# Bounding entanglement dimensionality from the covariance matrix

Shuheng Liu  State Key Laboratory for Mesoscopic Physics, School of Physics,
Frontiers Science Center for Nano-optoelectronics, Peking University, Beijing
100871, China  Vienna Center for Quantum Science and Technology, Atominstitut,
TU Wien, 1020 Vienna, Austria  Institute for Quantum Optics and Quantum
Information (IQOQI), Austrian Academy of Sciences, 1090 Vienna, Austria
Matteo Fadel  Department of Physics, ETH Zürich, 8093 Zürich, Switzerland
Qiongyi He<EMAIL_ADDRESS>State Key Laboratory for Mesoscopic Physics,
School of Physics, Frontiers Science Center for Nano-optoelectronics, Peking
University, Beijing 100871, China  Collaborative Innovation Center of Extreme
Optics, Shanxi University, Taiyuan, Shanxi 030006, China  Peking University
Yangtze Delta Institute of Optoelectronics, Nantong 226010, Jiangsu, China
Marcus Huber<EMAIL_ADDRESS>Vienna Center for Quantum Science and
Technology, Atominstitut, TU Wien, 1020 Vienna, Austria  Institute for Quantum
Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, 1090
Vienna, Austria  Giuseppe Vitagliano<EMAIL_ADDRESS>Vienna Center for
Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria
Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy
of Sciences, 1090 Vienna, Austria

###### Abstract

High-dimensional entanglement has been identified as an important resource in
quantum information processing, and also as a main obstacle for simulating
quantum systems. Its certification is often difficult, and most widely used
methods for experiment are based on fidelity measurements with respect to
highly entangled states. Here, instead, we consider covariances of collective
observables, as in the well-known Covariance Matrix Criterion (CMC) [1] and
present a generalization of the CMC criterion for determining the Schmidt
number of a bipartite system. This is potentially particularly advantageous in
many-body systems, such as cold atoms, where the set of practical measurements
is very limited and only variances of collective operators can typically be
estimated. To show the practical relevance of our results, we derive simpler
Schmidt-number criteria that require similar information as the fidelity-based
witnesses, yet can detect a wider set of states. We also consider case-study
criteria based on three orthogonal local spin covariances, which would unlock
experimentally feasible detection of high-dimensional entanglement in cold
atom systems. In that case, we are able to derive criteria that are valid for
mixtures of states with the same Schmidt bases.

Introduction.—Entanglement has been always seen as a crucial property of
quantum physics [2]. More recently, it is regarded also as an important
resource for quantum information tasks and, at the same time, as a main
obstacle for classical simulations of quantum systems [3, 4, 5]. In fact, a
lot of research has been devoted recently to the problem of distinguishing
separable from entangled states, and even quantifying entanglement as a
resource [6]. A proper resource-theoretic quantification of entanglement is
known to arise from an operational perspective, considering functionals of
quantum states that are monotonic under operations that cannot create
entanglement, for example, Local Operations and Classical Communications [7].
According to this paradigm, it turns out that for mixed states no unique
quantification of entanglement can be given, but rather one can define
independent figures of merit for specific tasks.

Relevant entanglement measures can be also given from a purely information-
theoretic approach considering entropies of marginals, optimal decompositions
of a state in terms of separable states, robustness to noise, distance to the
set of separable states, and so on [6]. All of these measures are in practice
very hard to estimate from experimental data, and one typically relies on
bounding them from so-called entanglement witnesses, i.e., observables that
have positive expectation values on separable states but can have negative
expectation value for some entangled states. Bounds on entanglement measures
have been discussed also in terms of nonlinear witnesses, typically involving
variances of collective operators [8, 9]. However, such bounds work well
either for very low-dimensional systems in the case of entropic measures [8]
or for measures related to the tolerance of the entangled state to noise [10,
11, 12, 9].

One particularly important measure that is relevant especially in the context
of classical simulation of quantum systems is the Schmidt number, or
entanglement dimensionality which, loosely speaking, quantifies the memory
needed to reproduce the correlations in the quantum state: Proving that a
quantum state as a certain entanglement dimensionality $r$ means that such a
state cannot be simulated with a system of dimension lower than $r$. At the
same time, genuine high-dimensional entanglement has been proven useful for
several practical tasks, ranging from improved security for quantum
cryptography [13, 14], to noise resistant quantum communication [15, 16] and
universal quantum computation [17, 18].

Such a measure, similar to other entropic measures like the concurrence or the
entanglement of formation, is typically especially hard to bound from
experimental data, as usual methods rely on fidelity measurements which must
be fine-tuned and, especially in many-body systems, typically inaccessible
[5]. In fact, entanglement dimensionality quantifications have been
successfully done experimentally only in certain two-photon systems, where
virtually arbitrary local measurements can be performed [19, 20, 21]. On the
other hand, it would be highly desirable to have more powerful tools that
would allow to detect the entanglement dimensionality in other quantum
platforms, e.g., cold atoms, which are suitable for the simulation of other
quantum systems. This raises the question on how to find bounds to the
entanglement dimensionality based on measurements routinely performed in such
systems, such as simple two-body correlators, or variances of simple
collective observables.

This is the problem that we address in this Letter, finding a generalization
of the well-known Covariance Matrix Criterion (CMC) to witness the Schmidt
number of a bipartite quantum state. After deriving the general criterion, we
specialize to simpler statements that are more readily applicable to quantum
experiments and compare them to the existing criteria based on fidelities with
respect to pure (high-dimensionally entangled) states. Afterwards, we also
look for criteria based on the covariances of three orthogonal local spin
observables, which would connect our results with the spin-squeezing methods
successfully applied in many experiments, especially with atomic gases [22,
23, 24, 25, 26, 27, 28]. In fact, bipartite entanglement has been recently
demonstrated between spatially separated atomic ensembles [29, 30, 31, 32,
33], and even a quantification of entanglement via monotones can be provided
[34, 35, 9], but not yet those associated with practical quantum information
applications, such as especially the entanglement dimensionality.

Methods.—Let us consider a bipartite system with Hilbert space
$\mathcal{H}=\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and a (bipartite) density
matrix $\varrho$ in the set of trace-class operators
$\mathcal{B}(\mathcal{H})$111Here for simplicity we restrict to bipartite
spaces with equal dimensions, but our main results can be straightforwardly
extend to nonequal dimensions.. The (symmetric) covariance matrix associated
to a set of observables $\boldsymbol{\mathfrak{M}}=(M_{1},\dots,M_{K})$ has
components given by:

$[{\rm
Cov}_{\varrho}(\boldsymbol{\mathfrak{M}})]_{jk}=\tfrac{1}{2}\langle{M_{j}M_{k}+M_{k}M_{j}}\rangle_{\varrho}-\langle{M_{j}}\rangle_{\varrho}\langle{M_{k}}\rangle_{\varrho}.$
(1)

Let us now consider a pair of
$\boldsymbol{\mathfrak{g}}=(\boldsymbol{\mathfrak{g}}_{a},\boldsymbol{\mathfrak{g}}_{b})$,
where
$\boldsymbol{\mathfrak{g}}_{a}=(g_{1}^{(a)}\otimes\mathbb{1},\dots,g_{d_{a}^{2}}^{(a)}\otimes\mathbb{1})$
and $\boldsymbol{\mathfrak{g}}_{b}=(\mathbb{1}\otimes
g_{1}^{(b)},\dots,\mathbb{1}\otimes g_{d_{b}^{2}}^{(b)})$ are orthonormal
basis of observables for party $a$ and $b$ respectively222Here we consider
basis operators (ortho)normalized as $\mathrm{tr}(g_{k}g_{l})=\delta_{kl}$..
Central to our work is the covariance matrix associated to (a canonical) pair
$\boldsymbol{\mathfrak{g}}$ calculated on a generic $\varrho$, namely

${\rm
Cov}_{\varrho}(\boldsymbol{\mathfrak{g}}):=\Gamma_{\varrho}=\left(\begin{array}[]{ll}\gamma_{a}&X_{\varrho}\\\
X_{\varrho}^{T}&\gamma_{b}\end{array}\right),$ (2)

where we emphasized that it has a block structure, in which the diagonals
$\gamma_{a}:={\rm Cov}_{\varrho_{a}}(\boldsymbol{\mathfrak{g}}_{a})$ and
$\gamma_{b}:={\rm Cov}_{\varrho_{b}}(\boldsymbol{\mathfrak{g}}_{b})$ are the
symmetric covariance matrices of each party, and the off-diagonal blocks333In
the following we will often simplify the notation and whenever clear from the
context omit the dependence on the density matrix and the local bases.

$\displaystyle(X_{\varrho})_{kl}$ $\displaystyle=\langle{g_{k}^{(a)}\otimes
g_{l}^{(b)}}\rangle_{\varrho}-\langle{g_{k}^{(a)}}\rangle_{\varrho}\langle{g_{l}^{(b)}}\rangle_{\varrho}$
(3)

are the cross-covariances between the two local observables vectors.
Concretely, as a canonical basis for each party we will consider
$\boldsymbol{\mathfrak{g}}=(\boldsymbol{\mathfrak{d}},\boldsymbol{\mathfrak{r}},\boldsymbol{\mathfrak{i}})$,
which is composed by diagonal operators
$\boldsymbol{\mathfrak{d}}=\\{\outerproduct{j}{j}\otimes\mathbb{1}\\}_{j=1}^{d}$,
real operators
$\boldsymbol{\mathfrak{r}}=\\{\tfrac{1}{\sqrt{2}}(\ket{j}\bra{k}+\ket{k}\bra{j})\otimes\mathbb{1}\\}_{j<k=1}^{d}$
and imaginary operators
$\boldsymbol{\mathfrak{i}}=\\{\tfrac{i}{\sqrt{2}}(\ket{j}\bra{k}-\ket{k}\bra{j})\otimes\mathbb{1}\\}_{j<k=1}^{d}$,
associated to a canonical Hilbert space basis $\\{\ket{j}\\}_{j=1}^{d}$. For
the components of such a basis we will often use a multi-index notation
$g_{\mu}$ with $\mu=(jk)$ and $1\leq j,k\leq d$, with the convention that
indices with $j=k$ correspond to the vector $\boldsymbol{\mathfrak{d}}$,
indices with $j<k$ correspond to $\boldsymbol{\mathfrak{r}}$ and indices with
$j>k$ correspond to $\boldsymbol{\mathfrak{i}}$.

It is well-known that every separable density matrix
$\sigma=\sum_{k}p_{k}\varrho_{a}\otimes\varrho_{b}$ with $p_{k}\geq 0$ and
$\sum_{k}p_{k}=1$ has to satisfy

$\Gamma_{\sigma}\geq\kappa_{a}\oplus\kappa_{b},$ (4)

where $\kappa_{a}=\sum_{k}p_{k}{\rm
Cov}_{\varrho_{a}}(\boldsymbol{\mathfrak{g}}_{a})$ and
$\kappa_{b}=\sum_{k}p_{k}{\rm
Cov}_{\varrho_{b}}(\boldsymbol{\mathfrak{g}}_{b})$ are positive matrices. In
other words, if a state is separable, there must exist positive matrices
$\kappa_{a}$ and $\kappa_{b}$, which are covariance matrices of single
particle states, such that Eq. 4 holds. Thus, given two positive matrices
$\kappa_{a}$ and $\kappa_{b}$ that lower bound $\Gamma_{\varrho}$ for a
certain density matrix $\varrho$ one has to be able to find pure local states
$\ket{\phi_{a}}$ and $\ket{\phi_{b}}$ and probabilities $p_{k}$ such that
$\kappa_{a}=\sum_{k}p_{k}{\rm Cov}_{\phi_{a}}(\boldsymbol{\mathfrak{A}})$ and
similarly for party $b$. If such states $\ket{\phi_{a}}$ and $\ket{\phi_{b}}$
(and the corresponding $\kappa_{a}$ and $\kappa_{b}$) are not found, then
$\varrho$ must be entangled. This is called covariance matrix criterion (CMC)
in the literature [1, 39, 8] and is known to be equivalent to all possible
entanglement criteria of the form $\sum_{k}(\Delta T_{k})^{2}_{\varrho}\geq
K_{a}+K_{b}$, where $T_{k}=A_{k}\otimes\mathbb{1}+\mathbb{1}\otimes B_{k}$ are
collective observables, $(\Delta
T)^{2}_{\varrho}=\langle{T^{2}}\rangle_{\varrho}-\langle{T}\rangle_{\varrho}^{2}$
is their variance and the bounds are given by
$K_{a}=\min_{\phi}\sum_{k}(\Delta A_{k})^{2}_{\phi}$ and similarly for $K_{b}$
[40]. The CMC follows from essentially two properties: (i) the concavity of
the covariance matrix under mixing quantum states, i.e., the inequality ${\rm
Cov}_{\varrho}(\boldsymbol{\mathfrak{M}})\geq\sum_{k}p_{k}{\rm
Cov}_{\varrho_{k}}(\boldsymbol{\mathfrak{M}})$ that holds for a mixed state
$\varrho=\sum_{k}p_{k}\varrho_{k}$ and for every observables’ set
$\boldsymbol{\mathfrak{M}}$; (ii) the fact that $\langle{A\otimes
B}\rangle_{\sigma}=\langle{A}\rangle_{\sigma}\langle{B}\rangle_{\sigma}$ holds
for all separable states $\sigma$ and for all local observables pairs $(A,B)$,
implying that $X_{\sigma}=0$. Note that we have
$\mathrm{tr}\left(\gamma_{n}\right)=d-\mathrm{tr}(\varrho^{2}_{n})$ and
$\mathrm{tr}(\kappa_{n})=d-1$ for $n=a,b$, where the second equality is valid
since $\kappa_{n}$ come from a pure state.

Another tool that is very important for what follows is the so-called Schmidt
decomposition: Any arbitrary bipartite pure state can be brought to the normal
form under local unitary operations $\ket{\psi}\mapsto U_{a}\otimes
U_{b}\ket{\psi}=\sum_{j}\sqrt{\lambda_{j}}\ket{j_{a}j_{b}}$, where
$\\{\ket{j_{n}}\\}$ with $n=a,b$ are orthonormal basis for the two local
Hilbert spaces and $\lambda_{j}$ are called Schmidt coefficients. Such a
Schmidt decomposition can be achieved essentially from the singular value
decomposition of the coefficient matrix of the state
$\ket{\psi}=\sum_{kl}c_{kl}\ket{k,l}$. In other words, the transformation that
brings the coefficient matrix $c_{kl}$ in its singular value decomposition
amounts to a local unitary transformation of $\ket{\psi}$, which in turn
provides the Schmidt decomposition. The Schmidt decomposition also gives the
reduced density matrix of each party as
$\varrho_{a}=\mathrm{tr}_{b}(\outerproduct{\psi}{\psi})=\sum_{j}\lambda_{j}\outerproduct{j_{a}}{j_{a}}$,
and similarly for $\varrho_{b}$. Clearly, such a transformation does not map a
product state into an entangled state nor vice-versa. Moreover, it cannot
change the amount of entanglement of the state, since all entanglement
monotones are by definition invariant under local unitary transformations. In
fact, the number $s(\ket{\psi})$ of nonzero Schmidt coefficients of a pure
state defines the entanglement monotone called Schmidt rank. This monotone can
be extended to all bipartite density matrices via convex-roof construction,
this way defining the Schmidt number:

$s(\varrho):=\inf_{\mathcal{D}(\varrho)}\max_{\ket{\psi_{i}}\in\mathcal{D}(\varrho)}r(\ket{\psi_{i}}),$
(5)

where the infimum is taken over all pure state decompositions
$\mathcal{D}(\varrho)=\\{p_{i},\ket{\psi_{i}}\\}$ of the density matrix
$\varrho=\sum_{i}p_{i}\ket{\psi_{i}}\bra{\psi_{i}}$. Roughly speaking, this
entanglement monotone is connected with the minimal dimension of the Hilbert
space needed to reproduce the states’ correlations. The Schmidt number can be
also lower bounded via entanglement witnesses, the most typical of which are
fidelities with respect to target states [41, 19, 20, 21]. Specifically, given
a certain pure target state
$\ket{\psi_{T}}=\sum_{j=1}\sqrt{\xi_{j}}\ket{j_{a}j_{b}}$ one can prove that
every other (mixed) state $\varrho$ of Schmidt number smaller than $r$ must
satisfy:

$F(\varrho,\psi_{T}):=\mathrm{tr}(\varrho|\psi_{T}\rangle\langle\psi_{T}|)\leq\sum_{j=1}^{r}\xi_{j},$
(6)

where clearly we are assuming that the target state has Schmidt rank larger or
equal than $r$. Here, the left-hand side is the fidelity between $\varrho$ and
the target state $\ket{\psi_{T}}$. As the most common example, considering a
target maximally entangled state
$\ket{\psi_{+}}=\sum_{j=1}^{d}\tfrac{1}{\sqrt{d}}\ket{j_{a}j_{b}}$, one
obtains $F(\varrho,\psi_{+})\leq\tfrac{r}{d}$.

A CMC criterion for entanglement dimensionality.— Here we present our main
results. First a CMC criterion with a tight bound that is valid for states
with a given Schmidt number $r$. Afterwards, we present a corollary that is
more practical for applications. Our main theorem follows:

######  Theorem 1.

Let us consider a bipartite density matrix
$\varrho\in\mathcal{B}(\mathcal{H})$ such that its Schmidt number is
$s(\varrho)\leq r$ and its canonical covariance matrix $\Gamma_{\varrho}={\rm
Cov}_{\varrho}(\boldsymbol{\mathfrak{g}})$. The matrix inequality

$\Gamma_{\varrho}\geq\sum_{k}p_{k}\Gamma^{(k)}_{r}$ (7)

must hold for some probability distributions $\\{p_{k}\\}$ and some positive
$\Gamma_{r}={\rm Cov}_{\psi_{r}}(\boldsymbol{\mathfrak{g}})$, where
$\psi_{r}=\sum_{j}\sqrt{\lambda_{j}}\ket{j_{a}j_{b}}$ are pure Schmidt-
rank-$r$ states.

Here $\Gamma_{r}$ has generally the following block form

$\Gamma_{r}=\left(\begin{array}[]{ll}\kappa_{a}&X_{r}\\\
X_{r}^{T}&\kappa_{b}\end{array}\right),$ (8)

and the singular values of the blocks satisfy

$\displaystyle\boldsymbol{\epsilon}(\kappa_{a})$
$\displaystyle=\boldsymbol{\epsilon}(\kappa_{b})=(\boldsymbol{\epsilon}(D),\\{\tfrac{1}{2}(\lambda_{j}+\lambda_{k})\\}_{j<k=1},\\{\tfrac{1}{2}(\lambda_{j}+\lambda_{k})\\}_{j>k=1}),$
(9) $\displaystyle\boldsymbol{\epsilon}(|X_{r}|)$
$\displaystyle=(\boldsymbol{\epsilon}(D),\\{\sqrt{\lambda_{j}\lambda_{k}}\\}_{j<k=1},\\{\sqrt{\lambda_{j}\lambda_{k}}\\}_{j>k=1})$

where $D$ is the $d\times d$ matrix with elements
$D_{jk}=\lambda_{j}\delta_{jk}-\lambda_{j}\lambda_{k}$.

Proof.—The statement follows from two properties: (i) concavity of
covariances, i.e., the fact that
$\Gamma_{\varrho}\geq\sum_{k}p_{k}\Gamma_{\varrho_{k}}$ holds for density
matrices decomposed as $\varrho=\sum_{k}p_{k}\varrho_{k}$; and (ii) the fact
that a generic covariance matrix of a pure Schmidt-rank-$r$ state has the
above block form and in the canonical basis constructed from its Schmidt
bases, the blocks are given by $\kappa_{a}=\kappa_{b}={\rm diag}(D,R,I)$ and
$X_{r}={\rm diag}(D,R_{C},I_{C})$ with
$D_{jk}=\lambda_{j}\delta_{jk}-\lambda_{j}\lambda_{k}$ being a $d\times
d$-dimensional matrix and $R=I={\rm diag}\\{\omega_{jk}\\}_{j<k=1}^{r}$ being
diagonal matrices of dimension $d(d-1)/2\times d(d-1)/2$ with eigenvalues
given by $\omega_{jk}=\tfrac{1}{2}(\lambda_{j}+\lambda_{k})$ and similarly
$R_{C}=-I_{C}={\rm diag}\\{\omega^{C}_{jk}\\}_{j<k=1}^{r}$ with
$\omega^{C}_{jk}=\sqrt{\lambda_{j}\lambda_{k}}$. This last fact can be
directly verified with simple algebra (see also [8]). With these two
ingredients, the proof goes as follows. Consider a generic Schmidt-number-$r$
density matrix $\varrho=\sum_{k}p_{k}\outerproduct{\psi_{k}}{\psi_{k}}$
decomposed in terms of pure Schmidt-rank-$r$ states
$\outerproduct{\psi_{k}}{\psi_{k}}$. Using concavity we have
$\Gamma_{\varrho}\geq\sum_{k}p_{k}\Gamma_{\psi_{k}}$ with each of the
$\Gamma_{\psi_{k}}$ having the form (8) in a particular basis. Finally, we use
the fact that arbitrary changes of local bases (which include local unitary
transformations on the quantum state) correspond to local orthogonal
transformations $O_{a}\oplus O_{b}\Gamma_{\varrho}(O_{a}\oplus O_{b})^{T}$. In
particular, such transformations cannot change the eigenvalues of $\kappa_{a}$
and $\kappa_{b}$ and the singular values of $X_{r}$. ∎

The matrix formulation of the criterion is per sè very hard to evaluate due to
the fact that one should in principle scan essentially all possible pure
Schmidt-rank-$r$-state covariance matrices $\Gamma_{\psi_{r}}$. However, it is
also possible to find corollaries to our main theorem which can be actually
readily applied, as soon as even just partial information about
$\Gamma_{\varrho}$ is available. One particular condition which is already
quite powerful is obtained taking the traces of the covariance matrix blocks.

######  Corollary 1.

For every Schmidt-number-$r$ density matrix $\varrho$ the following
inequalities hold

$\displaystyle(\mathrm{tr}|X_{\varrho}|-(r-1))^{2}$
$\displaystyle\leq[1-\mathrm{tr}(\varrho^{2}_{a})][1-\mathrm{tr}(\varrho^{2}_{b})],$
(10a)
$\displaystyle\mathrm{tr}(\varrho^{2}_{a})+\mathrm{tr}(\varrho^{2}_{b})+2\mathrm{tr}|X_{\varrho}|$
$\displaystyle\leq 2r,$ (10b)

where for Eq. 10a the condition $\mathrm{tr}|X_{\varrho}|\geq r-1$ must be
ensured. The inequality (10b) is strictly weaker than (10a)

Proof.—Let us multiply Eq. 7 with vectors
$\boldsymbol{t}_{\mu}=\left(\alpha\boldsymbol{u}_{\mu},-\beta\boldsymbol{v}_{\mu}\right)$,
where $\boldsymbol{u}_{\mu}$ and $\boldsymbol{v}_{\mu}$ are the singular
vectors of $X_{\varrho}$ and $\alpha,\beta>0$. We obtain
$\sum_{\mu}\boldsymbol{t}_{\mu}\Gamma_{\varrho}\boldsymbol{t}_{\mu}^{T}\geq\sum_{k}p_{k}\sum_{\mu}\left(\alpha^{2}\boldsymbol{u}_{\mu}\kappa^{(k)}_{a}\boldsymbol{u}_{\mu}^{T}+\beta^{2}\boldsymbol{v}_{\mu}\kappa^{(k)}_{b}\boldsymbol{v}_{\mu}^{T}-2\alpha\beta\boldsymbol{u}_{\mu}X^{(k)}_{r}\boldsymbol{v}_{\mu}^{T}\right)$,
and after summing over all $\mu$ (which span a whole basis for each party $a$
and $b$) we get
$\alpha^{2}\mathrm{tr}\gamma_{a}+\beta^{2}\mathrm{tr}\gamma_{b}-2\alpha\beta\mathrm{tr}|X_{\varrho}|=\alpha^{2}(d-\mathrm{tr}(\varrho_{a}^{2}))+\beta^{2}(d-\mathrm{tr}(\varrho_{b}^{2}))-2\alpha\beta\mathrm{tr}|X_{\varrho}|\geq\alpha^{2}\mathrm{tr}\kappa_{a}+\beta^{2}\mathrm{tr}\kappa_{b}-2\alpha\beta\sum_{\mu}|(X_{r})_{\mu\mu}|\geq\alpha^{2}\mathrm{tr}\kappa_{a}+\beta^{2}\mathrm{tr}\kappa_{b}-2\alpha\beta\mathrm{tr}|X_{r}|\geq\alpha^{2}(d-\mathrm{tr}(\varrho_{r}^{2}))+\beta^{2}(d-\mathrm{tr}(\varrho_{r}^{2}))-2\alpha\beta(r-\mathrm{tr}(\varrho_{r}^{2}))$,
where $\mathrm{tr}(\varrho_{r})$ is the trace of the single particle reduced
density matrix of a generic (optimal) pure Schmidt rank-$r$ state and we used
the fact that $\mathrm{tr}|X_{r}|\leq r-\mathrm{tr}(\varrho_{r}^{2})$ as it
can be directly verified using the generic expression of its singular values
from Eq. 9. This way, we have
$(\alpha-\beta)^{2}\geq(\alpha-\beta)^{2}\mathrm{tr}(\varrho_{r})\geq\alpha^{2}\mathrm{tr}(\varrho_{a}^{2})+\beta^{2}\mathrm{tr}(\varrho_{b}^{2})+2\alpha\beta(\mathrm{tr}|X_{\varrho}|-r)$.
Now, minimizing over all $\alpha$ and $\beta$ we have that the minimum is
achieved for
$\alpha/\beta=(\mathrm{tr}|X_{\varrho}|-r+1)/(1-\mathrm{tr}(\varrho_{a}^{2}))$,
whenever this is positive and results in Eq. 10a. Note that for the optimal
value of $\alpha/\beta$ to be positive we have to ensure that
$\mathrm{tr}|X_{\varrho}|\geq r-1$, which is an additional constraint for Eq.
10a to be a valid criterion. Instead, in the special case $\alpha=\beta=1$ we
get Eq. 10b, which is thus a weaker criterion than Eq. 10a. ∎

Applications.—To show the usefulness of our results we are going to observe
what states can be potentially detected. First, as a practical comparison with
the fidelity criterion, we can observe that Eq. 10b is stronger than the
criterion coming from the fidelity to the maximally entangled state.

######  Observation 1.

Every state $\varrho$ that violates the criterion in Eq. 6 with a target
maximally entangled state $\ket{\psi_{T}}=\ket{\psi_{+}}$ for some $r$, also
violates Corollary 1 for the same $r$.

Proof.—The Hilbert-Schmidt scalar product between a generic density matrix
$\varrho$ and $|\psi_{+}\rangle\langle\psi_{+}|$ can be expressed as
$F(\varrho,\psi_{+})=\mathrm{tr}(\varrho|\psi_{+}\rangle\langle\psi_{+}|)=\tfrac{1}{d}\sum_{\mu}\langle{g_{\mu}^{(a)}\otimes
g_{\mu}^{(b)}}\rangle_{\varrho}$. Then, let us consider the expression in Eq.
10b. We have that the left-hand side is
$\mathrm{tr}(\varrho^{2}_{a})+\mathrm{tr}(\varrho^{2}_{b})+2\mathrm{tr}|X_{\varrho}|\geq\sum_{jk}\langle{g^{(a)}_{jk}}\rangle^{2}+\langle{g^{(b)}_{jk}}\rangle^{2}-2\langle{g^{(a)}_{jk}}\rangle\langle{g^{(b)}_{jk}}\rangle+2\langle{g^{(a)}_{jk}\otimes
g^{(b)}_{jk}}\rangle\geq\sum_{jk}2\langle{g^{(a)}_{jk}\otimes
g^{(b)}_{jk}}\rangle=2dF(\varrho,\psi_{+})$ since
$\langle{g^{(a)}_{jk}}\rangle^{2}+\langle{g^{(b)}_{jk}}\rangle^{2}-2\langle{g^{(a)}_{jk}}\rangle\langle{g^{(b)}_{jk}}\rangle=(\langle{g^{(a)}_{jk}}\rangle-\langle{g^{(b)}_{jk}}\rangle)^{2}$
are positive terms. Thus, we have that $F(\varrho,\psi_{+})>r/d$ implies that
Eq. 10b is violated. ∎

In fact, Corollary 1 detects states that are not detected by Eq. 6. For
example consider the mixed state
$\varrho=\tfrac{1}{2}|\Psi_{3}\rangle\langle\Psi_{3}|+\tfrac{1}{4}(\ket{23}+\ket{32})(\bra{23}+\bra{32})$
with Schmidt number $s(\varrho)=3$, where
$\ket{\Psi_{3}}:=(1/\sqrt{3})(\ket{00}+\ket{11}+\ket{22})$. Its Schmidt number
certified by fidelity witness is at most $2$ [42], while Corollary 1 detects
the actual Schmidt number $3$. The advantage of Corollary 1 is that it is a
criterion which is invariant under a change of the local bases in the
definition of $\Gamma_{\varrho}$. In general, in fact, it is not obvious which
basis is more convenient to consider for the evaluation of theorem 1. A
special case in which this is actually clear is when the density matrix can be
written as a mixture of states with the same Schmidt bases, so called Schmidt-
correlated states [39]. For this particular class of states we can even
observe that our theorem provides a necessary and sufficient condition.

######  Observation 2.

A Schmidt-correlated state $\varrho=\sum_{ij}\varrho_{ij}\ket{ii}\bra{jj}$
satisfies $s(\varrho)>r$ if and only if it violates Eq. 7.

Proof.—The “if” direction is clear from theorem 1. The proof of the “only if”
part consists of two steps. First, one observes that for Schmidt-correlated
states the states $\ket{\psi^{(k)}_{r}}$ that provide the lower bound on the
right-hand side of Eq. 7 must have the same Schmidt bases as $\varrho$.
Second, one observes that whenever Eq. 7 is satisfied, these
$\ket{\psi^{(k)}_{r}}$ provide a decomposition of $\varrho$, which thus must
satisfy $s(\varrho)\leq r$ by definition (5). More details can be found in the
Supplemental Material [43]. ∎

The criterion Eq. 7 is a compact way of writing many individual entanglement
conditions in the form of variance-based uncertainty relations
$\sum_{k}(\Delta T_{k})^{2}_{\varrho}\geq\min_{\psi_{r}}(\Delta
T_{k})^{2}_{\psi_{r}}$, with $T_{k}=A_{k}\otimes\mathbb{1}+\mathbb{1}\otimes
B_{k}$ being collective observables. In fact, the criterion (10b) is nothing
but an uncertainty relation criterion for a full basis set, i.e., with
$T_{\mu}=g_{\mu}^{A}\otimes\mathbb{1}+\mathbb{1}\otimes g_{\mu}^{B}$, the
bound being $2(d-r)$. For other specific $T_{k}$, however, the minimization
over all pure Schmidt-rank-$r$ states might be quite challenging as $r$ grows.
A very important case consists of spin measurements along three orthogonal
directions, which is what is typically measured in, e.g., cold-atom
experiments [4, 5].

Hence, as a paradigmatic case let us consider three local observables
$\boldsymbol{\mathfrak{j}}=(j_{x},j_{y},j_{z})$ being spin component along
orthogonal axes in their irreducible representation with quantum number
$j=\tfrac{d-1}{2}$ for both parties. Taking a canonical observables’ basis
$\boldsymbol{\mathfrak{g}}=(\boldsymbol{\mathfrak{d}},\boldsymbol{\mathfrak{r}},\boldsymbol{\mathfrak{i}})$
with the diagonal operators sharing eigenbasis with $j_{z}$ we have that
$j_{x}$ and $j_{y}$ are combination of only the real and imaginary operators
respectively with the particular coefficients $x_{lk}=w_{l}\delta_{l+1,k}$ and
$y_{lk}=w_{k}\delta_{l,k+1}$ with
$w_{k}=\tfrac{1}{\sqrt{2}}\sqrt{j(j+1)-m_{k}(m_{k}+1)}$, while $j_{z}$ is just
combination of the diagonal operators with coefficients given by
$z_{lk}=m_{k}\delta_{lk}$, with $m_{k}=-j-1+k$ being the $j_{z}$ eigenvalues.
Next, we look for a lower bound to expressions of the form $(\Delta
J_{x}^{\pm})^{2}+(\Delta J_{y}^{\pm})^{2}+(\Delta J_{z}^{\pm})^{2}$, where
$J_{k}^{\pm}=j_{k}\otimes\mathbb{1}\pm\mathbb{1}\otimes j_{k}$ are collective
spin observables. Constructing the spin operators this way, we can derive a
statement that involves pure or Schmidt-correlated states, with measurements
which are aligned with the Schmidt bases of the state. In other words, every
Schmidt-correlated state $\varrho=\sum_{ij}\varrho_{ij}\ket{ii}\bra{jj}$ such
that $s(\varrho)\leq r$ must satisfy an uncertainty relation of the form
$\mathrm{tr}({\rm
Cov}_{\varrho_{a}}(\boldsymbol{\mathfrak{j}}_{a}))+\mathrm{tr}({\rm
Cov}_{\varrho_{b}}(\boldsymbol{\mathfrak{j}}_{b}))-2\sum_{i}|(X^{j}_{\varrho})_{ii}|\geq
B_{j,r}$, where $X^{j}_{\varrho}$ is the cross-covariance matrix between the
local spin operators and the bound is
$B_{j,r}=\min_{\boldsymbol{\lambda}}\sum_{k=1}^{2j}k(2j+1-k)\left(\sqrt{\lambda_{k}}-\sqrt{\lambda_{k+1}}\right)^{2}$.
Here, the minimization is taken over all vectors $\boldsymbol{\lambda}$ with
only $r$ nonzero and positive components which sum up to one, and $j_{z}$ have
eigenbases coinciding with the Schmidt bases of $\varrho$.

To prove such a bound, we again exploit the fact that for Schmidt correlated
states we know that the states providing the right-hand side of Eq. 7 must
have the same Schmidt bases. Consider the vectors
$t_{x}=(\boldsymbol{x},\pm\boldsymbol{x})$,
$t_{y}=(\boldsymbol{y},\pm\boldsymbol{y})$ and
$t_{z}=(\boldsymbol{z},\pm\boldsymbol{z})$, with component as above in the
canonical bases, which are constructed from the Schmidt bases of $\varrho$.
Multiplying Eq. 7 with the three vectors $\boldsymbol{t}_{k}$ and summing we
obtain
$\sum_{k}\boldsymbol{t}_{k}\Gamma_{\varrho}\boldsymbol{t}_{k}^{T}\geq\mathrm{tr}({\rm
Cov}_{\varrho_{a}}(\boldsymbol{\mathfrak{j}}_{a}))+\mathrm{tr}({\rm
Cov}_{\varrho_{b}}(\boldsymbol{\mathfrak{j}}_{b}))-2\sum_{i}|(X^{j}_{\varrho})_{ii}|=\sum_{\mu\nu}x_{\mu}x_{\nu}(2R_{\mu\nu}-2|(R_{C})_{\mu\nu}|)+y_{\mu}y_{\nu}(2I_{\mu\nu}-2|(I_{C})_{\mu\nu}|)\geq\min_{\ket{\psi_{r}}}B(\ket{\psi_{r}})$,
where the bound is taken over a generic Schmidt-rank-$r$ pure state. Here the
blocks are given by
$R=I=\frac{1}{2}\operatorname{diag}\left\\{\varrho_{jj}+\varrho_{kk}\right\\}_{j<k=1}^{d}$
and
$R_{C}=-I_{C}=\operatorname{diag}\left\\{\varrho_{jk}\right\\}_{j<k=1}^{d}$.
Since the $\ket{\psi_{r}}$ providing the bound must have Schmidt bases
coinciding with our computational bases (due to Lemma 1 in the Supplemental
Material [43]) and since both local $j_{z}$ are diagonal in such computational
bases, we get similarly that
$B(\ket{\psi_{r}})=\sum_{\mu\nu}x_{\mu}x_{\nu}(2R^{\prime}_{\mu\nu}-2|(R^{\prime}_{C})_{\mu\nu}|)+y_{\mu}y_{\nu}(2I^{\prime}_{\mu\nu}-2|(I^{\prime}_{C})_{\mu\nu}|)$,
with diagonal blocks with singular values as in theorem 1. Substituting the
expressions in Eq. 9 we finally get
$B(\ket{\psi_{r}})=\sum_{k}2w_{k}^{2}(\sqrt{\lambda_{k}}-\sqrt{\lambda_{k+1}})^{2}$
with $2w_{k}^{2}=k(2j+1-k)$, which must be then minimized over all pure
states, i.e., all Schmidt vectors $\boldsymbol{\lambda}$.

The bound $B_{j,r}$ is quite difficult to calculate analytically for a given
pair $(j,r)$. Still, in the case $r=2$ such a bound can be actually calculated
for all $j$, and moreover further (non-optimal) bounds can be derived for all
$r$. In fact, we have
$B_{j,r}\geq\left(r\sum_{k=1}^{r}k^{-1}(2j+1-k)^{-1}\right)^{-1}\geq\frac{2j}{r^{2}}$.
Furthermore, for $r=2$ we have $B_{j,2}=4j-1-\sqrt{1-4j+8j^{2}}$, which is the
tight bound. A detailed proof of this bound can be found in the Supplemental
Material [43]. Note that these bounds formally work only for Schmidt-
correlated states which have Schmidt bases aligned with the eigenvectors of
one spin direction. However, numerical search shows that actually for general
spin observables the bound is still satisfied, thus hinting at the fact that
the construction above is optimal also for the general case. Furthermore, one
can observe that typical spin states prepared in experiments, such as singlet
states, Dicke states and spin squeezed states are detected, at least for a
small amount of noise. A detailed study of such a general inequality with spin
operators is left for a subsequent publication.

Conclusions.—In conclusion, generalizing the work of Ref. [1] we have derived
a covariance matrix criterion for the entanglement dimensionality. We have
also discussed weaker corollaries of such a general criterion and their
application to detect general states, as well as collective spin states. The
latter are routinely prepared in atomic ensembles, and collective spin
variances are almost the only observables that can be measured in those
experiments, which gave also a further motivation for the present work and for
subsequent investigations. As an outlook, many theoretical questions are
uncovered by the present work and left for future investigation, including a
further improvement of our theorem with non-unitary filtering operations [1,
39] and the relation with other criteria based on collective variances, e.g.,
Ref. [22]. These questions would have implications also on the application
side, to find further concrete inequalities that have the sufficient noise
tolerance to be employed in actual experiments.

Acknowledgments.—This work is supported by the National Natural Science
Foundation of China (Grants No. 12125402). SL acknowledges financial support
from the China Scholarship Council (Grant No. 202006010302). MF was supported
by the Swiss National Science Foundation, and by The Branco Weiss Fellowship –
Society in Science, administered by the ETH Zürich. QH acknowledges the
Beijing Natural Science Foundation (Z190005) and the Key R&D Program of
Guangdong Province (Grant No. 2018B030329001). GV acknowledges financial
support from the Austrian Science Fund (FWF) through the grants P 35810-N
(Stand-Alone) and ZK 3 (Zukunftskolleg). MH acknowledges funding from the
European Research Council (Consolidator grant ’Cocoquest’ 101043705).

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##  Appendix A Supplemental Material

In this supplement we provide formal proofs corroborating some of the
statements in the main text.

######  Lemma 1.

Let us consider a Schmidt-correlated density matrix
$\varrho=\sum_{ij}\varrho_{ij}|ii\rangle\langle jj|$ and the associated
covariance matrix $\Gamma_{\varrho}$ in the canonical operator basis
constructed from its Schmidt basis. The matrix
$\Delta:=\Gamma_{\varrho}-\sum_{k}p_{k}\Gamma_{\psi_{k}}$ as in Eq. 7 in the
main text can be positive semidefinite only if all states $\ket{\psi_{k}}$
have the same Schmidt bases as $\varrho$.

Proof.—Let us consider generic states
$\ket{\psi_{k}}=\sum_{i,j}m_{ij}^{(k)}|ij\rangle$, expanded in the Schmidt
bases of $\varrho$ and let us consider the principal minor of $\Delta$
corresponding to the operators
$\\{\ket{i}\bra{i}\otimes\mathbb{1},\mathbb{1}\otimes\ket{i}\bra{i}\\}$:

$\Delta_{(\nu\nu)}=\left(\begin{array}[]{ll}D_{ii}-\sum_{k}p_{k}a_{k}&D_{ii}-\sum_{k}p_{k}c_{k}\\\
D_{ii}-\sum_{k}p_{k}c_{k}&D_{ii}-\sum_{k}p_{k}b_{k}\end{array}\right),\text{
with }\nu=(ii),D_{ii}=\varrho_{ii}-\varrho_{ii}^{2},$ (1)

where the generic elements $a_{k},b_{k},c_{k}$ are given by:
$a_{k}=\sum_{j}|m_{ij}^{(k)}|^{2}-\left(\sum_{j}|m_{ij}^{(k)}|^{2}\right)^{2}$,
$b_{k}=\sum_{j}|m_{ji}^{(k)}|^{2}-\left(\sum_{j}|m_{ji}^{(k)}|^{2}\right)^{2}$
and
$c_{k}=|m_{ii}^{(k)}|^{2}-\left(\sum_{j}|m_{ij}^{(k)}|^{2}\right)\left(\sum_{j}|m_{ji}^{(k)}|^{2}\right)$.
From the positivity of $\Gamma_{\psi_{k}}$ we have

$\sqrt{\left(\sum_{k}p_{k}a_{k}\right)\left(\sum_{k}p_{k}b_{k}\right)}\geqslant\sum_{k}p_{k}\sqrt{a_{k}b_{k}}\geqslant\sum_{k}p_{k}c_{k}$
(2)

Then notice that $\Delta_{(\nu\nu)}$ is positive only when
$\sum_{k}p_{k}a_{k}=\sum_{k}p_{k}b_{k}=\sum_{k}p_{k}c_{k}\leqslant D_{ii}$. In
this case Eq. 2 saturates, which means that $a_{k}=b_{k}=c_{k}$ for all $k$.
Imposing that $a_{k}=b_{k}$ and $a_{k}=c_{k}$ gives $m^{(k)}_{ij}=0$ for all
$j\neq i$. This means that $\ket{\psi_{k}}=\sum_{i}m^{(k)}_{ii}|ii\rangle$ for
all $k$, which proves the claim. ∎

With this Lemma, we are now in the position to provide a more formal proof of
Observation 2 in the main text.

Proof of Observation 2. The coefficients $\varrho_{ij}$ can be written in
computational basis as
$\varrho_{ij}=\sum_{u}q_{u}\sqrt{\Lambda_{i}^{(u)}\Lambda_{j}^{(u)}}$ where
$\Lambda_{i}^{(u)}$ are Schmidt coefficients of every pure state component of
$\varrho$. The blocks in the covariance matrix of $\varrho$ are given by
$\gamma_{a}=\gamma_{b}=\operatorname{diag}(D,R,I)$ and
$X_{\varrho}=\operatorname{diag}\left(D,R_{C},I_{C}\right)$ with
$D_{jk}=\varrho_{jj}\delta_{jk}-\varrho_{jj}\varrho_{kk}$,
$R=I=\frac{1}{2}\operatorname{diag}\left\\{\varrho_{jj}+\varrho_{kk}\right\\}_{j<k=1}^{d}$
and
$R_{C}=-I_{C}=\operatorname{diag}\left\\{\varrho_{jk}\right\\}_{j<k=1}^{d}$.
Using Lemma 1 above, we know that Eq. 7 can be satisfied only if all the
$\ket{\psi^{(k)}_{r}}$ share the same Schmidt bases as $\varrho$. Let us
consider the principal minor of
$\Delta=\Gamma_{\varrho}-\sum_{k}p_{k}\Gamma_{r}^{(k)}$ corresponding to the
operators $\left\\{\frac{1}{\sqrt{2}}(|j\rangle\langle k|+|k\rangle\langle
j|)\otimes\mathbb{1},\mathbb{1}\otimes\frac{1}{\sqrt{2}}(|j\rangle\langle
k|+|k\rangle\langle j|)\right\\}$:

$\Delta_{(\nu\nu)}=\left(\begin{array}[]{ll}R_{\nu\nu}-\sum_{u}p_{u}R_{\nu\nu}^{(u)}&(R_{C})_{\nu\nu}-\sum_{u}p_{u}(R_{C}^{(u)})_{\nu\nu}\\\
(R_{C})_{\nu\nu}-\sum_{u}p_{u}(R_{C}^{(u)})_{\nu\nu}&R_{\nu\nu}-\sum_{u}p_{u}R_{\nu\nu}^{(u)}\end{array}\right),\text{
with }\nu=(jk),j<k.$ (3)

From the positivity of $\Delta_{(\nu\nu)}$ we have
$R_{\nu\nu}-\sum_{u}p_{u}R_{\nu\nu}^{(u)}\geq 0$ for all $\nu$. Since
$\sum_{j<k}R_{jk,jk}=\frac{1}{2}(d-1)\sum_{j}\varrho_{jj}=\frac{1}{2}(d-1)$
and
$\sum_{j<k}R_{jk,jk}^{(u)}=\frac{1}{2}(d-1)\sum_{j}\lambda_{j}=\frac{1}{2}(d-1)$,
then we have that $\Delta_{(\nu\nu)}$ is the null matrix and therefore
$\varrho_{jj}=\sum_{u}p_{u}\lambda_{j}^{(u)}$ and
$\varrho_{jk}=\sum_{u}p_{u}\sqrt{\lambda_{j}^{(u)}\lambda_{k}^{(u)}}$. By
setting $\Lambda=\lambda$ and $q=p$, we can then find a set of pure states
$\ket{\psi_{r}^{(k)}}$ which satisfy Eq. 7, which also provides a
decomposition of $\varrho$. In summary, if Eq. 7 is violated, it means one
cannot find a decomposition using $\ket{\psi_{r}^{(k)}}$ whose Schmidt rank
$\leq r$, i.e., the Schmidt number of $\varrho$ is greater than $r$.
Conversely, if the Schmidt number of $\varrho$ is greater than $r$, then one
cannot have $\varrho_{jj}=\sum_{u}p_{u}\lambda_{j}^{(u)}$ and
$\varrho_{jk}=\sum_{u}p_{u}\sqrt{\lambda_{j}^{(u)}\lambda_{k}^{(u)}}$, which
means that Eq. 7 cannot be satisfied. In fact, whenever these last equations
are satisfied, then one can find a decomposition of $\varrho$ in terms of
Schmidt-rank-$r$ pure states. In conclusion, the violation of Eq. 7 is
necessary and sufficient for $s(\varrho)>r$ to hold. ∎

Finally, we prove the lower bound to $B_{j,r}$ valid for the inequality with
the spin operators.

######  Lemma 2.

The lower bound $B_{j,r}$ satisfies

$B_{j,r}\geq\left(r\sum_{k=1}^{r}k^{-1}(2j+1-k)^{-1}\right)^{-1}\geq\frac{2j}{r^{2}}.$
(4)

when $r<d$. Furthermore, for $r=2$ we have

$B_{j,2}=4j-1-\sqrt{1-4j+8j^{2}}.$ (5)

Proof.—By using Cauchy–Schwarz inequality, we have
$B_{j,r}=\min_{\boldsymbol{\lambda}}\sum_{k=1}^{r}2w^{2}_{k}\left(\sqrt{\lambda_{k}}-\sqrt{\lambda_{k+1}}\right)^{2}\geq\left(\sum_{k=1}^{r}(2w^{2}_{k})^{-1}\right)^{-1}\min_{\boldsymbol{\lambda}}\left(\sum_{k=1}^{r}\left|\sqrt{\lambda_{k}}-\sqrt{\lambda_{k+1}}\right|\right)^{2}\geq\frac{1}{r}\left(\sum_{k=1}^{r}(2w^{2}_{k})^{-1}\right)^{-1}$,
with $2w^{2}_{k}=k(2j+1-k)$. The last bound is because
$\sum_{k=1}^{r}\left|\sqrt{\lambda_{k}}-\sqrt{\lambda_{k+1}}\right|$ reaches
the minimum when $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{r}$ and
$\lambda_{1}\geq\frac{1}{r}$. Then we have
$B_{j,r}\geq\frac{1}{r}\left(\sum_{k=1}^{r}\left(k(2j+1-k)\right)^{-1}\right)^{-1}\geq\left(r\sum_{k=1}^{r}(2j)^{-1}\right)^{-1}=\frac{2j}{r^{2}}$.
When $r=2<d$, the lowest variance is
$\min_{\lambda_{1}}2j\left(\sqrt{\lambda_{1}}-\sqrt{1-\lambda_{1}}\right)^{2}+(4j-2)(1-\lambda_{1})$
by taking $\lambda_{2}=1-\lambda_{1}$. The solution is
$4j-1-\sqrt{1-4j+8j^{2}}$. ∎


# Constructing totally $p$-adic numbers of small height

S. Checcoli  S. Checcoli, Institut Fourier, Université Grenoble Alpes, 100 rue
des Mathématiques, 38610 Gières, France<EMAIL_ADDRESS>and  A. Fehm  A. Fehm, Institut für Algebra, Fakultät Mathematik, Technische
Universität Dresden, 01062 Dresden, Germany<EMAIL_ADDRESS>
###### Abstract.

Bombieri and Zannier gave an effective construction of algebraic numbers of
small height inside the maximal Galois extension of the rationals which is
totally split at a given finite set of prime numbers. They proved, in
particular, an explicit upper bound for the lim inf of the height of elements
in such fields. We generalize their result in an effective way to maximal
Galois extensions of number fields with given local behaviour at finitely many
places.

##  1\. Introduction

Let $h$ denote the absolute logarithmic Weil height on the field
$\overline{\mathbb{Q}}$ of algebraic numbers. We are interested in explicit
height bounds for elements of $\overline{\mathbb{Q}}$ with special local
behaviour at a finite set of primes. The first result in this context is due
to Schinzel [Sch73] who proved a height lower bound for elements in the field
of totally real algebraic numbers $\mathbb{Q}^{\rm tr}$, the maximal Galois
extension of $\mathbb{Q}$ in which the infinite prime splits totally. More
precisely, he showed that every $\alpha\in\mathbb{Q}^{\rm tr}$ has either
$h(\alpha)=0$ or

$h(\alpha)\geq\frac{1}{2}\log\left(\frac{1+\sqrt{5}}{2}\right).$

Explicit upper and lower bounds for the limit infimum of the height of
algebraic integers in $\mathbb{Q}^{\rm tr}$ are given in [Smy80, Smy81,
Fla96].

In [BZ01] Bombieri and Zannier investigate the analogous problem for the
$p$-adic numbers. More precisely, in [BZ01, Theorem 2] they prove the
following:

######  Theorem 1.1 (Bombieri–Zannier).

Let $p_{1},\dots,p_{n}$ be distinct prime numbers, for each $i$ let $E_{i}$ be
a finite Galois extension of $\mathbb{Q}_{p_{i}}$, and $L$ the maximal Galois
extension of $\mathbb{Q}$ contained in all $E_{i}$. Denote by $e_{i}$ and
$f_{i}$ the ramification index and inertia degree of
$E_{i}/\mathbb{Q}_{p_{i}}$. Then

$\liminf_{\alpha\in
L}h(\alpha)\;\geq\;\frac{1}{2}\cdot\sum_{i=1}^{n}\frac{\log(p_{i})}{e_{i}(p_{i}^{f_{i}}+1)}.$

In the special case $E_{i}=\mathbb{Q}_{p_{i}}$, Bombieri and Zannier in [BZ01,
Example 2] show that the lower bound in Theorem 1.1 is almost optimal. More
precisely:

######  Theorem 1.2 (Bombieri–Zannier).

Let $p_{1},\dots,p_{n}$ be prime numbers and let $L$ be the maximal Galois
extension of $\mathbb{Q}$ contained in all $\mathbb{Q}_{p_{i}}$. Then

$\liminf_{\alpha\in
L}h(\alpha)\;\leq\;\sum_{i=1}^{n}\frac{\log(p_{i})}{p_{i}-1}.$

Other proofs, refinements and generalizations were given in [Fil14, Pot15,
FP15, FP19, PS19] See also [Smy07] for a general survey on the height of
algebraic numbers. In Remark 1.4 we will discuss in detail the contribution
[Fil14] and how it compares to our work.

The goal of this note is to generalize in an effective way the upper bound
Theorem 1.2 to general $E_{i}$, and to further replace the base field
$\mathbb{Q}$ by an arbitrary number field. Our main result is the following:

######  Theorem 1.3.

Let $K$ be a number field and let $\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}$ be
distinct primes ideals of the ring of integers $\mathcal{O}_{K}$ of $K$. For
each $i$, let $E_{i}$ be a finite Galois extension of the completion $F_{i}$
of $K$ at $\mathfrak{p}_{i}$. Denote by $e_{i}$ and $f_{i}$ the ramification
index and the relative inertia degree of $E_{i}/F_{i}$ and write
$q_{i}=|\mathcal{O}_{K}/\mathfrak{p}_{i}|=p_{i}^{{f(\mathfrak{p}_{i}|p_{i})}}$.
Then for the maximal Galois extension $L$ of $K$ contained in all $E_{i}$,

$\liminf_{\alpha\in
L}h(\alpha)\;\leq\;\sum_{i=1}^{n}\frac{f(\mathfrak{p}_{i}|p_{i})}{[K:\mathbb{Q}]}\cdot\frac{\log(p_{i})}{e_{i}(q_{i}^{f_{i}}-1)}.$

More precisely, let

$C=\max\left\\{[K:\mathbb{Q}],|\Delta_{K}|,\max_{i}(e_{i}f_{i}),\max_{i}q_{i}^{f_{i}}\right\\}$

where $\Delta_{K}$ is the absolute discriminant of $K$. Then for every
$0<\epsilon<1$ there exist infinitely many $\alpha\in\mathcal{O}_{L}$ of
height

(1.1)
$h(\alpha)\leq\sum_{i=1}^{n}\frac{f(\mathfrak{p}_{i}|p_{i})}{[K:\mathbb{Q}]}\cdot\frac{\log(p_{i})}{e_{i}(q_{i}^{f_{i}}-1)}+13nC^{2n+2}\frac{\log\left([K(\alpha):K]\right)}{[K(\alpha):K]}+\begin{cases}0,&n=1\\\
n\epsilon,&n>1\end{cases}.$

Namely, for every $\rho\geq 3C^{n}$ there exists such $\alpha$ of degree

(1.2) $\rho\leq[K(\alpha):K]\leq\begin{cases}C\rho,&n=1\\\ \rho^{\frac{(4\log
C)^{n+1}}{\log^{n}(1+\epsilon)}},&n>1\end{cases}.$

Note that in the special case $K=\mathbb{Q}$ and $E_{i}=\mathbb{Q}_{p_{i}}$ we
reobtain Theorem 1.2, except that Theorem 1.3 appears stronger in that the
result is effective and the $\liminf$ can be taken over algebraic integers.
However, an inspection of the proof of Bombieri and Zannier shows that it is
effective as well and does in fact produce algebraic integers.

######  Remark 1.4.

Theorem 1.3 provides an effective version of a result of Fili [Fil14, Theorem
1.2]. The bound in that result seems to differ from ours by the factor
$e(\mathfrak{p}_{i}|p_{i})$, and [Fil14, Theorem 1.1] (and similarly [FP15,
Theorem 9]) also states a variant of Theorem 1.1 which contradicts our Theorem
1.3, but according to Paul Fili (personal communication) this is merely an
error in normalization in [Fil14] and [FP15] that became apparent when
comparing to our result, and the $e_{v}$ in the denominator of Theorems 1.1,
1.2, and Conjecture 1 of [Fil14] (and similarly in the statements of [FP15])
should have been the absolute instead of the relative ramification index. When
this correction is made, the lower bound of [Fil14, Theorem 1.1] agrees with
the one in [FP19, Theorem 13], and the upper bound of [Fil14, Theorem 1.2]
agrees with the one in Theorem 1.3.

In any case, Fili’s proof of [Fil14, Theorem 1.2] uses capacity theory on
analytic Berkovich spaces and does not provide explicit bounds on the degree
and the height of a sequence of integral elements in the $\liminf$. Instead,
our effective proof is more elementary and is inspired by Bombieri and
Zannier’s effective proof of Theorem 1.2. To the best of our knowledge,
Theorem 1.3 is the only result currently available that gives a bound on the
height in terms of the degree of such a sequence of $\alpha$, except for the
case where $K=\mathbb{Q}$ and $E_{i}=\mathbb{Q}_{p_{i}}$ for all $i$, where
such a bound can be deduced from [BZ01].

We also remark that our use of [Fil14, Theorem 1.2] in [CF21] is limited to
the cases where [Fil14, Theorem 1.2] agrees with Theorem 1.3.

The paper is organised as follows. In Section 2 we collect all the preliminary
results needed to prove Theorem 1.3, namely: a consequence of Dirichlet’s
theorem on simultaneous approximation (Proposition 2.1), a bound for the size
of representatives in quotient rings of rings of integers (Proposition 2.2), a
variant of Hensel’s lemma (Proposition 2.3), a bound for the height of a root
of a polynomial defined over a number field in terms of its coefficients
(Proposition 2.4), and a construction of special Galois invariant sets of
representatives of residue rings of local fields (Proposition 2.5).

The proof of Theorem 1.3 is carried out in Section 3. We briefly sketch it
here for clarity. Following Bombieri and Zannier’s strategy, given $\rho\geq
3C^{n}$ we construct a monic irreducible polynomial $g\in\mathcal{O}_{K}[X]$
such that

  1. (i)

its degree is upper and lower bounded in terms of $\rho$ as the degree of
$\alpha$ in Theorem 1.3,

  2. (ii)

the complex absolute value of all conjugates of its coefficients is
sufficiently small, and

  3. (iii)

all its roots are contained in all $E_{i}$.

In Bombieri and Zannier’s proof of Theorem 1.2, (i) and (ii) were achieved by
using the Chinese Remainder Theorem to deform the polynomial
$\prod_{i=1}^{\rho}(X-i)$ into an irreducible polynomial of the same degree
with coefficients small enough to give the desired bound for the height of the
roots. Then a variant of Hensel’s lemma was applied to show that the roots of
the constructed polynomial are still in $\mathbb{Q}_{p_{i}}$ for each $i$.

In our generalisation, the degree of the polynomial is carefully chosen to
obtain (i) in Section 3.1 via Proposition 2.1 (necessary only if $n>1$, which
leads to the better bounds in the case $n=1$). The polynomial $g$ satisfying
(ii) is then constructed in Section 3.2: We start with polynomials
$\prod_{\alpha\in\tilde{A}_{i}}(X-\alpha)$, where now
$\tilde{A}_{i}\subseteq\mathcal{O}_{E_{i}}$ is a set constructed using
Proposition 2.5. These polynomials are then merged into an irreducible
polynomial $g$ by applying the Chinese Remainder Theorem and Proposition 2.2
to bound the size of its coefficients. Property (iii) is verified in Section
3.3, using Proposition 2.3. Finally, Proposition 2.4 is applied to show that
$g$ has a root $\alpha$ of height bounded from above as desired.

##  2\. Notation and preliminaries

We fix some notation. If $K$ is a number field or a non-archimedean local
field we let $\mathcal{O}_{K}$ denote the ring of integers of $K$. For an
ideal $\mathfrak{a}$ of $\mathcal{O}_{K}$ we denote by
$N(\mathfrak{a})=|\mathcal{O}_{K}/\mathfrak{a}|$ its norm. For a nonzero prime
ideal $\mathfrak{p}$ of $\mathcal{O}_{K}$, we denote by $v_{\mathfrak{p}}$ the
discrete valuation on $K$ with valuation ring
$(\mathcal{O}_{K})_{\mathfrak{p}}$ normalized such that
$v_{\mathfrak{p}}(K^{\times})=\mathbb{Z}$. If $L/K$ is an extension of number
fields and $\mathfrak{P}$ is a prime ideal of $\mathcal{O}_{L}$ lying above a
prime ideal $\mathfrak{p}$ of $\mathcal{O}_{K}$ we denote by
$e(\mathfrak{P}|\mathfrak{p})$ and $f(\mathfrak{P}|\mathfrak{p})$ the
ramification index and the inertia degree. For an extension $E/F$ of non-
archimedean local fields we denote the ramification index and the inertia
degree also by $e(E/F)$ and $f(E/F)$.

###  2.1. Auxiliary results

In this section we collect the preliminary results we need to prove Theorem
1.3. These results are not related to each other and we list them in this
section following their order of appearance in the proof of Theorem 1.3.

######  Proposition 2.1.

Let $x_{1},\dots,x_{n}$ be integers greater than $1$. For every $\rho\geq 3$
and $0<\epsilon<1$ there exist positive integers $r,k_{1},\dots,k_{n}$ such
that $r\geq\rho$ and, for all $i$, $r\leq x_{i}^{k_{i}}\leq(1+\epsilon)r$ and

$k_{i}\leq\frac{2^{2n+1}\log^{n}(\max_{j}x_{j})\log(\rho)}{\log(x_{i})\log^{n}(1+\epsilon)}.$

###### Proof.

Say $x_{1}=\max_{i}x_{i}$. Let $\alpha_{i}=2\log(\rho)/\log(x_{i})$ and
$Q=\lceil 2\log(x_{1})/\log(1+\epsilon)\rceil$. By the simultaneous Dirichlet
approximation theorem [Sch80, Chapter II, Section 1, Theorem 1A] there exist
positive integers $q,k_{1},\dots,k_{n}$ with $1\leq q<Q^{n}$ such that
$|q\alpha_{i}-k_{i}|\leq Q^{-1}$ for all $i$, and thus
$|2\log(\rho)q-\log(x_{i}^{k_{i}})|\leq\log(1+\epsilon)/2$. Letting
$r=\min_{i}x_{i}^{k_{i}}$, one has $\log(r)\geq 2\log(\rho)q-1\geq\log(\rho)$
for $q\geq 1$ and $\rho\geq 3$. In addition, for all $i$,
$0\leq\log(x_{i}^{k_{i}})-\log(r)\leq\log(1+\epsilon)$, hence $r\leq
x_{i}^{k_{i}}\leq(1+\epsilon)r$. Finally $k_{i}\leq q\alpha_{i}+1\leq
2q\alpha_{i}\leq 2\log(\rho)Q^{n}\log(x_{i})^{-1}$ and replacing $Q$ we get
the desired bound. ∎

The next proposition deals with bounds for the absolute value of small
representatives for quotient rings.

######  Proposition 2.2.

Let $K$ be a number field of degree $m=[K:\mathbb{Q}]$. Given a nonzero ideal
$\mathfrak{a}$ of $\mathcal{O}_{K}$, there exists a set of representatives $A$
of $\mathcal{O}_{K}/\mathfrak{a}$ such that, for every $a\in A$ and every
$\sigma\in{\rm Hom}(K,\mathbb{C})$, one has

$|\sigma(a)|\leq\delta_{K}N(\mathfrak{a})^{1/m}$

where $\delta_{K}=m^{\frac{3}{2}}{2}^{\frac{m(m-1)}{2}}\sqrt{|\Delta_{K}|}$.

###### Proof.

This is an immediate consequence of well-known results on lattice reduction.
For instance, [BFH17, Proposition 15] gives for every
$\alpha\in\mathcal{O}_{K}$, an element $a\in\mathcal{O}_{K}$ with
$\alpha-a\in\mathfrak{a}$ such that

$\sqrt{\sum_{\sigma}|\sigma(a)|^{2}}\leq
m^{\frac{3}{2}}{\ell}^{\frac{m(m-1)}{2}}\sqrt{|\Delta_{K}|}N(\mathfrak{a})^{1/m}$

where $\ell$ depends on certain parameters $\eta\in\
(1/2,1),\delta\in(\eta^{2},1)$ and $\theta>0$ coming from applying a variant
of the LLL-reduction algorithm as in [CSV12, Theorem 5.4] and [NSV11, Theorem
7] (see also [BFH17, §4, p.595]) to a $\mathbb{Z}$-basis of $\mathfrak{a}$. In
particular, choosing $\eta=2/3,\delta=7/9$ and $\theta=(\sqrt{19}-4)/3$, we
have $\ell=2$, which gives the claimed upper bound for $|\sigma(a)|$. ∎

The following proposition is a variant of Hensel’s lemma.

######  Proposition 2.3.

Let $E$ be a finite extension of $\mathbb{Q}_{p}$, $\mathfrak{P}$ the maximal
ideal of $\mathcal{O}_{E}$ and $v=v_{\mathfrak{P}}$. Let $f\in E[X]$ and
$x_{0}\in E$. Assume there exist $a,b\in\mathbb{Z}$ such that

  1. (i)

$v(f(x_{0}))>a+b$,

  2. (ii)

$v(f^{\prime}(x_{0}))\leq a$,

  3. (iii)

$v(f^{(\nu)}(x_{0})/\nu!)\geq a-(\nu-1)b$ for every $\nu\geq 2$.

Then there exists $x\in E$ with $f(x)=0$ and $v(x-x_{0})>b$.

###### Proof.

This can be proved precisely as the special case $E=\mathbb{Q}_{p}$ in [BZ01,
Lemma 1]. Alternatively, one can reduce this to one of the standard forms of
Hensel’s lemma as follows. Let $\beta\in E$ with $v(\beta)=b$. Then
$g(X):=(\beta f^{\prime}(x_{0}))^{-1}f(\beta X+x_{0})$ is in
$\mathcal{O}_{E}[X]$ by $(i)$-$(iii)$ and has a simple zero $X=0$ modulo
$\mathfrak{P}$ by $(i)$ and $(ii)$, hence by Hensel’s lemma $g$ has a zero
$x^{\prime}\in\mathfrak{P}$, and $x=\beta x^{\prime}+x_{0}$ is then the
desired zero of $f$. ∎

The final proposition in this subsection gives a bound for the height of the
roots of a polynomial with small algebraic coefficients.

######  Proposition 2.4.

Let $K$ be a number field and let
$f(X)=X^{m}+a_{m-1}X^{m-1}+\ldots+a_{0}\in\mathcal{O}_{K}[X]$. If $B\geq 1$
with $|\sigma(a_{i})|<B$ for every $i$ and every $\sigma\in{\rm
Hom}(K,\mathbb{C})$, then $f$ has a root $\alpha$ with

$h(\alpha)\leq\frac{\log(B\sqrt{m+1})}{m}.$

###### Proof.

Let $M_{K}=M_{K}^{0}\cup M_{K}^{\infty}$ be the set of (finite and infinite)
places of $K$ and let $d=[K:\mathbb{Q}]$. For a place $v\in M_{K}$, denote by
$d_{v}=[K_{v}:\mathbb{Q}_{v}]$ the local degree. Let

$\hat{h}(f)=\log\left(\prod_{v\in M_{K}}M_{v}(f)^{d_{v}/d}\right)$

where if $v$ is non-archimedean $M_{v}(f)=\max_{i}(|a_{i}|_{v})$, while if $v$
is archimedean and corresponds to the embedding $\sigma\in{\rm
Hom}(K,\mathbb{C})$, $M_{v}(f)$ is the Mahler measure $M(\sigma(f))$ of the
polynomial $\sigma(f)$.

By [Zan09, Appendix A, section A.2, pag. 210] we have that, if
$\alpha_{1},\ldots,\alpha_{m}\in\overline{\mathbb{Q}}$ are the roots of $f$
(with multiplicities), then $\hat{h}(f)=\sum_{i=1}^{m}h(\alpha_{i})$ where $h$
denotes the usual logarithmic Weil height. Thus, if $\alpha$ is a root of $f$
of minimal height, then

(2.1) $h(\alpha)\leq\frac{\hat{h}(f)}{m}.$

Let $\sigma_{1},\ldots,\sigma_{r}$ and
$\tau_{1},\overline{\tau_{1}},\ldots,\tau_{s},\overline{\tau_{s}}$ be,
respectively, the real and pairwise conjugate complex embeddings of $K$ in
$\mathbb{C}$, so that $d=r+2s$. As $f$ has coefficients in $\mathcal{O}_{K}$,
$M_{v}(f)\leq 1$ if $v$ is non-archimedean and we have that

$\displaystyle\hat{h}(f)$ $\displaystyle\leq\log\left(\prod_{v\in
M_{K}^{\infty}}M_{v}(f)^{d_{v}/d}\right)=\log\left(\prod_{i=1}^{r}M(\sigma_{i}(f))^{1/d}\cdot\prod_{j=1}^{s}M(\tau_{j}(f))^{2/d}\right)=$
$\displaystyle=\log\left(\prod_{\sigma\in{\rm
Hom}(K,\mathbb{C})}M(\sigma(f))^{1/d}\right).$

By [Zan09, Section 3.2.2, formula (3.7)] and by our hypothesis on $B$, we have
that $M(\sigma(f))\leq B\sqrt{m+1}$ for all $\sigma\in{\rm
Hom}(K,\mathbb{C})$, thus $\hat{h}(f)\leq\log(B\sqrt{m+1})$ and, plugging this
bound into (2.1), we conclude. ∎

###  2.2. Representatives of residue rings of local fields

This subsection contains the technical key result needed to construct the
local polynomials in the proof of Theorem 1.3. Let $E/F$ be a Galois extension
of non-archimedean local fields with Galois group $G$. Let $\mathfrak{p}$ be
the maximal ideal of $\mathcal{O}_{F}$, $\mathfrak{P}$ be the maximal ideal of
$\mathcal{O}_{E}$ and for $k\in\mathbb{N}$ denote by
$\pi_{k}:\mathcal{O}_{E}\rightarrow\mathcal{O}_{E}/\mathfrak{P}^{k}$ the
residue map.

It is known that one can always find a $G$-invariant set of representatives of
the residue field $\mathcal{O}_{E}/\mathfrak{P}$, e.g. the Teichmüller
representatives. As long as the ramification of $E/F$ is tame, one can also
find $G$-invariant sets of representatives of each residue ring
$\mathcal{O}_{E}/\mathfrak{P}^{k}$, but if the ramification is wild, this is
not necessarily so. We will therefore work with the following substitute for
such a $G$-invariant set of representatives:

######  Proposition 2.5.

Let $E/F$ be a Galois extension of non-archimedean local fields and define
$G,\mathfrak{p},\mathfrak{P},\pi_{k}$ as above. Let $d$ be a multiple of
$|G|$. There exists a constant $c$ such that for every $k$ there is
$A\subseteq\mathcal{O}_{E}$ such that

  1. (1)

$A$ is $G$-invariant,

  2. (2)

all orbits of $A$ have length $|G|$,

  3. (3)

$\pi_{k}|_{A}:A\rightarrow\mathcal{O}_{E}/\mathfrak{P}^{k}$ is $d$-to-1 and
onto, and

  4. (4)

$\pi_{k+c}|_{A}$ is injective.

Moreover, if $F$ is a $p$-adic field, one can choose

$c\leq
e(\mathfrak{P}|\mathfrak{p})\left(d+|G|+\frac{e(\mathfrak{p}|p)}{p-1}+1\right).$

###### Proof.

Note that $G$ naturally acts on $\mathcal{O}_{E}$ and on
$\mathcal{O}_{E}/\mathfrak{P}^{k}$, and that $\pi_{k}$ is $G$-equivariant. Fix
some primitive element $\alpha\in\mathcal{O}_{E}^{\times}$ of $E/F$ and a
uniformizer $\theta\in\mathcal{O}_{F}$ of $v_{\mathfrak{p}}$, and let

$\displaystyle e$ $\displaystyle=$ $\displaystyle
e(\mathfrak{P}|\mathfrak{p}),$ $\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle\max_{1\neq\sigma\in
G}v_{\mathfrak{P}}(\alpha-\sigma\alpha),(\mbox{with }c_{0}=0\mbox{ if }G=1),$
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\lceil|G|+c_{0}/e\rceil,\mbox{ and }$ $\displaystyle c$
$\displaystyle=$ $\displaystyle e(d+c_{1}).$

Let $k\in\mathbb{N}$ be given. The desired set $A$ is obtained by applying the
following Claim in the case $X=\mathcal{O}_{E}/\mathfrak{P}^{k}$:

######  Claim.

For every $G$-invariant subset $X\subseteq\mathcal{O}_{E}/\mathfrak{P}^{k}$
there exists a $G$-invariant subset $A\subseteq\mathcal{O}_{E}$ with all
orbits of length $|G|$ such that $\pi_{k+c}|_{A}$ is injective and
$\pi_{k}|_{A}$ is $d$-to-$1$ onto $X$.

We prove the Claim by induction on $|X|$: If $X=\emptyset$, $A=\emptyset$
satisfies the claim. If $X\neq\emptyset$ take $x\in X$ and let
$X^{\prime}=X\setminus Gx$, where $Gx$ denotes the orbit of $x$ under $G$. By
the induction hypothesis there exists $A^{\prime}\subseteq\mathcal{O}_{E}$
satisfying the claim for $X^{\prime}$. Choose $a\in\pi_{k}^{-1}(x)$ and let
$k_{0}=\lceil\frac{k}{e}\rceil$. Then

$n_{0}:=\min\\{n\geq 0:v_{\mathfrak{P}}(a-\sigma a)\neq
e(k_{0}+n)+v_{\mathfrak{P}}(\alpha-\sigma\alpha)\;\forall 1\neq\sigma\in
G\\}<|G|,$

as $v_{\mathfrak{P}}(a-\sigma a)-v_{\mathfrak{P}}(\alpha-\sigma\alpha)$
attains less than $|G|$ many distinct values. Thus for $1\neq\sigma\in G$,

$\displaystyle
v_{\mathfrak{P}}((a+\theta^{n_{0}+k_{0}}\alpha)-\sigma(a+\theta^{n_{0}+k_{0}}\alpha))$
$\displaystyle=$ $\displaystyle\min\\{v_{\mathfrak{P}}(a-\sigma
a),e(n_{0}+k_{0})+v_{\mathfrak{P}}(\alpha-\sigma\alpha)\\}$
$\displaystyle\leq$ $\displaystyle e(n_{0}+k_{0})+c_{0}$ $\displaystyle<$
$\displaystyle k+ec_{1},$

so if we replace $a$ by $a+\theta^{n_{0}+k_{0}}\alpha$, we can assume without
loss of generality that $\pi_{k+ec_{1}}$ is injective on $Ga$ and that
$|Ga|=|G|$. If we now let

$A=A^{\prime}\cup\\{\sigma(a)+\theta^{k_{0}+c_{1}+j}:\sigma\in G,0\leq
j<d/|G_{x}|\\}$

where $G_{x}$ is the stabilizer of $x$, then $\pi_{k}|_{A}$ is $d$-to-$1$ onto
$X=X^{\prime}\cup Gx$ and $A$ is $G$-invariant with all orbits of length
$|G|$. As

$k+ec_{1}\leq e(k_{0}+c_{1}+j)<k+c,$

we have that $\pi_{k+c}|_{A}$ is injective.

Now, if $F$ is a $p$-adic field and if we chose $\alpha\in\mathcal{O}_{E}$ to
be also a generator of $\mathcal{O}_{E}$ as a $\mathcal{O}_{F}$-algebra, by
[Ser80, Chap.IV, Ex. 3(c)], one has the explicit bound $c_{0}\leq
e(\mathfrak{P}/{p})/(p-1)$ which implies the stated bound for $c$. ∎

######  Remark 2.6.

Note that if (4) holds for some $c,k,A$, then also for $c^{\prime},k,A$ for
any $c^{\prime}\geq c$.

##  3\. Proof of Theorem 1.3

Using the notation of Theorem 1.3, for every $1\leq i\leq n$, let
$\mathfrak{P}_{i}$ be the maximal ideal of $\mathcal{O}_{E_{i}}$, $v_{i}$ the
extension of $v_{\mathfrak{P}_{i}}$ to an algebraic closure of $E_{i}$,
$G_{i}={\rm Gal}(E_{i}/F_{i})$ and $d=\prod_{i=1}^{n}|G_{i}|$. Let $C$ be the
constant from Theorem 1.3, note that $C\geq 2$, and let $c=4C^{n+1}$. Fix an
integer $\rho\geq 3C^{n}$ and note that $\rho/d\geq 3$ since $d\leq C^{n}$. If
$n=0$ let $\epsilon=0$, otherwise fix $0<\epsilon<1$.

###  3.1. Choosing the right degree

If $n>1$ we apply Proposition 2.1 to $x_{i}={q_{i}}^{f_{i}}$ to obtain
positive integers $r>\rho/d$ and $k_{1},\ldots,k_{n}$ such that for every $i$,

  1. (i)

$r\leq q_{i}^{f_{i}k_{i}}\leq(1+\epsilon)r$ and

  2. (ii)

$k_{i}\leq 2^{2(n+1)}(\log C)^{n}\frac{\log(\rho/d)}{\log^{n}(1+\epsilon)}$,

where we used that, for every $i$, $\log
2\leq\log(x_{i})=\log({q_{i}^{f_{i}}})\leq\log C$. It follows that

$\log(\rho/d)\leq\log(r)\leq(4\log
C)^{n+1}\frac{\log(\rho/d)}{\log^{n}(1+\epsilon)}.$

If $n=1$ we instead set $r=q_{1}^{f_{1}k_{1}}$, where
$k_{1}=\lceil\log(\rho/d)/\log(q_{1}^{f_{1}})\rceil$, so that (i) holds with
$\epsilon=0$, and

$\log(\rho/d)\leq\log(r)\leq\log(\rho/d)+\log(q_{1}^{f_{1}}).$

Using that $(4\log C)^{n+1}\geq\log^{n}(1+\epsilon)$, we conclude

(3.1) $\rho\leq dr\leq\begin{cases}C\rho,&n=1\\\ \rho^{\frac{(4\log
C)^{n+1}}{\log^{n}(1+\epsilon)}},&n>1\end{cases}$

###  3.2. Construction of the polynomial $g$

We first want to prove the following:

######  Claim.

For every $i$, there exists a polynomial $g_{i}\in\mathcal{O}_{K}[X]$ of
degree $dr$ whose set of roots ${A_{i}}$ satisfies

  1. (a)

$A_{i}\subseteq E_{i}$,

  2. (b)

$v_{i}({\alpha}-{\beta})<k_{i}+c$ for all ${\alpha},{\beta}\in{A_{i}}$ with
${\alpha}\neq{\beta}$, and

  3. (c)

$v({g_{i}}^{\prime}(\alpha))\leq
d\left(\frac{q_{i}^{f_{i}k_{i}}-1}{q_{i}^{f_{i}}-1}+c\right)$ for every
${\alpha}\in{A_{i}}$.

###### Proof of the claim.

As
$e(\mathfrak{P}_{i}|\mathfrak{p}_{i})(d+|G_{i}|+\frac{e(\mathfrak{p}_{i}|p_{i})}{p_{i}-1}+1)\leq
C(C^{n}+C+C+1)\leq 4C^{n+1}=c$, by Proposition 2.5 there is a
$G_{i}$-invariant set $A_{i}^{\prime}\subseteq\mathcal{O}_{E_{i}}$ with all
orbits of length $|G_{i}|$ such that
${A_{i}^{\prime}}\rightarrow\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}}$ is
$d$-to-$1$ and
${A_{i}^{\prime}}\rightarrow\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}+c}$ is
injective. As $|{A_{i}^{\prime}}|=dq_{i}^{f_{i}k_{i}}$, $|G_{i}|$ divides $d$,
and $r\leq q_{i}^{f_{i}k_{i}}$, there exists a $G_{i}$-invariant subset
${\tilde{A}_{i}}\subseteq{A_{i}^{\prime}}$ with $|\tilde{A_{i}}|=dr$. Let

$\tilde{g_{i}}=\prod_{\alpha\in\tilde{A_{i}}}(X-\alpha)\in\mathcal{O}_{F_{i}}[X].$

We first prove that conditions (a)-(c) hold for $\tilde{g_{i}}$ and the set
$\tilde{A_{i}}$, instead of $g_{i}$ and $A_{i}$. Note that
$\tilde{g_{i}}\in\mathcal{O}_{F_{i}}[X]$ is monic of degree $dr$ and that
condition (a) holds for $\tilde{A_{i}}$ by construction. Moreover, as the map
$\tilde{A_{i}}\rightarrow\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}+c}$ is
injective, we have that condition (b) is also satisfied for $\tilde{A_{i}}$.
As for condition (c), note that the valuation $v_{\mathfrak{P}_{i}}$ on
$\mathcal{O}_{E_{i}}$ induces a map
$\bar{v}:(\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}})\setminus\\{0\\}\rightarrow\\{0,\dots,k_{i}-1\\}$
such that ${v}_{\mathfrak{P}_{i}}(\gamma)=\bar{v}(\pi_{k_{i}}(\gamma))$ for
all $\gamma\in\mathcal{O}_{E_{i}}\setminus\mathfrak{P}_{i}^{k_{i}}$, where
$\pi_{k_{i}}$ denotes the residue map
$\mathcal{O}_{E_{i}}\rightarrow\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}}$.
Now

$\displaystyle v_{\mathfrak{P}_{i}}(\tilde{g_{i}}^{\prime}(\alpha))$
$\displaystyle=$
$\displaystyle\sum_{\alpha\neq\beta\in\tilde{A_{i}}}v_{\mathfrak{P}_{i}}(\alpha-\beta)$
$\displaystyle\leq$ $\displaystyle\sum_{\alpha\neq\beta\in
A_{i}^{\prime}}v_{\mathfrak{P}_{i}}(\alpha-\beta)$ $\displaystyle=$
$\displaystyle\sum_{\stackrel{{\scriptstyle\alpha\neq\beta\in
A_{i}^{\prime}}}{{\pi_{k_{i}}(\alpha)=\pi_{k_{i}}(\beta)}}}v_{\mathfrak{P}_{i}}(\alpha-\beta)+\sum_{\stackrel{{\scriptstyle\alpha\neq\beta\in
A_{i}^{\prime}}}{{\pi_{k_{i}}(\alpha)\neq\pi_{k_{i}}(\beta)}}}v_{\mathfrak{P}_{i}}(\alpha-\beta)$
$\displaystyle<$ $\displaystyle(d-1)\cdot(k_{i}+c)+d\cdot\sum_{{0\neq
a\in\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}}}}\bar{v}(a)$

and

$\displaystyle\sum_{{0\neq
a\in\mathcal{O}_{E_{i}}/\mathfrak{P}_{i}^{k_{i}}}}\bar{v}(a)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k_{i}-1}|\\{a:\bar{v}(a)=j\\}|\cdot
j\quad=\quad\sum_{j=0}^{k_{i}-1}\sum_{l=1}^{j}|\\{a:\bar{v}(a)=j\\}|$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{k_{i}-1}\sum_{j=l}^{k_{i}-1}|\\{a:\bar{v}(a)=j\\}|\quad=\quad\sum_{l=1}^{k_{i}-1}|\\{a:\bar{v}(a)\geq
l\\}|$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{k_{i}-1}(q_{i}^{f_{i}(k_{i}-l)}-1)\quad=\quad\sum_{l=0}^{k_{i}-1}q_{i}^{f_{i}l}-k_{i}\quad=\quad\frac{1-q_{i}^{f_{i}k_{i}}}{1-q_{i}^{f_{i}}}-k_{i}$

and plugging this into the previous inequality gives condition (c) for
$\tilde{g}_{i}$.

As $\mathcal{O}_{K}$ is dense in $\mathcal{O}_{F_{i}}$ with respect to
$v_{i}$, we obtain a monic polynomial ${g_{i}}\in\mathcal{O}_{K}[X]$ of degree
$dr$ arbitrarily close to $\tilde{g_{i}}$. Let ${A_{i}}$ be the set of roots
of ${g_{i}}$. By the continuity of roots [EP05, Theorem 2.4.7] we can achieve
that the roots of ${g_{i}}$ are arbitrarily close to the roots of
$\tilde{g_{i}}$, in particular that conditions (b) and (c) are satisfied by
$g_{i}$ and $A_{i}$. Moreover, by Krasner’s lemma [Lan94, Ch.II, §2,
Proposition 4], we can in addition achieve condition (a), completing the proof
of the claim. ∎

Now, let $p_{0}$ be the smallest prime number not in the set
$\\{p_{1},\ldots,p_{n}\\}$ and let $\mathfrak{p}_{0}$ be a prime ideal of
$\mathcal{O}_{K}$ above $p_{0}$. Fix a monic polynomial
$g_{0}\in\mathcal{O}_{K}[X]$ of degree $dr$ whose reduction modulo
$\mathfrak{p}_{0}$ is irreducible. Let

(3.2)
$m_{i}=\frac{d}{e_{i}}\left(\frac{q_{i}^{f_{i}k_{i}}-1}{q_{i}^{f_{i}}-1}+k_{i}+2c\right)$

and

$\mathfrak{a}=\mathfrak{p}_{0}\mathfrak{p}_{1}^{m_{1}}\cdots\mathfrak{p}_{n}^{m_{n}}.$

By the Chinese Remainder Theorem and Proposition 2.2 there exists a monic
polynomial $g\in\mathcal{O}_{K}[X]$ such that

  1. (1)

$\deg g=dr$,

  2. (2)

$g\equiv g_{0}\mbox{ mod }\mathfrak{p}_{0}[X]$,

  3. (3)

$g\equiv{g_{i}}\mbox{ mod }\mathfrak{p}_{i}^{m_{i}}[X]$ for $i=1,\dots,n$, and

  4. (4)

$|\sigma(a)|\leq\delta_{K}N(\mathfrak{a})^{1/[K:\mathbb{Q}]}$ for every
coefficient $a$ of $g$ and every $\sigma\in{\rm Hom}(K,\mathbb{C})$,

where
$\delta_{K}=[K:\mathbb{Q}]^{\frac{3}{2}}{2}^{\frac{[K:\mathbb{Q}]([K:\mathbb{Q}]-1)}{2}}\sqrt{|\Delta_{K}|}$.

Note that (2) implies that $g$ is irreducible. In particular, we get from (1)
and (3.1) that every root $\alpha$ of $g$ satisfies the degree bound (1.2) of
Theorem 1.3.

###  3.3. The roots of $g$ are in $E_{i}$ for every $i$.

We claim that the conditions of Proposition 2.3 hold for the field $E_{i}$,
the polynomial $g$ and $x_{0}=\alpha\in{A_{i}}$ (which lies in $E_{i}$ by
condition (a)) by setting $a=v_{i}({g_{i}}^{\prime}(\alpha))$ and
$b=k_{i}+c-1$. Indeed, by (c)

(3.3) $\displaystyle a=v_{i}({g_{i}}^{\prime}(\alpha))\leq
d\cdot\frac{q_{i}^{f_{i}k_{i}}-1}{q_{i}^{f_{i}}-1}+dc<e_{i}m_{i}-b,$

and, writing $g-{g_{i}}=t_{i}$ with $t_{i}\in\mathfrak{p}_{i}^{m_{i}}[X]$ by
(3) of Section 3.2, we have $g(\alpha)=t_{i}(\alpha)$ and therefore
$v_{i}(g(\alpha))\geq e_{i}m_{i}>a+b$, so condition (i) holds. Similarly for
condition (ii), we have
$g^{\prime}(\alpha)=t_{i}^{\prime}(\alpha)+{g_{i}}^{\prime}(\alpha)$ and since

$v_{i}(t_{i}^{\prime}(\alpha))\geq
e_{i}m_{i}>v_{i}({g_{i}}^{\prime}(\alpha)),$

we conclude that
$v_{i}(g^{\prime}(\alpha))=v_{i}({g_{i}}^{\prime}(\alpha))=a$. Now for
$\nu\geq 2$ write

${g_{i}}^{(\nu)}(\alpha)=\nu!{g_{i}}^{\prime}(\alpha)\sum_{\begin{subarray}{c}B\subseteq{A_{i}}\\\
|B|=\nu-1\\\ \alpha\notin B\end{subarray}}\prod_{\beta\in
B}(\alpha-\beta)^{-1}.$

Thus

$v_{i}({{g_{i}}^{(\nu)}(\alpha)}/{\nu!})\geq
a-(\nu-1)\max_{\beta\neq\alpha}v_{i}(\alpha-\beta){\geq}a-(\nu-1)b$

where the last inequality holds by (b). Moreover, using (3.3), we get

$v_{i}({t_{i}^{(\nu)}(\alpha)}/{\nu!})\geq e_{i}m_{i}-v_{i}(\nu!)\geq
a+b-\frac{e(\mathfrak{P}_{i}|p_{i})\nu}{p_{i}-1}\geq a-(\nu-1)b$

where the last inequality holds since $b\geq
c\geq\frac{e(\mathfrak{P}_{i}|p_{i})}{p_{i}-1}$. Thus

$v_{i}({g^{(\nu)}(\alpha)}/{\nu!})\geq\min\left\\{v_{i}({{g_{i}}^{(\nu)}(\alpha)}/{\nu!}),v_{i}({t_{i}^{(\nu)}(\alpha)}/{\nu!})\right\\}\geq
a-(\nu-1)b$

fulfilling condition (iii).

So Proposition 2.3 gives ${\alpha}^{\prime}\in E_{i}$ with
$g({\alpha}^{\prime})=0$ and $v_{i}(\alpha^{\prime}-\alpha)>b$. As
$v_{i}(\alpha-\beta)\leq b$ for all $\beta\in{A_{i}}\setminus\\{\alpha\\}$ by
(b), we conclude that ${\alpha}^{\prime}\neq{\beta}^{\prime}$ for all
$\alpha\neq\beta$. Hence $g$ has precisely $|{A_{i}}|=dr$ many roots in
$E_{i}$. As this holds for every $i$ and $g$ totally splits in the maximal
Galois extension $L$ of $K$ that is contained in all $E_{i}$. Moreover, as
$g\in\mathcal{O}_{K}[X]$, all roots of $g$ are actually in $\mathcal{O}_{L}$.

###  3.4. Bounding the height of the roots of $g$

From condition (4) of Section 3.2, for every coefficient $a$ of $g$ and every
$\sigma\in{\rm Hom}(K,\mathbb{C})$, we have

$|\sigma(a)|\leq
B:=\delta_{K}N(\mathfrak{p}_{0})^{1/[K:\mathbb{Q}]}\cdot\prod_{i=1}^{n}N(\mathfrak{p}_{i})^{m_{i}/[K:\mathbb{Q}]}.$

By Proposition 2.4, $g$ has a root $\alpha\in\mathcal{O}_{L}$ with $h(\alpha)$
bounded by

(3.4) $\frac{\log(B\sqrt{\deg g+1})}{\deg
g}\leq\frac{\log\left(\delta_{K}N(\mathfrak{p}_{0})^{1/[K:\mathbb{Q}]}\sqrt{\deg
g+1}\right)}{\deg g}+\sum_{i=1}^{n}\frac{m_{i}}{\deg
g}\cdot\frac{\log(q_{i})}{[K:\mathbb{Q}]}.$

By the definition of $m_{i}$ in (3.2) and recalling $\deg g=dr$ from (1), we
have

(3.5) $\displaystyle\frac{m_{i}}{\deg g}$ $\displaystyle=$
$\displaystyle\frac{1}{e_{i}}\cdot\frac{1}{q_{i}^{f_{i}}-1}\cdot\frac{q_{i}^{f_{i}k_{i}}-1}{r}+\frac{k_{i}}{e_{i}r}+\frac{2c}{e_{i}r}.$

Condition (i) of Section 3.2 implies that

$\frac{q_{i}^{f_{i}k_{i}}-1}{r}\leq 1+\epsilon.$

Moreover,

$\frac{k_{i}}{e_{i}r}=\frac{\log(q_{i}^{f_{i}k_{i}})}{e_{i}rf_{i}\log(q_{i})}\leq\frac{2d}{e_{i}f_{i}\log(q_{i})}\cdot\frac{\log(\deg
g)}{\deg g}\leq 3C^{n}\frac{\log(\deg g)}{\deg g}$

Finally,

$\frac{2c}{e_{i}r}\leq\frac{8dC^{n+1}}{e_{i}\deg g}\leq\frac{8C^{2n+1}}{\deg
g}.$

Therefore, substituting in (3.5), and recalling that $C\geq 2$ and $\rho\geq
3$, we have

$\frac{m_{i}}{\deg
g}\leq\frac{1}{e_{i}(q_{i}^{f_{i}}-1)}+\frac{\epsilon}{e_{i}(q_{i}^{f_{i}}-1)}+11C^{2n+1}\frac{\log(\deg
g)}{\deg g}.$

Thus the second summand in (3.4) can be bounded as

$\displaystyle\sum_{i=1}^{n}\frac{m_{i}}{\deg
g}\cdot\frac{\log(q_{i})}{[K:\mathbb{Q}]}\leq\sum_{i=1}^{n}$
$\displaystyle\frac{f(\mathfrak{p}_{i}|p_{i})}{[K:\mathbb{Q}]}\cdot\frac{\log(p_{i})}{e_{i}(q_{i}^{f_{i}}-1)}+n\epsilon+11nC^{2n+2}\frac{\log(\deg
g)}{\deg g}.$

As for the first summand in (3.4), note that
$N(\mathfrak{p}_{0})^{1/[K:\mathbb{Q}]}\leq p_{0}$ where $p_{0}$ is the
smallest prime number not in the set $\\{p_{1},\ldots,p_{n}\\}$, which, by
Bertrand’s postulate, can be bounded by $p_{0}<2\max_{i}p_{i}\leq 2C$.
Moreover,

$\delta_{K}=[K:\mathbb{Q}]^{\frac{3}{2}}{2}^{\frac{[K:\mathbb{Q}]([K:\mathbb{Q}]-1)}{2}}\sqrt{|\Delta_{K}|}\leq
C^{2}\cdot 2^{\frac{C(C-1)}{2}}$

and thus, as $C\geq 2$,

$\displaystyle\frac{\log\left(\delta_{K}N(\mathfrak{p}_{0})^{1/[K:\mathbb{Q}]}\sqrt{\deg
g+1}\right)}{\deg g}$ $\displaystyle\leq$
$\displaystyle\frac{\log\left(C^{3}2^{\frac{C^{2}-C+2}{2}}\sqrt{\deg
g+1}\right)}{\deg g}\leq 2nC^{2n+1}\frac{\log(\deg g)}{\deg g}.$

Therefore, from (3.4) we get

$h(\alpha)\leq\sum_{i=1}^{n}\frac{f(\mathfrak{p}_{i}|p_{i})}{[K:\mathbb{Q}]}\cdot\frac{\log(p_{i})}{e_{i}(q_{i}^{f_{i}}-1)}+n\epsilon+13nC^{2n+2}\frac{\log(\deg
g)}{\deg g},$

so $\alpha$ satisfies the height bound (1.1) of Theorem 1.3 (recalling that
$\epsilon=0$ if $n=1$).

## Acknowledgments

The authors thank Lukas Pottmeyer for pointing out the results in [Fil14,
FP15, FP19], Paul Fili for the exchange regarding [Fil14], and Philip Dittmann
for helpful discussions on $p$-adic fields as well as for suggesting the short
proof of Proposition 2.3. The first author’s work has been funded by the ANR
project Gardio 14-CE25-0015.

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