fillings, which were not reported before. The
results are summarized in Fig. 6. Further research is needed to define whether
these incompressible walls are due to the Stark many-body localization [34].
The steeper the slope of the external potential, the narrower the domain
walls. These phases and domain walls can be moved around by varying a global
chemical potential, thus providing a possible functionality of this kind of
systems. Cold-atom chains placed in an external electric field are suggested
as experimental realizations of our system.

Figure 6: Schematic plot of the various phases observed in the chain subjected
to a linear potential, for three different regimes defined by the nearest-
neighbor interaction, $V$. The sequence of numbers, 212121 and 202020, refer
to the occupations of neighboring sites. MI: Mott insulator, BI: band
insulator, IIF: incompressible incommensurate-filling phase.

Acknowledgments.— NAB and KH acknowledge support from ICTP through the STEP
and Associates Programmes respectively, and from the PICT 2018-01546 grant of
the ANPCyT. The authors thank useful discussions with Carlos Balseiro.

## References

  * [1] F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper, and V. E. Korepin, The one-dimensional Hubbard model, (Cambridge University Press, 2009).
  * [2] P. W. Anderson, Antiferromagnetism: Theory of Superexchange Interaction, Phys. Rev. 79, 350 (1950).
  * [3] T. Giamarchi, Mott transition in one dimension, Physica B: Condensed Matter 230-232, 975 (1997).
  * [4] F. Mila and X. Zotos, Phase Diagram of the One-Dimensional Extended Hubbard Model at Quarter-Filling, Euro. Phys. Lett. 24, 133 (1993).
  * [5] Y. Z. Zhang, Dimerization in a Half-Filled One-Dimensional Extended Hubbard Model, Phys. Rev. Lett. 92, 246404 (2004).
  * [6] S. Ejima, and S. Nishimoto, Phase Diagram of the One-Dimensional Half-Filled Extended Hubbard Model, Phys. Rev. Lett. 99, 216403 (2007).
  * [7] S. Glocke, A. Klümper, and J. Sirker, Half-filled one-dimensional extended Hubbard model: Phase diagram and thermodynamics, Phys. Rev. B 76, 155121 (2007).
  * [8] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  * [9] X. W. Guan, M. Batchelor, and C. Lee, Fermi gases in one dimension: From Bethe Ansatz to experiments, Rev. Mod. Phys. 85, 1633 (2013).
  * [10] U. Khanna, Y. Gefen, O. Entin-Wohlman, and A. Aharony, Edge Reconstruction of a Time-Reversal Invariant Insulator: Compressible-Incompressible Stripes, Phys. Rev. Lett. 128, 186801 (2022) and references therein.
  * [11] J. E. Hirsch, Charge-Density-Wave to Spin-Density-Wave Transition in the Extended Hubbard Model, Phys. Rev. Lett. 53, 2327 (1984).
  * [12] T. Nishino, Charge Excitation Gap of the Extended Hubbard Model, J. Phys. Soc. Jpn 61, 3651 (1992).
  * [13] M. Tsuchiizu and A. Furusaki, Ground-state phase diagram of the one-dimensional half-filled extended Hubbard model, Phys. Rev. B 69, 035103 (2004).
  * [14] J. Spalding, S-W. Tsai, and D. K. Campbell, Critical entanglement for the half-filled extended Hubbard model,  Phys. Rev. B 99, 195445 (2019).
  * [15] e.g., Y-Y. Xiang, X-J. Liu, Y-H. Yuan, J. Cao, and C-M. Tang, Doping dependence of the phase diagram in one-dimensional extended Hubbard model: a functional renormalization group study, J. Phys.: Condens. Matter 31 125601 (2019) and references therein.
  * [16] L. Philoxene, V. H. Dao, and R. Frésard, Spin and charge modulations of a half-filled extended Hubbard model, Phys. Rev. B 106, 235131 (2022).
  * [17] A. Recati, P. O. Fedichev, W. Zwerger, and P. Zoller, Spin-Charge Separation in Ultracold Quantum Gases, Phys. Rev. Lett. 90, 020401 (2003).
  * [18] F. Heidrich-Meisner, G. Orso, and A. E. Feiguin, Phase separation of trapped spin-imbalanced Fermi gases in one-dimensional optical lattices, Phys. Rev. A 81, 053602, (2010).
  * [19] M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L. Pollet, I. Bloch, and C. Gross, Spin- and density-resolved microscopy of antiferromagnetic correlations in Fermi-Hubbard chains, Science 353, 1257 (2016).
  * [20] D. Greif, M. F. Parsons, A. Mazurenko, C. S. Chiu, S. Blatt, F. Huber, G. Ji, and M. Greiner, Site-resolved imaging of a fermionic Mott insulator, Science 351, 953 (2016).
  * [21] M. Udono, T. Kaneko, and K. Sugimoto, Wannier-Stark Ladders and Stark Shifts of Excitons in Mott Insulators, Phys. Rev. B 108, L081304, (2023).
  * [22] S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  * [23] S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  * [24] K. A. Hallberg, New trends in density matrix renormalization,  Adv. Phys. 55, 477 (2006).
  * [25] U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
  * [26] U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  * [27] M. C. Bañuls, Tensor Network Algorithms: A Route Map, Annual Review of Cond. Matt. Phys. 14, 173 (2023).
  * [28] The series of numbers represent the average occupations.
  * [29] P. Bak, Commensurate phases, incommensurate phases and the devil’s staircase, Rep. Prog. Phys. 45, 587 (1982).
  * [30] See also K. Inagaki, K. Nakatsugawa, and S. Tanda, Lock-in transition of the charge-density waves of quasi-one-dimensional conductors: origin of the first-order transition, arXiv:2304.11525.
  * [31] D. Cho, G. Gye, J. Lee, S-H. Lee, L. Wang, S-W. Cheong, and H. W. Yeom, Correlated electronic states at domain walls of a Mott-charge-density-wave insulator $1T-TaS_{2}$, Nat. Comm. 8, 392 (2017).
  * [32] M. Nakamura, Tricritical behavior in the extended Hubbard chains, Phys. Rev. B 61, 16377 (2000).
  * [33] See Supplemental Material at [URL will be inserted by publisher] for details.
  * [34] M. Schulz, C. A. Hooley, R. Moessner, and F. Pollmann, Stark Many-Body Localization,  Phys. Rev. Lett. 122, 040606 (2019).

##  I Supplemental material for "Coexistence of insulating phases in confined
fermionic chains with a Wannier-Stark potential"

N. Aucar Boidi

K. Hallberg

Amnon Aharony

Ora Entin-Wohlman

Size of the Mott and band insulator regions.— Figure S1 presents the size of
the Mott region as a function of $1/\mu_{0}$ for two different values of $U$
and $L$ and for $V=0$, as a function of $1/\mu_{0}$. It confirms the linear
dependence on $1/\mu_{0}$, as found in Eq. (3) of the main text.

Figure S1: The size of the Mott region, scaled by $(U/2-2)(L-1)$ for various
values of $L$ and $V$. The dotted line is a guide to the eye, indicating the
linear dependence on $1/\mu_{0}$.

Local density of states.— Introducing the lesser and greater Green’s functions
at site $j$,

$\displaystyle A^{>}_{j\sigma}(\omega)$
$\displaystyle=-\frac{1}{\pi}\text{Im}\langle
c_{j\sigma}(\omega+i\eta-H+E_{0})^{-1}c^{\dagger}_{j\sigma}\rangle\ ,$
$\displaystyle A^{<}_{j\sigma}(\omega)$
$\displaystyle=-\frac{1}{\pi}\text{Im}\langle
c_{j\sigma}^{\dagger}(\omega+i\eta-H+E_{0})^{-1}c_{j\sigma}\rangle\ ,$ (S1)

where the expectation value refers to the ground state of the system whose
energy is $E_{0}$ and $\eta$ is a small Lorentzian broadening, the local
density of states is given by

$\displaystyle
A_{j}\left(\omega\right)=\sum_{\sigma}\big{[}A^{>}_{j\sigma}\left(\omega\right)+A^{<}_{j\sigma}\left(-\omega\right)\big{]}\
.$ (S2)

This quantity is found numerically, exploiting the density-matrix
renormalization group technique.

Below we show the detailed local density of states calculated for $U=10$,
$V=0$ and $V=8$ (all energies are measured in units of $t$ and relative to the
Fermi energy $E_{F}$, see the main text).

  1. i.

$V=0$: Figure S2 shows the local density of states at different sites. The
upper (UHB) and lower (LHB) Hubbard bands are clearly seen, separated by an
energy $~{}U$. The UHB crosses the Fermi energy for sites $6\leq i\leq 14$,
but it always displays a small gap at $\omega=0$, (more details for a smaller
$\eta=0.01$ are given in Fig. 3 of the main text), indicating an
incompressible behavior in the presence of the Wannier-Stark potential, unlike
the metallic behavior observed in the uniform case (Fig. 3 in the main text).
For $14<i\leq 20$ the Fermi energy lies within the Mott gap and this sector of
the chain is a Mott insulator (Fig. 2 in the main text). It is particle-hole
symmetric for $20\leq i\leq 40$. The wiggles are due to the finite size of
each region. It is interesting to note that the density of states at
$\omega<E_{F}=0$ at $i=6$ resembles the free particle situation. This is due
to the fact that the system is nearly completely filled at that site and the
dynamics of one hole can be understood by that of a nearly free particle. Also
interesting is the evolution of the LHB which begins with a very low weight at
low energies (close to $\omega=-15$ for $i=6$) and acquires more weight and
gets wider towards the middle of the chain.

Figure S2: Local density of states at different sites (shifted vertically) for
$\mu_{0}=10$, $V=0$, $\eta=0.1$ and $L=41$ sites. The Fermi energy is
$E_{F}=0$ Figure S3: Local density of states for $\mu_{0}=16$, $V=8$ and
$L=41$ sites at different sites of the chain (shifted vertically). Symbols
indicate excitations energies for different processes.

  2. ii.

Case $V=8$: Figure S3 presents the density of states for the different CDW
regions, all showing a gap at the Fermi energy. To understand these results we
calculate the energy of having different configurations in one atom taking
into account its nearest neighbors ($t=0$) and adding/removing one particle at
the atomic (middle) site $i$. The different states are encoded in the three
sites as $(n_{1},n_{2},n_{3})$, where $n_{i}=0,1,2$ indicates the local
charge. The corresponding transitions are marked in Fig. S3 and their energies
are calculated below. We use $\mu_{0}=16$, $V=8$ and $U=10$.

  * •

$i=3$ (diamonds in Fig. S3):

$\displaystyle E(222)-E(212)$ $\displaystyle=$ $\displaystyle
U/2-\mu+\mu_{i}+2V=7.4$ $\displaystyle E(202)-E(212)$ $\displaystyle=$
$\displaystyle U/2+\mu-\mu_{i}-2V=-2.6$

  * •

$i=4$ (star):

$E(121)-E(111)=U/2-\mu+\mu_{i}=-7.8\\\ $ (S3)

  * •

$i=19$ (black dot):

$E(212)-E(202)=-U/2-\mu+\mu_{i}+2V=10.2\\\ $ (S4)

  * •

$i=38$ (empty triangles):

$\displaystyle E(020)-E(010)$ $\displaystyle=$ $\displaystyle
U/2-2V-\mu+\mu_{i}=3.4$ $\displaystyle E(010)-E(000)$ $\displaystyle=$
$\displaystyle-U/2-2V-\mu+\mu_{i}=-6.6$