# Common Coherence Witnesses and Common Coherent States

Bang-Hai Wang1<EMAIL_ADDRESS>Zi-Heng Ding2,3 Zhihao Ma2,3
<EMAIL_ADDRESS>Shao-Ming Fei4,5<EMAIL_ADDRESS>1School of Computer
Science and Technology, Guangdong University of Technology, Guangzhou 510006,
China
2School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University,
Shanghai 200240, China
3Shenzhen Institute for Quantum Science and Engineering, Southern University
of Science and Technology, Shenzhen 518055, China
4School of Mathematical Sciences, Capital Normal University, Beijing 100048,
China
5Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

###### Abstract

We show the properties and characterization of coherence witnesses. We show
methods for constructing coherence witnesses for an arbitrary coherent state.
We investigate the problem of finding common coherence witnesses for certain
class of states. We show that finitely many different witnesses
$W_{1},W_{2},\cdots,W_{n}$ can detect some common coherent states if and only
if $\sum_{i=1}^{n}t_{i}W_{i}$ is still a witnesses for any nonnegative numbers
$t_{i}(i=1,2,\cdots,n)$. We show coherent states play the role of high-level
witnesses. Thus, the common state problem is changed into the question of when
different high-level witnesses (coherent states) can detect the same coherence
witnesses. Moreover, we show a coherent state and its robust state have no
common coherence witness and give a general way to construct optimal coherence
witnesses for any comparable states.

###### pacs:

03.65.Ud, 03.65.Ca, 03.67.Mn, 03.67.-a

##  I I. Introduction

Originating from the fundamental superposition principle in quantum mechanics,
quantum coherence Streltsov17 ; Hu2018  plays a crucial role in quantum
metrology Giovannetti04 ; Giovannetti11 , quantum algorithms Hillery 16 ,
nanoscale thermodynamics Aberg14 ; Gour15 ; Lostaglio15 ; Narasimhachar15 ;
Francica19  and energy transportation in the biological systems Lloyd11 ;
Huelga13 ; Lambert13 ; Romero14 . Detecting and quantifying quantum coherence,
therefore, become fundamental problems in the emerging quantum areas. Numerous
impressive schemes on measures of quantum coherence have been presented
Baumgratz14 ; Yuan2015 ; Fang2018 ; Winter2016 ; Streltsov2015 ; Du2015 ;
Qi2017 ; Napoli16 ; Bu2017 ; Huang2017 ; Jin2018 ; Xi2019 .

The coherence witness, inspired by entanglement witnesses, is arguably a
powerful tool for coherence detection in experiments Piani16 ; Napoli16 ;
Wang17 ; Zheng18 ; Ringbauer18 ; Nie19 ; Ma19  and coherence quantification in
theory Wang21 ; Ma21 . It directly detects any coherent states and gives rise
to measures of quantum coherence without state tomography. Compared with the
entanglement witness, the coherence witness has many different characteristics
deserving to be investigated extensively.

Two natural questions arise that when different coherence witnesses can detect
some common coherent states and when different coherent states can be detected
by some common coherence witnesses in finite-dimensional systems. Although
these two similar questions related to entanglement witnesses have been well
solved, separably Wu06 ; Wu07 ; Hou10a , the problems of common coherence
witnesses and common coherent states remain unsolved.

In this paper we systematically investigate and solve the problems of common
coherence witnesses and common coherent states. This paper is organized as
follows. In section II, we review the concept of coherence witnesses and the
methods of constructing coherence witnesses. In section III we show sufficient
and necessary conditions for any given two or many coherence witnesses to be
incomparable, and deal with problem of common coherence witnesses. In section
IV, we characterize coherent states based on high-level witnesses and solve
the problem when different coherent states can be detected by common coherence
witnesses. Summary and discussions are given in section IV.

##  II II. Common coherence witnesses

With respect to a fixed basis $\\{|i\rangle\\}_{i=1,2,\cdots,d}$ of the
$d$-dimensional Hilbert Space $\mathcal{H}$, a state is called incoherent if
it is diagonal in this basis. Denote $\mathcal{I}$ the set of incoherent
states. The density operator of an arbitrary incoherent state
$\delta\in\mathcal{I}$ is of the form,

$\delta=\sum_{i=1}^{d}\delta_{i}|i\rangle\langle i|.$ (1)

Clearly, the set of incoherent states $\mathcal{I}$ is convex and compact.
Since the set of all incoherent states is convex and compact, there must exist
a hyperplane which separates a arbitrary given coherent state from the set of
all incoherent states by the Hahn-Banach theorem Edwards65 . We call this
hyperplane a coherence witness Piani16 ; Napoli16 . A coherence witness is an
Hermitian operator, $W=W^{\dagger}$, such that (i) $\text{tr}(W\delta)\geq 0$
for all incoherent states $\delta\in\mathcal{I}$, and (ii) there exists a
coherent state $\pi$ such that $\text{tr}(W\pi)<0$. More precisely, an
Hermitian operator $W$ on $\mathcal{H}$ is a coherence witness if (i’) its
diagonal elements are all non-negative, and (ii’) there is at least one
negative eigenvalue. Following the definition of incoherent states and the
Hahn-Banach theorem, we can restrict the condition (i) to $tr(W\delta)=0$ and
relax (ii) to $tr(W\pi)\neq 0$ Napoli16 ; Ren2017 ; Wang21 . As coherence
witnesses are hermitian quantum mechanical observables, they can be
experimentally implemented Wang17 ; Zheng18 ; Ringbauer18 ; Nie19 ; Ma19 .

Since the density matrix of an entangled quantum state can not be diagonal,
from the definition (1) an entangled quantum state must be a coherent state.
Therefore, the entanglement witnesses are also kinds of coherence witnesses
with respect to a fixed basis. We denote $\mathbf{S}$ the set of all separable
states, $\mathbf{E}$ the set of all entangled states, $\mathbf{I}$ the set of
all incoherent states and $\mathbf{C}$ the set of all coherent states. Fig. 1
(a) illustrates the schematic picture of the relations between entanglement
and coherence. Therefore, we can construct coherence witnesses in a similar
way of constructing entanglement witnesses Guhne09 ; Horodecki09 .

Figure 1: (Color online) (a) With respective to a fixed basis, all
entanglement witnesses are also coherence witnesses. (b) We denote
$\mathbf{Q}$ the set of all quantum states and $\mathbf{W}$ the set of all
coherence witnesses. Coherent states play the role of high-level witnesses
witnessing coherence witnesses.

For a given coherent state $|\psi\rangle\langle\psi|$, one has coherence
witness,

$W=\alpha I-|\psi\rangle\langle\psi|,$ (2)

where $I$ is the identity matrix and
$\alpha=\max\mathrm{Tr}(\delta|\psi\rangle\langle\psi|)$ with the maximal
running over all incoherent state $\delta$. Coherence witnesses can also be
constructed from geometrical methods,

$W=\frac{1}{N}(\delta-\rho+\mathrm{Tr}(\delta(\rho-\delta))I),$ (3)

where $\delta$ is the closest incoherent state to $\rho$, $N=\|\rho-\delta\|$
and $\|A\|\equiv\sqrt{Tr(A{\dagger}A)}$. Recently, a general way of
constructing a coherence witness for an arbitrary state has been provided
Wang21 ; Ma21 : $W_{\rho}=-\rho+\Delta(\rho)$ is an optimal coherence witness
to detect the coherence of $\rho$, where $\Delta(\rho)=\sum_{i=0}^{d-1}\langle
i|\rho|i\rangle|i\rangle\langle i|$ is the dephasing operation in the
reference basis $\\{|i\rangle\\}_{i=0}^{d-1}$. More general constructions of
coherence witnesses are also given in Wang21 ; Ma21 .

For a coherence witness $W$, we define
$D_{W}=\\{\rho\,|\,\textrm{tr}\left(\rho W\right)<0\\}$, namely, the set of
all coherent states “ witnessed ” by $W$. Give two coherence witnesses $W_{1}$
and $W_{2}$, we say that $W_{2}$ is finer than $W_{1}$ if $D_{W_{1}}\subseteq
D_{W_{2}}$, that is, if all the coherent states “ witnessed ” by $W_{1}$ are
also “ witnessed ” by $W_{2}$. We call $W$ optimal if there exists no other
coherence witness which is finer than it. It is shown that a coherent witness
is optimal if and only if its diagonal elements are all zero Wang21 . For
normalization we set $\|W\|_{\infty}=1$ as there exist traceless coherence
witnesses.

Moreover, given two coherence witnesses $W_{1}$ and $W_{2}$, we say that
$W_{2}$ and $W_{1}$ are incomparable if $D_{W_{1}}\cap D_{W_{2}}=\emptyset$.
Two coherence witnesses $W_{1}$ and $W_{2}$ can detect some common coherent
states if $D_{W_{1}}\cap D_{W_{2}}\neq\emptyset$. To proceed, we need the
following lemma.

Lemma 1: If $W_{2}$ and $W_{1}$ are incomparable, i.e., $D_{W_{1}}\cap
D_{W_{2}}=\emptyset$ and if $D_{W}\subset D_{W_{1}}\cup D_{W_{2}}$, then
either $D_{W}\subset D_{W_{1}}$or $D_{W}\subset D_{W_{2}}$ .

Proof.— On the contrary, suppose that both $D_{W_{1}}\cap D_{W}$ and
$D_{W_{2}}\cap D_{W}$ are nonempty. Take $\rho_{i}\in D_{W_{i}}\cap D_{W}$ ,
$i=1,2$. Consider the segment $[\rho_{1},\rho_{2}]$ consising of
$\rho_{t}=(1-t)\rho_{1}+t\rho_{2}$, where $0\leq t\leq 1$. As $D_{W}$ is
convex, we obtain

$[\rho_{1},\rho_{2}]\subset D_{W}\subset D_{W_{1}}\cup D_{W_{2}}.$ (4)

Thus we have

$[\rho_{1},\rho_{2}]=(D_{W_{1}}\cap[\rho_{1},\rho_{2}])\cup(D_{W_{2}}\cap[\rho_{1},\rho_{2}]),$
(5)

which means that $[\rho_{1},\rho_{2}]$ can be divided into two convex parts.
It follows that there is $0<t_{0}<1$ such that $\\{\rho_{t}:0\leq
t<t_{0}\\}\subset D_{W_{1}}$ , $\\{\rho_{t}:t_{0}<t\leq 1\\}\subset D_{W_{2}}$
and either $\rho_{t_{0}}\in D_{W_{1}}$ or $\rho_{t_{0}}\in D_{W_{2}}$.

Assume that $\rho_{t_{0}}\in D_{W_{1}}$; then
$\text{tr}(W_{1}\rho_{t_{0}})<0$. Thus, for sufficiently small $\varepsilon>0$
with $t_{0}+\varepsilon\leq 1$, we have

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\text{tr}(\rho_{t_{0}+\varepsilon}W_{1})$ $\displaystyle=$
$\displaystyle\text{tr}(\rho_{t_{0}}W_{1})+\varepsilon[\text{tr}(\rho_{2}W_{1})-\text{tr}(\rho_{1}W_{1})]<0$

which leads to a contradiction. Similarly, $\rho_{t_{0}}\in D_{W_{2}}$ leads
to a contradiction as well. This completes the proof. $\blacksquare$

Theorem 1. $W_{2}$ and $W_{1}$ are incomparable (no common coherent states can
be detected) if and only if there exist $a>0$ and $b>0$ such that
$W_{a,b}=aW_{1}+bW_{2}$ is positive.

Proof.— Obviously, if $W_{2}$ is finer than $W_{1}$, then $W_{2}$ is finer
than $W_{a,b}$ and $W_{a,b}$ is finer than $W_{1}$ for positive $a$ and $b$.
Hence, $D_{W_{1}}\cap D_{W_{2}}\subseteq D_{W}=\emptyset$ since
$W_{a,b}=aW_{1}+bW_{2}$ for some $a>0$ and $b>0$. Take $t=\frac{a}{b}$.

By Lemma 1, we have $D_{W_{a,b}}\subset D_{W_{1}}$ or $D_{W_{a,b}}\subset
D_{W_{2}}$for all $a>0$ and $b>0$. Then
$D_{W_{a,b}}=D_{\frac{1}{b}W_{a,b}}=D_{tW_{1}+W_{2}}=D_{\frac{t}{1+t}W_{1}+\frac{1}{1+t}W_{2}}$.
Hence, we obtain $D_{W_{a,b}}=D_{\lambda W_{1}+(1-\lambda)W_{2}}\doteq
W_{\lambda}$ by taking $\lambda=\frac{t}{1+t}$, where $\lambda\in(0,1)$. We
now can consider $W_{\lambda}$ as $W_{a,b}$. When $t$ varies from $0$ to
$\infty$ continuously, then $\lambda$ varies from $0$ to $1$ continuously,
which means that $D_{W_{\lambda}}$ also varies form $D_{W_{2}}$ to $D_{W_{1}}$
continuously. Take $\lambda_{0}=\text{sup}\\{\lambda:D_{W_{\lambda}}\subset
D_{W_{2}}\\}$.

We claim that if $D_{W_{\lambda_{0}}}\subset D_{W_{2}}$ then there exist
$0<\varepsilon<1-\lambda_{0}$ such that $W_{\lambda_{0}+\varepsilon}$ is a
positive operator. Otherwise, if for all $0<\varepsilon<1-\lambda_{0}$,
$D_{W_{\lambda_{0}+\varepsilon}}\neq\emptyset$, then we have
$D_{W_{\lambda_{0}}}\subset D_{W_{2}},D_{W_{\lambda_{0}+\varepsilon}}\subset
D_{W_{1}}$, and for all $\rho\in D_{W_{\lambda_{0}}}$

$\text{tr}(W_{\lambda_{0}}\rho)<0,\text{tr}(W_{\lambda_{0}}\rho)+\varepsilon(\text{tr}(W_{1}\rho)-\text{tr}(W_{2}\rho))\geq
0.$ (6)

Note that $\text{tr}(W_{1}\rho)\geq 0$ and $\text{tr}(W_{2}\rho)<0$, the
second part of the last inequality is positive, and $\varepsilon$ is any small
positive number, so the last inequality is impossible.

On the other hand, if $D_{W_{\lambda_{0}}}\subset D_{W_{1}}$ then there exist
$0<\varepsilon<\lambda_{0}$ such that $D_{W_{\lambda_{0}-\varepsilon}}$ is a
positive operator. Otherwise, if for all $0<\varepsilon<\lambda_{0}$,
$D_{W_{\lambda_{0}-\varepsilon}}\neq\emptyset$, then we have
$D_{W_{\lambda_{0}}}\subset D_{W_{1}}$ ,
$D_{W_{\lambda_{0}-\varepsilon}}\subset D_{W_{2}}$, and for all $\rho\in
D_{W_{\lambda_{0}}}$, we have

$\text{tr}(W_{\lambda_{0}}\rho)<0,\text{tr}(W_{\lambda_{0}}\rho)+\varepsilon(\text{tr}(W_{2}\rho)-\text{tr}(W_{1}\rho))\geq
0.$ (7)

For the similar reason of Eq.(6), Eq.(7) is impossible as well.

To sum up the previous discussion, no matter $D_{W_{\lambda_{0}}}\subset
D_{W_{1}}$ or $D_{W_{\lambda_{0}}}\subset D_{W_{2}}$, there exists
$\lambda\geq 0$, or equivalently $t>0$ ($a>0$ and $b>0$) such that
$W_{\lambda}$ ($W_{a,b}$) is a positive operator, which completes the proof of
the theorem. $\blacksquare$

Corollary 1. $W_{2}$ and $W_{1}$ are not incomparable if and only if
$W_{a,b}=aW_{1}+bW_{2}$ are witnesses for all $a>0$ and $b>0$.

Theorem 1 can be generalized to the case of finitely many witnesses. We have
the following result.

Theorem 2. $W_{1},W_{2},\cdots,W_{n}$ are incomparable if and only if there
exist $t_{i}>0$ $(i=1,2,\cdots,n)$ such that $W=\sum_{i=1}^{n}t_{i}W_{i}$ is
positive.

Proof.— (i) The “if” part. If $W=\sum_{i=1}^{n}t_{i}W_{i}\geq 0$ for
$t_{i}\geq 0$, then $D_{W}=\emptyset$. Let $S=\\{W_{i}|1\leq i\leq n\\}$ and
the convex hull of $cov(S)=\\{\sum_{i=1}^{k}t_{i}W_{i}|t_{i}\geq
0,\sum_{i=1}^{K}t_{i}=1,W_{i}\in S,K\in\mathbb{N}\\}$. Without loss of
generality we assume that any subsect of $S$ can detect some coherent states
simultaneously. For $n=2$, Theorem 2 holds as it reduces to the Theorem 1. Now
assume that the Theorem 2 holds for $K\leq n-1$. We prove that Theorem 2 holds
for $K=n$. Indeed we only need to prove the case of $n=3$. The case of
arbitrary $n$ can be proved in a similar way.

By the assumption, we have $D_{W_{1}}\neq\emptyset$, $D_{W_{1}}\cap
D_{W_{2}}\neq\emptyset$ and $D_{W_{1}}\cap D_{W_{3}}\neq\emptyset$. But
$D_{W_{1}}\cap D_{W_{2}}\cap D_{W_{3}}\neq\emptyset$, that is, $(D_{W_{1}}\cap
D_{W_{2}})\cap D_{W_{1}}\cap D_{W_{3}}\neq\emptyset$. Let
$W_{b,c}=bW_{2}+cW_{3}$, where $b>0$ and $c>0$. We have $D_{W_{1}}\cap
D_{W_{b,c}}\subset(D_{W_{1}}\cap D_{W_{2}})\cup(D_{W_{1}}\cap D_{W_{3}})$.
Since $(D_{W_{1}}\cap D_{W_{2}})$ and $(D_{W_{1}}\cap D_{W_{3}})$ are disjoint
and $D_{W_{1}}\cap D_{W_{b,c}}$ is convex, $D_{W_{1}}\cap D_{W_{b,c}}$ varies
from $(D_{W_{1}}\cap D_{W_{3}})$ to $(D_{W_{1}}\cap D_{W_{2}})$, whenever
$\frac{b}{c}$ varies from 0 to $\infty$. By the similar argument to that in
the proof of Theorem 1, we conclude that there exist $\frac{b_{0}}{c_{0}}>0$
such that $D_{W_{1}}\cap D_{W_{b,c}}=\emptyset$. Therefore,
$W=a^{\prime}W_{1}+b^{\prime}W_{b,c}=aW_{1}+b^{\prime}bW_{2}+b^{\prime}cW_{3}\geq
0$ for some $a^{\prime}>0$ and $b^{\prime}>0$. By induction on $n$ we complete
the proof of (i).

(ii) The “only if” part is clear. If $D_{W}=\emptyset$, then there exist $W$
such that $W\geq 0$ $(W\in cov(S))$ from the proof in (i). It follows that $W$
is not a witness, which gives a contraction. $\blacksquare$

##  III III. Common coherent states

A framework which assembles hierarchies of “witnesses” has been proposed in
Wang18 . In this framework, a coherence witness can witness coherent states,
and on the other hand, a coherent state can also act as a “high-level-witness
” of coherence witnesses which witnesses coherence witnesses. Concretely, when
a coherence witness $W$ detects a coherent state $\rho$, we say that $W$
“witnesses” the coherence of the state $\rho$. A question naturally arises.
What “witnesses” coherence witnesses. It is known that the set of quantum
states (incoherent states and coherent states) is also convex and compact.
Thus, by the Hahn-Banach theorem, there is at least one “high-level” witness
“witnessing” a coherence witness, see Fig. 1 (b).

For a high-level witness of coherence witnesses $\Pi$, one has (i”)
$\text{tr}(\Pi\varrho)\geq 0$ for all quantum states $\varrho$, and (ii”)
there exists at least one coherence witness $W$ such that $\text{tr}(\Pi
W)<0$. Coherence witnesses “witness” coherent states and coherent states
“witness”coherence witnesses. Coherent states play the role of witnesses.
Since coherent states are also (high-level) witnesses, the question when
different coherent states can be detected by some common coherence witnesses
can be transformed into the question when different high-level witnesses
(coherent states) can detect the same coherence witnesses. From the high-
level-witness role played by coherent states and the Theorem 1, we have the
following result.

Theorem 3. Two coherent states $\rho_{1}$ and $\rho_{2}$ are incomparable,
i.e., $D_{\rho_{1}}\cap D_{\rho_{2}}=\emptyset$, if and only if there exists
$0<t<1$ such that $\rho_{t}=t\rho_{1}+(1-t)\rho_{2}$ is an incoherent state.

The robust of coherence $\mathscr{C}_{\mathcal{R}}(\rho)$ Piani16 ; Napoli16
of a coherent state $\rho\in{\mathcal{D}}(\mathbb{C}^{d})$ is defined as

$\mathscr{C}_{\mathcal{R}}(\rho)=\min_{\tau\in{\mathscr{D}}(\mathbb{C}^{d})}\left\\{s\geq
0\ \Big{|}\ \frac{\rho+s\ \tau}{1+s}=\delta\in{\mathcal{I}}\right\\},$ (8)

where ${\mathscr{D}}(\mathbb{C}^{d})$ stands for the convex set of density
operators acting on a $d$-dimensional Hilbert space. We have the following
conclusions.

Corollary 2: Any coherent state $\rho$ and the state minimizing $s$ in (8))
$\tau$ have no common coherence witnesses.

Corollary 3. Two coherent states $\rho_{1}$ and $\rho_{2}$ are not
incomparable if and only if there does not exist $0<t<1$ such that
$\rho_{t}=t\rho_{1}+(1-t)\rho_{2}$ is an incoherent state.

From the general construction of optimal coherence witnesses for an arbitrary
coherent state Wang21 ; Ma21  and Corollary 3, there also exists a general way
of constructing a common optimal coherence witness for different coherent
states.

Corollary 4. For two given not incomparable coherent states $\rho_{1}$ and
$\rho_{2}$, the optimal coherence witness $W=aW_{\rho_{1}}+bW_{\rho_{2}}$
detects both the coherence of $\rho_{1}$ and $\rho_{2}$, where $a>0$, $b>0$
and $W_{\rho_{i}}=-\rho_{i}+\Delta(\rho_{i})$ $(i=1,2)$.

It is also not difficult to generalize Theorem 3 to the case for finitely many
coherent states.

Theorem 4. The coherent states $\rho_{1},\rho_{2},\cdots,\rho_{n}$ are
incomparable if and only if there exist $\sum_{i=1}^{n}t_{i}=1$, $t_{i}>0$
$(i=1,2,\cdots,n)$ such that $\rho=\sum_{i=1}^{n}t_{i}\rho_{i}$ is an
incoherent state.

##  IV IV. Summary and Discussions

To summarize, we have investigated the properties of coherent witnesses and
the methods of constructing coherence witnesses for any arbitrarily given
coherent states. We have presented the conditions for different witnesses to
detect the same coherent states, as well as the conditions for a set of
different coherent states whose coherence can be detected by a common set of
coherence witnesses. Here, we mainly considered the case of discrete quantum
systems in finite-dimensional Hilbert spaces. In fact our results hold also
for infinite-dimensional cases, since our main results are proved without the
additional assumption $tr(W_{1})=tr(W_{2})$. However, the coherence in
continuous variable systems (such as light modes) is significantly different
from the case of the discrete systems. For instance, the set of Gaussian
states must be closed and convex, but not necessarily bounded by the Hahn-
Banach theorem Bruss02 ). Our investigations may highlight further researches
on these related problems.

##  V ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China
under Grant Nos. 62072119, 61672007 and 12075159, Guangdong Basic and Applied
Basic Research Foundation under Grant No. 2020A1515011180, Shenzhen Institute
for Quantum Science and Engineering, Southern University of Science and
Technology (Grant Nos. SIQSE202005, SIQSE202001), Natural Science Foundation
of Shanghai (Grant No. 20ZR1426400), Beijing Natural Science Foundation
(Z190005), the Academician Innovation Platform of Hainan Province, and Academy
for Multidisciplinary Studies, Capital Normal University.

## References

  * (1) A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod. Phys. 89 041003 (2017). arXiv:quant-ph/1609.02439.
  * (2) M.-L. Hu, X. Hu, J.-C. Wang, Y. Peng, Y.-R. Zhang, and H. Fan, Phys. Rep. 762-764 1 (2018).
  * (3) V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-Enhanced Measurements: Beating the Standard Quantum Limit, Science 306, 1330 (2004).
  * (4) V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nature Photonics 5, 222 (2011).
  * (5) M. Hillery, Phys. Rev. A 93, 012111 (2016).
  * (6) J. Åberg, Phys. Rev. Lett. 113, 150402 (2014).
  * (7) G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern, The Resource Theory of Informational Nonequilibrium in Thermodynamics, Physics Reports, 583,1 (2015).
  * (8) M. Lostaglio, D. Jennings, and T. Rudolph, Nature Communications 6, 6383 (2015).
  * (9) V. Narasimhachar and G. Gour, Nature Communications 6, 7689 (2015).
  * (10) G. Francica, J. Goold, and F. Plastina, Phys. Rev. E 99, 042105 (2019).
  * (11) S. Lloyd, J. Phys.: Conf. Ser. 302, 012037 (2011).
  * (12) S. F. Huelga and M. B. Plenio, Contemporary Physics 54, 181 (2013).
  * (13) N. Lambert, Y.-N. Chen, Y.-C. Cheng, C.-M. Li, G.-Y. Chen, and F. Nori, Nat Phys 9, 10 (2013).
  * (14) E. Romero, R. Augulis, V. I. Novoderezhkin, M. Ferretti, J. Thieme, D. Zigmantas, and R. van Grondelle, Nature Physics 10, 676 (2014).
  * (15) T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113 140401 (2014).
  * (16) X. Yuan, H. Zhou, Z. Cao, and X. Ma, Phys. Rev. A 92 022124 (2015).
  * (17) K. Fang, X. Wang, L. Lami, B. Regula, and G. Adesso, Phys. Rev. Lett. 121 070404 (2018).
  * (18) A. Winter and D. Yang, Phys. Rev. Lett. 116 120404 (2016).
  * (19) A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Phys. Rev. Lett. 115 020403 (2015).
  * (20) S. Du, S. Bai, and X. Qi,Quantum Inf. Comput. 15 1307 (2015).
  * (21) X. Qi, T. Gao, and F. Yan, J. Phys. A: Math. Theor. 50 285301 (2017).
  * (22) K. F. Bu, U. Singh, S. M. Fei, A. K. Pati, and J. D. Wu, Phys. Rev. Lett. 119 150405 (2017).
  * (23) Z. Huang and H. Situ, Ann. Phys. 377 484 (2017).
  * (24) Z.-X. Jin and S.-M. Fei, Phys. Rev. A 97 062342 (2018).
  * (25) Z. Xi and S. Yuwen, Phys. Rev. A 99 022340 (2019).
  * (26) Carmine Napoli, Thomas R. Bromley, Marco Cianciaruso, Marco Piani, Nathaniel Johnston, and Gerardo Adesso, Phys. Rev. Lett. 116 150502 (2016).
  * (27) M. Piani, M. Cianciaruso, T. R. Bromley, C. Napoli, N. Johnston, G. Adesso, Phys. Rev. A 93 042107 (2016).
  * (28) Y-Q Nie, H. Zhou, J.-Y. Guan, Q. Zhang, X. Ma, J. Zhang, and J.-W. Pan, Phys. Rev. Lett. 123 090502 (2019).
  * (29) Y.-T. Wang, J.-S. Tang, Z.-Y. Wei, S. Yu, Z.-J. Ke, X.- Y. Xu, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 118 020403 (2017).
  * (30) W. Zheng, Z. Ma, H. Wang, S.-M. Fei, and X. Peng, Phys. Rev. Lett. 120 230504 (2018).
  * (31) M. Ringbauer, T. R. Bromley, M. Cianciaruso, L. Lami, W. Y. Sarah Lau, G. Adesso, A. G. White, A. Fedrizzi, and M. Piani, Phys. Rev. X 8 041007 (2018).
  * (32) J. Ma, A. Hakande, X. Yuan, and X. Ma, Phys. Rev. A 99 022328 (2019).
  * (33) B.-H. Wang, S.-Q. Zhou, Z. Ma, S.-M. Fei, Tomographic Witnessing and Holographic Quantifying of Coherence. Quant. Inform. Proc. 20 181 (2021).
  * (34) Z. Ma, Z. Zhang, Y. Dai, Y. Dong, C. Zhang, Phys. Rev. A 103 012409 (2021).
  * (35) YuChun Wu, YongJian Han, and GuangCan Guo, Phys. Lett. A 356 402 (2006).
  * (36) Y.-C. Wu and G.-C. Guo, Phys. Rev. A 75 052333 (2007).
  * (37) J. Hou and Y. Guo, Phys. Rev. A 82 052301 (2010).
  * (38) R.E. Edwards, Functional analysis, theory and application (Holt, Rinehart and Winston. New York, 1965).
  * (39) Huizhong Ren, Anni Lin, Siying He, Xueyuan Hu, Annals of Physics 387 281-289 (2017).
  * (40) O. Gühne and G. Tóth, Phys. Rep.474 1 (2009).
  * (41) R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys.  81 865 (2009).
  * (42) B.-H. Wang, Phys. Rev. A 97 050302(R) (2018).
  * (43) D. Bruß,J. I. Cirac, P. Horodecki, F. Hulpke, B. Kraus, M. Lewenstein, and A. Sanpera, J. Mod. Opt.,49 1399 (2002).


(because the transformation $\Phi_{j}$ is measure preserving), so we changed
the bad derivative to the good one on a set of measure $(1-\varepsilon)|K|$
which is much larger than the measure $\varepsilon|K|$ of the set where the
good derivative turned bad. We iterate this procedure infinitely many times in
such a way that the measure of the set of good points converges to the measure
of the cube $\mathscr{Q}$.

In fact, in the actual construction, we will use not only one cube $K$ to
modify $\Phi_{j}$, but a finite family of cubes that approximates well the
measure of the set $\Phi_{j}(B)$.

By taking sufficiently small cubes we will guarantee that $\Phi_{j}$ is a
Cauchy sequence in the uniform metric $d$. Thus, the sequence will converge to
a homeomorphism $\Phi$ and it will follow that $\Phi$ satisfies (6.9).

Using the above idea, we will construct a sequence of measure preserving
homeomorphisms $\\{\Phi_{j}\\}_{j=0}^{\infty}$,
$\Phi_{j}:\mathscr{Q}\to\mathscr{Q}$, $\Phi_{j}|_{\partial\mathscr{Q}}={\rm
id\,}$, that are approximately differentiable almost everywhere and for all
$j\geq 1$ satisfy:

(6.10) $D_{\rm
a}\Phi_{j}(x)\in\\{\mathcal{I},\mathcal{R}\\}=\\{\lambda(x),\mathcal{R}\lambda(x)\\}\quad\text{a.e.
in $\mathscr{Q}$,}$ (6.11) $|B_{j}|\leq 2^{-j},$ (6.12)
$d(\Phi_{j},\Phi_{j+1})\leq 2^{-j+1},\quad|L_{j}|\leq
2^{-j+1},\quad\Big{|}B_{j}\setminus\bigcup_{\ell=j}^{\infty}L_{\ell}\Big{|}=0,$

where $d$ is the uniform metric,
$L_{j}:=\overline{\\{x\in\mathscr{Q}:\,\Phi_{j}\neq\Phi_{j+1}\\}}$, and the
set

$B_{j}:=\\{x\in\mathscr{Q}:\,D_{\rm a}\Phi_{j}(x)=\mathcal{R}\lambda(x)\\}$

is the set of points where the approximate derivative of $\Phi_{j}$ is bad.

Before we construct such a sequence, we show that a sequence satisfying the
above conditions converges to a homeomorphism $\Phi$ that has all the required
properties, except for being a limit of volume preserving diffeomorphisms (we
will take care of it at the end of the proof).

Clearly, (6.12) and Lemma 3.20 imply convergence in the uniform metric to some
homeomorphism $\Phi:\mathscr{Q}\to\mathscr{Q}$ that is identity on the
boundary.

Note that by (6.12),

$A_{j}:=\mathscr{Q}\setminus\bigcup_{\ell=j}^{\infty}L_{\ell}.$

is an increasing sequence of measurable sets that exhaust $\mathscr{Q}$ up to
a set of measure zero.

We have that $\Phi=\Phi_{j}$ on $A_{j}$, because
$\Phi_{j}=\Phi_{j+1}=\Phi_{j+2}=\ldots$ on $A_{j}$, and hence

$D_{\rm a}\Phi=D_{\rm a}\Phi_{j}=\lambda\quad\text{almost everywhere in
}A_{j}.$

The first equality follows from Lemma 6.3 and the last equality follows
(6.10), from the definition of $B_{j}$ and the fact that $|A_{j}\cap B_{j}|=0$
(which is a consequence of (6.12) and the definition of $A_{j}$). Since the
sets $A_{j}$ exhaust $\mathscr{Q}$ up to a set of measure zero, $D_{\rm
a}\Phi=\lambda$ a.e. in $\mathscr{Q}$.

Therefore, it remains to construct the sequence $\Phi_{j}$ with the properties
described above and after the construction is completed to prove that the
homeomorphisms $\Phi_{j}$ can be approximated by volume preserving
diffeomorphisms.

We will construct a sequence $\Phi_{j}$ by induction as a sequence of measure
preserving homeomorphisms that are identity on the boundary, and satisfy
properties (6.10) and (6.11). Then, the properties listed in (6.12) will be
verified directly, but they are not needed to run the induction.

In the initial step we choose $\Phi_{0}={\rm id\,}:\mathscr{Q}\to\mathscr{Q}$,
so obviously $B_{0}=E$ (up to a set of measure zero) and conditions (6.10) and
(6.11) are satisfied. Now suppose that we already constructed homeomorphisms
$\Phi_{\ell}$ for $\ell\leq j$; we will describe construction of $\Phi_{j+1}$
as a modification of $\Phi_{j}$.

The homeomorphism $\Phi_{j}^{-1}$ is uniformly continuous on $\mathscr{Q}$, so
let $\delta_{j}>0$ be such that $|\Phi_{j}^{-1}(x)-\Phi_{j}^{-1}(y)|\leq
2^{-j}$ whenever $|x-y|\leq\delta_{j}$.

Next, we choose a finite family of closed cubes $K^{j}_{i}$,
$i=1,\ldots,m_{j}$, with pairwise disjoint interiors, with
$\operatorname{diam}K_{i}^{j}\leq\min(\delta_{j},2^{-j})$, and such that
$\bigcup_{i=1}^{m_{j}}K_{i}^{j}$ well approximates the set $\Phi_{j}(B_{j})$
measurewise:

(6.13)
$\bigg{|}\Phi_{j}(B_{j})\vartriangle\bigcup_{i=1}^{m_{j}}K_{i}^{j}\bigg{|}\leq
2^{-(j+1)}.$

We set $L_{j}=\Phi^{-1}_{j}(\bigcup_{i=1}^{m_{j}}K_{i}^{j})$; then (6.13) and
the fact that $\Phi_{j}$ is a measure preserving homeomorphism yields
$|B_{j}\vartriangle L_{j}|\leq 2^{-(j+1)}$.

Note that with this choice

$|L_{j}|\leq|B_{j}|+|B_{j}\vartriangle L_{j}|\leq 2^{-j}+2^{-(j+1)}<2^{-j+1}.$

Let $\kappa_{i}^{j}:K_{i}^{j}\to\mathscr{Q}$ be the standard similarity maps
between the cubes and define

$\Psi_{j}(y):=\begin{cases}(\kappa_{i}^{j})^{-1}\circ\Psi\circ\kappa_{i}^{j}(y)&\text{
if }y\in K_{i}^{j},\\\ y&\ \text{ if
}y\in\mathscr{Q}\setminus\bigcup_{i=1}^{m_{j}}K_{i}^{j}.\end{cases}$

Clearly,

(6.14) $D_{\rm a}\Psi_{j}(y)=\mathcal{R}\qquad\text{for almost all
}y\in\bigcup_{i=1}^{m_{j}}K_{i}^{j}=\Phi_{j}(L_{j}).$

Since the cubes $K_{i}^{j}$ have pairwise disjoint interiors, it follows that
$\Psi_{j}$ is a measure preserving homeomorphism. Moreover,
$\Psi_{j}|_{\partial\mathscr{Q}}={\rm id\,}$. Then we define
$\Phi_{j+1}=\Psi_{j}\circ\Phi_{j}$ and clearly $\Phi_{j+1}$ is measure
preserving, too, with $\Phi_{j+1}|_{\partial\mathscr{Q}}={\rm id\,}$. Note
that $\Psi_{j}$ is approximately differentiable a. e. and $\det D_{\rm
a}\Psi=\pm 1$ a. e. on $\mathscr{Q}$ so by Lemma 6.6, $\Phi_{j+1}$ which is
the composition of $\Psi_{j}$ with $\Phi_{j}$, is approximately differentiable
a. e. on $\mathscr{Q}$.

The homeomorphisms $\Phi_{j}$ and $\Phi_{j+1}$ differ only on the set
$L_{j}=\bigcup_{i}\Phi_{j}^{-1}(K_{i}^{j})$. Since both
$\operatorname{diam}\Phi_{j}^{-1}(K_{i}^{j})$ and
$\operatorname{diam}K_{i}^{j}$ are at most $2^{-j}$, we have
$d(\Phi_{j},\Phi_{j+1})\leq 2^{-j+1}$.

Also, the definition of the set $L_{j}$ and the construction of $\Phi_{j+1}$
shows that

$L_{j}=\overline{\\{x\in\mathscr{Q}:\,\Phi_{j}(x)\neq\Phi_{j+1}(x)\\}}.$

It follows from Lemma 6.5 and (6.14) that

$D_{\rm a}\Phi_{j+1}(x)=D_{\rm a}\Psi_{j}(\Phi_{j}(x))\cdot D_{\rm
a}\Phi_{j}(x)=\mathcal{R}D_{\rm a}\Phi_{j}(x)\quad\text{for almost all }x\in
L_{j}.$

Thus, $D_{\rm a}\Phi_{j+1}=\lambda$ on $L_{j}\cap B_{j}$. Also,

$D_{\rm a}\Phi_{j+1}(x)=D_{\rm a}\Phi_{j}(x)\quad\text{for almost all
}x\in\mathscr{Q}\setminus L_{j},$

so $D_{\rm a}\Phi_{j+1}=\lambda$ on $(\mathscr{Q}\setminus L_{j})\setminus
B_{j}$. This means that the set $B_{j+1}$ of points where $D_{\rm
a}\Phi_{j+1}=\mathcal{R}\lambda$ is contained (modulo a set of measure zero)
in the complement of the union of these two sets, which is $B_{j}\vartriangle
L_{j}$, and hence

$|B_{j+1}|\leq|B_{j}\vartriangle L_{j}|\leq 2^{-(j+1)}.$

This completes the proof that $\Phi_{j+1}$ satisfies the induction hypothesis.
What is left to prove is the last part of (6.12),
$\Big{|}B_{j}\setminus\bigcup_{\ell=j}^{\infty}L_{\ell}\Big{|}=0$.

To see that, it suffices to realize that for all $k>j$ the mappings $\Phi_{k}$
have bad derivative $D_{\rm a}\Phi_{k}=\mathcal{R}\lambda$ at almost all
points of $B_{j}\setminus\bigcup_{\ell=j}^{\infty}L_{\ell}$, because
$\Phi_{k}$ is obtained from $\Phi_{j}$ by a sequence of modifications which
happen only in $\bigcup_{\ell=j}^{\infty}L_{\ell}$. Thus
$B_{j}\setminus\bigcup_{\ell=j}^{\infty}L_{\ell}\subset\bigcap_{k\geq j}B_{k}$
(modulo a set of measure zero). Since $|B_{k}|\to 0$, it follows that
$\Big{|}B_{j}\setminus\bigcup_{\ell=j}^{\infty}L_{\ell}\Big{|}=0$.

Now it remains to show that the homeomorphism $\Phi$ can be approximated in
the metric $d$ by volume preserving diffeomorphisms that are identity in a
neighborhood of $\partial Q$.

Let $\Psi_{k}\overset{d}{\to}\Psi$ be a sequence of volume preserving
diffeomorphisms, $\Psi_{k}={\rm id\,}$ near $\partial\mathscr{Q}$, from
Theorem 1.5. Note that the diffeomorphisms

$\Psi_{j,k}:=\begin{cases}(\kappa_{i}^{j})^{-1}\circ\Psi_{k}\circ\kappa_{i}^{j}(y)&\text{
if }y\in K_{i}^{j},\\\ y&\ \text{ if
}y\in\mathscr{Q}\setminus\bigcup_{i=1}^{m_{j}}K_{i}^{j}\end{cases}$

are volume preserving and $\Psi_{j,k}={\rm id\,}$ near $\partial\mathscr{Q}$.
Clearly, $\Psi_{j,k}\overset{d}{\to}\Psi_{j}$ as $k\to\infty$.

An easy induction on $j$ shows that $\Phi_{j}$ can be approximated in the
metric $d$ by $C^{\infty}$ volume preserving diffeomorphisms $\Phi_{j,k}$ that
are identity near $\partial\mathscr{Q}$. If $j=0$, then $\Phi_{0}={\rm id\,}$
and we can take $\Phi_{0,k}:=\Phi_{0}$. If the claim is true for $j$,
$\Phi_{j,k}\overset{d}{\to}\Phi_{j}$, then, it is easy to check that
$\Phi_{j+1,k}:=\Psi_{j,k}\circ\Phi_{j,k}\overset{d}{\to}\Psi_{j}\circ\Phi_{j}=\Phi_{j+1}$.
Therefore, for each $j$, we can find $k_{j}$ such that
$d(\Phi_{j,k_{j}},\Phi_{j})<2^{-j}$ and hence
$\Phi_{j,k_{j}}\overset{d}{\to}\Phi$. ∎

##  7\. Proof of Theorem 1.4

Proof of Theorem 1.4.

###  7.1. Reduction to the case $\det T>0$

It suffices to prove Theorem 1.4 under an additional assumption that $\det
T>0$ a.e. Indeed, assume that we have already proven Theorem 1.4 under this
assumption and, for a general $T$, let

$\hat{T}(x)=\begin{cases}T(x)&\text{ if }\det T\geq 0,\\\
\mathcal{R}T(x)&\text{ if }\det T<0,\end{cases}$

where $\mathcal{R}$ is defined in (6.8). Then $\det\hat{T}=|\det T|>0$ a.e.
Let $\hat{\Phi}$ be the almost everywhere approximately differentiable
homeomorphism provided by Theorem 1.4 with $\hat{T}$ in place of $T$, $D_{\rm
a}\hat{\Phi}=\hat{T}$ a.e.

Let $E:=\\{\det T<0\\}$. Note that since $\hat{\Phi}$ satisfies the Lusin (N)
condition, the set $\hat{E}:=\hat{\Phi}(E)$ is measurable (because $E$ is the
union of a Borel set and a set of measure zero). Theorem 6.8 yields an a.e.
approximately differentiable homeomorphism $\Phi^{\prime}$ such that
$\Phi^{\prime}|_{\partial\mathscr{Q}}={\rm id\,}$ and

$D_{\rm a}\Phi^{\prime}(x)=\begin{cases}\mathcal{R}&\text{ for a.e.
}x\in\hat{E},\\\ \mathcal{I}&\text{ for a.e.
}x\in\mathscr{Q}\setminus\hat{E}.\end{cases}$

Then Lemma 6.5 implies that the composition
$\Phi:=\Phi^{\prime}\circ\hat{\Phi}$ is a.e. approximately differentiable and
satisfies the chain rule (6.2), so for a.e. $x\in\mathscr{Q}$ we have

(7.1) $\begin{split}D_{\rm a}\Phi(x)=D_{\rm
a}\Phi^{\prime}(\hat{\Phi}(x))\,D_{\rm
a}\hat{\Phi}(x)&=\left.\begin{cases}\mathcal{R}\,\mathcal{R}T(x)&\text{ for
a.e. }x\in E,\\\ \mathcal{I}\,T(x)&\text{ for a.e. }x\in\mathscr{Q}\setminus
E\\\ \end{cases}\right\\}=T(x).\end{split}$

Obviously, $\Phi|_{\partial\mathscr{Q}}={\rm id\,}$ and, since $\Phi^{\prime}$
is measure preserving (see Remark 6.9) and $\hat{\Phi}$ preserves sets of
measure zero, $\Phi$ preserves sets of measure zero as well.

Since by Theorem 6.8 and Theorem 1.4 the homeomorphisms $\Phi^{\prime}$ and
$\hat{\Phi}$ can be approximated in the uniform metric $d$ by
$C^{\infty}$-diffeomorphisms $\Phi^{\prime}_{k}$ and $\hat{\Phi}_{k}$,
$\Phi^{\prime}_{k}=\hat{\Phi}_{k}={\rm id\,}$ near $\partial\mathscr{Q}$, one
can easily check that

$\Phi^{\prime}_{k}\circ\hat{\Phi}_{k}\overset{d}{\to}\Phi^{\prime}\circ\hat{\Phi}=\Phi.$

It remains to check that $\Phi^{-1}$ is approximately differentiable a. e. on
$\mathscr{Q}$ and $D_{\rm a}\Phi^{-1}(y)=T^{-1}(\Phi^{-1})(y)$ for a. e.
$y\in\mathscr{Q}$. Indeed, $\Phi^{-1}=\hat{\Phi}^{-1}\circ\Phi^{\prime-1}$ and
both $\hat{\Phi}^{-1}$ (by Theorem 1.4) and $\Phi^{\prime-1}$ (see Remark 6.9)
are approximately differentiable a. e. on $\mathscr{Q}$. Since $\det D_{\rm
a}(\Phi^{\prime-1})\neq 0$ a. e. (Remark 6.9), $\Phi^{-1}$ is approximately
differentiable a. e. by Lemma 6.5. Then applying the chain rule (6.2) to
$\Phi^{-1}\circ\Phi=\mathcal{I}$ yields that the derivative of $\Phi^{-1}$ has
the required form.

This concludes the proof of Theorem 1.4 in the general case, provided we can
prove it for $T$ such that $\det T>0$ a.e. in $\mathscr{Q}$.

Thus, from now on, we assume that $\det T>0$ a.e.

###  7.2. General outline of the construction

The general plan of the proof is as follows: To construct a homeomorphism
$\Phi$ such that $\Phi|_{\partial\mathscr{Q}}={\rm id\,}$ and $D_{\rm
a}\Phi=T$ a. e., we shall iterate the construction from Theorem 1.2 on the
smaller and smaller subsets of $\mathscr{Q}$ on which the derivative is not
yet as required.

We inductively show that there exists a family of orientation preserving
$C^{1}$-diffeomorphisms $\Phi_{k}$ of $\mathscr{Q}$ and Borel sets
$C_{k}\subset\mathscr{Q}$ with the following properties for $k\geq 1$:

  1. (i)

$\Phi_{k}=\mathrm{id}$ near $\partial\mathscr{Q}$;

  2. (ii)

$\Phi_{k+1}=\Phi_{k}$ on $C_{k}$;

  3. (iii)

$D\Phi_{k}=T$ on $C_{k}$;

  4. (iv)

$C_{k}$ is an increasing family of sets, $C_{1}\subset C_{2}\subset\cdots$,
with $\lim_{k\to\infty}|C_{k}|=1$;

  5. (v)

$d(\Phi_{k},\Phi_{k+1})<2^{-(k-1)}$ for $k\geq 2$.

The limit map $\Phi:=\lim_{k\to\infty}\Phi_{k}$ is the required homeomorphism.
Indeed, property (v) implies that $(\Phi_{k})$ is a Cauchy sequence in the
uniform metric $d$, hence its limit is a homeomorphism as shown in Lemma 3.20.
By (i), $\Phi=\mathrm{id}$ on $\partial\mathscr{Q}$. Note that $\Phi=\Phi_{k}$
on $C_{k}$ by (ii) and (iv). Therefore, Lemma 6.3 and (iii) imply that

$D_{\rm a}\Phi=T\text{ a.\,e.\ on }\bigcup_{k=1}^{\infty}C_{k},$

and hence $D_{\rm a}\Phi=T$ a.e. on $\mathscr{Q}$, because $|C_{k}|\to 1$.

At this point, let us stress that Lemma 6.6 implies that $\Phi$ and
$\Phi^{-1}$ satisfy the Lusin (N) condition which is part (b) of Theorem 1.4
and property (a) follows directly from Lemma 6.7.

Finally, $\Phi_{k}$ are $C^{1}$-diffeomorphisms, identity near
$\partial\mathscr{Q}$, that converge to $\Phi$ in the uniform metric $d$, but
according to Lemma 3.21, each $\Phi_{k}$ can be approximated in the metric $d$
by $C^{\infty}$-diffeomorphisms and (c) follows.

This completes the proof of Theorem 1.4 and it remains to construct
diffeomorphisms $\Phi_{k}$ and Borel sets $C_{k}$ satisfying (i)-(v).

The construction of the family $\Phi_{k}$ and of the sets $C_{k}$ is
complicated. In fact, the sets $C_{k}$ are not constructed inductively, but
defined only at the end, when all the steps of the inductive construction are
concluded.

The actual inductive construction provides a quadruple
$(\Phi_{k},\mathscr{P}_{k},E_{k},L_{k})_{k=1}^{\infty}$:

  * •

a diffeomorphism $\Phi_{k}$ of $\mathscr{Q}$ onto itself; the diffeomorphism
$\Phi_{k}$ is constructed by a modification of $\Phi_{k-1}$,

  * •

a partition $\mathscr{P}_{k}$ of the unit cube; the corrections leading from
$\Phi_{k-1}$ to $\Phi_{k}$ are done at a small scale, i.e., within the
elements of the partition $\mathscr{P}_{k-1}$.

  * •

a large set $E_{k}$, on which $D\Phi_{k}=T$,

  * •

a small set $L_{k}\subset E_{k-1}$ such that $\Phi_{k}=\Phi_{k-1}$ on
$E_{k-1}\setminus L_{k}$ (although $D\Phi_{k-1}=T$ on $E_{k-1}$, for technical
reasons in the construction of $\Phi_{k}$ from $\Phi_{k-1}$ we alter
$\Phi_{k}$ on the subset $L_{k}$ of $E_{k-1}$),

  * •

the sets $C_{k}$ are constructed at the very end by (7.5).

To be more precise, we have, for $k\geq 1$,

  1. (a)

a $C^{1}$-diffeomorphism $\Phi_{k}:\mathscr{Q}\to\mathscr{Q}$,
$\Phi_{k}=\mathrm{id}$ near $\partial\mathscr{Q}$;

  2. (b)

a compact set $E_{k}\subset\mathscr{Q}$ such that $D\Phi_{k}=T$ on $E_{k}$;

  3. (c)

$2^{-(k+1)}<|\mathscr{Q}\setminus E_{k}|<2^{-k}$;

  4. (d)

a Borel set $L_{k}\subset E_{k-1}$ for $k\geq 2$, such that
$\Phi_{k}=\Phi_{k-1}$ on $E_{k-1}\setminus L_{k}$ and $|L_{k}|<2^{-k}$;

  5. (e)

a partition $\mathscr{P}_{k}$ of the unit cube,
$\mathscr{P}_{k}=\\{P_{ki}\\}_{i=1}^{M_{k}}$ for $k\geq 2$, such that

(7.2) $\Phi_{k}(P_{k-1,i})=\Phi_{k-1}(P_{k-1,i})\quad\text{ for }k\geq 3,$
(7.3) $|\Phi_{k}(P_{ki})|=\int_{P_{ki}}\det T(x)\,dx\quad\text{ for }k\geq 2,$
(7.4)
$\operatorname{diam}{P_{ki}}<2^{-k},\quad\operatorname{diam}{\Phi_{k}(P_{ki})}<2^{-k}\quad\text{
for }k\geq 2.$

We will show that the family of diffeomorphisms $(\Phi_{k})$ with properties
(a)-(e) satisfies conditions (i)-(v) for

(7.5) $C_{k}:=\bigcap_{j=k}^{\infty}\left(E_{j}\setminus L_{j+1}\right).$

Conditions (i) and (a) are the same. Clearly, $C_{k}\subset E_{k}$, which
means that condition (iii) is satisfied. Applying (d) with $k+1$ in place of
$k$, we get $\Phi_{k+1}=\Phi_{k}$ on $E_{k}\setminus L_{k+1}$ and since
$C_{k}\subset E_{k}\setminus L_{k+1}$, condition (ii) also holds. Since the
sets $C_{k}$ form an increasing family of sets, in order to show (iv), it
remains to check that $\lim_{k\to\infty}|C_{k}|=1$.

For the sake of this calculation, set $A_{j}:=E_{j}\setminus L_{j+1}$. Since
$L_{j+1}\subset E_{j}$ and in view of (c) and (d),

(7.6) $|\mathscr{Q}\setminus A_{j}|=|\mathscr{Q}\setminus
E_{j}|+|L_{j+1}|<2^{-j}+2^{-(j+1)}=3\cdot 2^{-(j+1)}.$

Therefore,

$|\mathscr{Q}\setminus
C_{k}|=\Big{|}\mathscr{Q}\setminus\bigcap_{j=k}^{\infty}A_{j}\Big{|}=\Big{|}\bigcup_{j=k}^{\infty}(\mathscr{Q}\setminus
A_{j})\Big{|}\leq\sum_{j=k}^{\infty}|\mathscr{Q}\setminus
A_{j}|<3\sum_{j=k}^{\infty}2^{-(j+1)}=3\cdot 2^{-k}.$

This implies that

$|C_{k}|>1-3\cdot 2^{-k}$

and since $|C_{k}|\leq 1$, this shows that $|C_{k}|\to 1$ and finishes the
proof of (iv). It remains to show (v).

It follows from condition (e) that for $k\geq 3$ the diffeomorphism $\Phi_{k}$
is a modification of $\Phi_{k-1}$ in each of the sets $P_{k-1,i}$, i.e.,
$\Phi_{k}(P_{k-1,i})=\Phi_{k-1}(P_{k-1,i})$. Hence

$\|\Phi_{k}-\Phi_{k-1}\|_{\infty}\leq\max_{i}\big{\\{}\operatorname{diam}\Phi_{k-1}(P_{k-1,i})\big{\\}}<2^{-(k-1)},$
$\|\Phi_{k}^{-1}-\Phi_{k-1}^{-1}\|_{\infty}\leq\max_{i}\\{\operatorname{diam}P_{k-1,i}\\}<2^{-(k-1)},$

so $d(\Phi_{k},\Phi_{k-1})<2\cdot 2^{-(k-1)}=2^{-(k-2)}$ and (v) follows. The
proof of properties (i)-(v) is complete.333In the construction of $\Phi_{2}$
from $\Phi_{1}$, we modify $\Phi_{1}$ in $\mathscr{Q}$, so in this step we use
the trivial partition $\mathscr{P}_{1}=\\{\mathscr{Q}\\}$.

###  7.3. Construction of $\Phi_{1}$ and $\Phi_{2}$.

By a direct application of Theorem 1.2, we obtain a diffeomorphism $\Phi_{1}$
of the unit cube, $\Phi_{1}=\mathrm{id}$ near $\partial\mathscr{Q}$, and a
compact set $E_{1}$ such that $D\Phi_{1}=T$ on $E_{1}$ and
$1/4<|\mathscr{Q}\setminus E_{1}|<1/2$. Note that $\Phi_{1}$ and $E_{1}$
satisfy conditions (a)-(e), because conditions (d) and (e) do not apply to
$k=1$.

We shall now describe in detail the construction of $\Phi_{2}$, which
demonstrates all the crucial aspects of the construction of $\Phi_{k}$ based
on $\Phi_{k-1}$. The induction step for general $k$ will be described later.

In the course of the proof we use two numbers $\alpha=1/2$, $\beta=3/4$. We
write $\alpha$, $\beta$ instead of the actual fractions as we believe it makes
it easier to transfer this argument to the proof for arbitrary $k$. Note that
$|E_{1}|>1-\alpha$.

We begin with correcting the way $\Phi_{1}$ distributes measure so that we are
later able to apply Theorem 1.2 in sets of smaller diameter. To this end, we
use Proposition 5.8 for the compact set $\Phi_{1}(E_{1})$ and functions

$f(y)=\det T(\Phi_{1}^{-1}(y))\det D\Phi_{1}^{-1}(y)\quad\text{and}\quad
g(y)=1.$

We check that (5.16) is satisfied. By change of variables,

$\int_{\mathscr{Q}}\det T(\Phi_{1}^{-1}(y))\det
D\Phi_{1}^{-1}(y)\,dy=\int_{\Phi_{1}^{-1}(\mathscr{Q})}\det
T(x)\,dx=\int_{\mathscr{Q}}\det T(x)\,dx=\int_{\mathscr{Q}}1\,dx.$

Moreover, since $D\Phi_{1}=T$ on $E_{1}$, we have $f(y)=g(y)$ for all
$y\in\Phi_{1}(E_{1})$. Therefore, the assumptions of Proposition 5.8 are
satisfied.

By Proposition 5.8, we get a diffeomorphism $\Psi:\mathscr{Q}\to\mathscr{Q}$,
a partition $\mathscr{R}=\\{R_{2i}\\}_{i=1}^{2^{nN_{2}}}$ of $\mathscr{Q}$ and
a compact set $\widetilde{K}\subset\Phi_{1}(E_{1})$ with properties described
below.

Partition $\mathscr{R}$ is a diffeomorphic dyadic partition and satisfies

(7.7) $\operatorname{diam}\Psi(R_{2i})<1/4\text{ and
}\operatorname{diam}(\Phi_{1}^{-1}(R_{2i}))<1/4$

(we use here the uniform continuity of $\Phi_{1}^{-1}$) and

$\int_{R_{2i}}f(y)\,dy=|\Psi(R_{2i})|,$

which after a change of variables in the integral becomes

(7.8) $\int_{\Phi_{1}^{-1}(R_{2i})}\det T(x)\,dx=|\Psi(R_{2i})|.$

We find $\eta>0$ such that for any measurable set $A\subset\mathscr{Q}$ with
$|A|<\eta$, $\int_{A}\det\,D\Phi^{-1}_{1}<\alpha^{2}/2$. We choose
$\widetilde{K}\subset\Phi_{1}(E_{1})$ so that

$|\Phi_{1}(E_{1})\setminus\widetilde{K}|<\eta.$

Moreover, $\Psi=\mathrm{id}$ near $\widetilde{K}\cup\partial\mathscr{Q}$. In
view of the inequality above, setting $K:=\Phi_{1}^{-1}(\widetilde{K})\subset
E_{1}$, we have

(7.9) $|E_{1}\setminus
K|=|\Phi_{1}^{-1}\,(\Phi_{1}(E_{1})\setminus\widetilde{K})|<\alpha^{2}/2.$

Eventually, we set

$\widetilde{\Phi}_{2}:=\Psi\circ\Phi_{1}\quad\text{and}\quad\mathscr{P}_{2}:=\\{P_{2i}\\}_{i=1}^{M_{2}},\text{
where }M_{2}=2^{nN_{2}},\text{ and }P_{2i}:=\Phi^{-1}_{1}(R_{2i}).$

Then (7.7) and (7.8) become

(7.10)
$\operatorname{diam}{\widetilde{\Phi}_{2}(P_{2i})}<1/4,\quad\operatorname{diam}({P_{2i}})<1/4,\quad\int_{P_{2i}}\det
T(x)\,dx=|\widetilde{\Phi}_{2}(P_{2i})|,$

i. e., diffeomorphism $\widetilde{\Phi}_{2}$ satisfies (e) for $k=2$
(condition (7.2) does not apply to $k=2$). Observe also that
$\widetilde{\Phi}_{2}=\Phi_{1}$ near $K\cup\partial\mathscr{Q}$. Consequently,

(7.11) $D\widetilde{\Phi}_{2}=T\text{ on
}K,\quad\widetilde{\Phi}_{2}=\mathrm{id}\text{ near
}\partial\mathscr{Q}\quad\text{and}\quad\widetilde{\Phi}_{2}=\Phi_{1}\text{ on
}K\subset E_{1}.$

We construct $\Phi_{2}$ by replacing $\widetilde{\Phi}_{2}$ inside each
$P_{2i}$ with a diffeomorphism $\Phi_{2i}$ which has correct derivative
$D\Phi_{2i}=T$ on a larger set. To this end, we would like to keep
$\widetilde{\Phi}_{2}$ on $K$ unchanged and apply Theorem 1.2 to the open set
$\mathring{P}_{2i}\setminus K$. Unfortunately, this set need not be connected
and Theorem 1.2 cannot be applied. To overcome this difficulty we use Lemma
5.12 to find a compact set $\widetilde{E}_{2i}\subset\mathring{P}_{2i}\cap K$
so that $\mathring{P}_{2i}\setminus\widetilde{E}_{2i}$ is connected and

(7.12) $|(\mathring{P}_{2i}\cap
K)\setminus\widetilde{E}_{2i}|<\alpha^{2}/(2M_{2}).$

Set

$\widetilde{E}_{2}:=\bigcup_{i=1}^{M_{2}}\widetilde{E}_{2i}\subset
E_{1}\quad\text{and}\quad L_{2}:=E_{1}\setminus\widetilde{E}_{2}.$

Clearly, $\widetilde{E}_{2}$ is compact and summing (7.12) over
$i=1,\ldots,M_{2}$ yields

$|K\setminus\widetilde{E}_{2}|<\alpha^{2}/2.$

By (7.9), we arrive at

(7.13) $|L_{2}|=|E_{1}\setminus\widetilde{E}_{2}|=|E_{1}\setminus
K|+|K\setminus\widetilde{E}_{2}|<\alpha^{2}.$

Consequently,

(7.14) $|\widetilde{E}_{2}|=|E_{1}|-|L_{2}|>1-\alpha-\alpha^{2}.$

Let us stress that $\widetilde{E}_{2}\subset K$ so (7.11) yields

(7.15) $D\widetilde{\Phi}_{2}=T\text{ on }\widetilde{E}_{2}$

and

(7.16) $\widetilde{\Phi}_{2}=\Phi_{1}\text{ on
}\widetilde{E}_{2}=E_{1}\setminus L_{2}.$

Although $D\Phi_{1}=T$ on $E_{1}$, as explained earlier, the set $L_{2}$ is
the set on which we will have spoiled the already prescribed derivative of
$\Phi_{1}$. This is the cost we bear in order to be able to prescribe the
derivative farther. One can also think of the set $\widetilde{E}_{2}$ as the
set of points in which the prescribed derivative survives the transition from
$\Phi_{1}$ to $\Phi_{2}$.

Let us now focus on applying Theorem 1.2, i. e. , correcting the derivative of
$\widetilde{\Phi}_{2}$. The set

$\Omega_{2i}:=\mathring{P}_{2i}\setminus\widetilde{E}_{2i}$

is open and connected. Observe that, setting
$\Omega_{2}:=\bigcup_{i}\Omega_{2i}$,

$|\Omega_{2}|=|\mathscr{Q}\setminus\widetilde{E}_{2}|,$

since the set $\Omega_{2}$ coincides with
$\mathscr{Q}\setminus\widetilde{E}_{2}$ up to a set of measure zero which
consists of boundaries of the diffeomorphic cubes $P_{2i}$. We check that
$\Omega_{2i}$ satisfies

(7.17)
$\displaystyle\begin{split}|\widetilde{\Phi}_{2}(\Omega_{2i})|&=|\widetilde{\Phi}_{2}(P_{2i})|-|\widetilde{\Phi}_{2}(\widetilde{E}_{2i})|\overset{\eqref{eq92}}{=}\int_{P_{2i}}\det
T(x)\,dx-\int_{\widetilde{E}_{2i}}\det D\widetilde{\Phi}_{2}(x)\,dx\\\
&\overset{\eqref{eq97}}{=}\int_{P_{2i}}\det
T(x)\,dx-\int_{\widetilde{E}_{2i}}\det T(x)\,dx=\int_{\Omega_{2i}}\det
T(x)\,dx.\end{split}$

We are now in position to use Theorem 1.2 for the domain $\Omega_{2i}$ and the
diffeomorphism $\widetilde{\Phi}_{2}$. We find a diffeomorphism
$\Phi_{2i}:\Omega_{2i}\to\widetilde{\Phi}_{2}(\Omega_{2i})$ and a compact set
$E^{\prime}_{2i}\subset\Omega_{2i}$ such that

(7.18) $D\Phi_{2i}=T\text{ on
}E^{\prime}_{2i},\quad\Phi_{2i}=\widetilde{\Phi}_{2}\text{ near
}\partial\Omega_{2i}\quad\text{and}\quad|E^{\prime}_{2i}|>\beta|\Omega_{2i}|.$

Let

$E^{\prime}_{2}:=\bigcup_{i=1}^{M_{2}}E^{\prime}_{2i}.$

Clearly, $E^{\prime}_{2}$ is compact. By the third condition in (7.18), we
have

$|E^{\prime}_{2}|>\beta|\Omega_{2}|=\beta|\mathscr{Q}\setminus\widetilde{E}_{2}|.$

We replace the diffeomorphism $\widetilde{\Phi}_{2}$ with $\Phi_{2i}$ inside
each $\Omega_{2i}$, setting

$\Phi_{2}:=\begin{cases}\Phi_{2i}&\text{on }\Omega_{2i},\\\
\widetilde{\Phi}_{2}&\text{on }\mathscr{Q}\setminus\Omega_{2}.\end{cases}$

Observe that thanks to the second condition in (7.18), $\Phi_{2}$ is indeed a
diffeomorphism. Moreover, $\Phi_{2}=\widetilde{\Phi}_{2}$ near
$\bigcup_{i}\partial P_{2i}$. Since
$\partial\mathscr{Q}\subset\bigcup_{i}\partial P_{2i}$, $\Phi_{2}={\rm id\,}$
near $\partial\mathscr{Q}$, whence (a) holds.

Note that $\widetilde{E}_{2}\subset\mathscr{Q}\setminus\Omega_{2}$ and
$\Phi_{2}=\widetilde{\Phi}_{2}$ in a neighborhood of $\widetilde{E}_{2}$.
Hence, the first condition in (7.18) together with (7.15) imply that

$D\Phi_{2}=T\text{ on }\hat{E}_{2}:=E^{\prime}_{2}\cup\widetilde{E}_{2}.$

We calculate

$\displaystyle\begin{split}|\hat{E}_{2}|&=|E^{\prime}_{2}|+|\widetilde{E}_{2}|>\beta|\mathscr{Q}\setminus\widetilde{E}_{2}|+|\widetilde{E}_{2}|=\beta|\mathscr{Q}|+(1-\beta)|\widetilde{E}_{2}|\\\\[5.69054pt]
&\overset{\eqref{eq96}}{>}\beta+(1-\beta)(1-\alpha-\alpha^{2})=1-\alpha(1+\alpha)(1-\beta)\\\\[5.69054pt]
&>1-\alpha^{2},\end{split}$

since $\beta>1/(1+\alpha)$. Therefore,
$|\mathscr{Q}\setminus\hat{E}_{2}|<\alpha^{2}=1/4$. By discarding some points
from $\hat{E}_{2}$ if necessary, we obtain a compact set
$E_{2}\subset\hat{E}_{2}$ such that $1/8<|\mathscr{Q}\setminus E_{2}|<1/4$.
Clearly, $D\Phi_{2}=T$ on $E_{2}$.

Let us check if $\Phi_{2}$ satisfies properties (a)-(e). We have already
verified conditions (a), (b) and (c). Moreover, in view of (7.10), (e) holds
as well, because $\Phi_{2}(P_{2i})=\widetilde{\Phi}_{2}(P_{2i})$ ((7.2) does
not apply to $k=2$). Since $\Phi_{2}=\widetilde{\Phi}_{2}$ on
$\widetilde{E}_{2}=E_{1}\setminus L_{2}$, recalling (7.16) and (7.13), we see
that (d) also holds.

###  7.4. Construction of $\Phi_{k}$ given $\Phi_{k-1}$.

Let $k\geq 3$. As in the construction of $\Phi_{2}$, we use $\alpha=1/2$ and
$\beta=3/4$. We assume that we have found $\Phi_{k-1}$ satisfying properties
(a)-(e) for $k-1$ instead of $k$ and we show how to construct $\Phi_{k}$. In
fact, we only need the following properties from the previous step:

  * •

$\Phi_{k-1}:\mathscr{Q}\to\mathscr{Q}$, $\Phi_{k-1}={\rm id\,}$ near
$\partial\mathscr{Q}$,

  * •

$D\Phi_{k-1}=T$ on a compact set $E_{k-1}$, $\alpha^{k}<|\mathscr{Q}\setminus
E_{k-1}|<\alpha^{k-1}$,

  * •

$\Phi_{k-1}(P_{k-1,i})=\int_{P_{k-1,i}}\det T(x)\,dx$.

In the construction of $\Phi_{2}$ from $\Phi_{1}$, we alter $\Phi_{1}$ inside
$\mathscr{Q}$ while keeping $\Phi_{2}=\Phi_{1}$ near $\partial\mathscr{Q}$. In
the construction of $\Phi_{k}$ from $\Phi_{k-1}$ we repeat the same
construction, but we alter $\Phi_{k-1}$ inside each diffeomorphic closed cube
$P_{k-1,i}$ while keeping $\Phi_{k}=\Phi_{k-1}$ near $\partial P_{k-1,i}$.
Therefore, the crucial though technical difference between the construction
for $k\geq 3$ is that we do not use Proposition 5.8 (stated for a closed cube)
but Proposition 5.11 (stated for a diffeomorphic closed cube).

Define $E_{k-1,i}:=E_{k-1}\cap P_{k-1,i}$ for $i=1,\ldots,M_{k-1}$. Note that
$E_{k-1,i}$ is compact and that $|\bigcup_{i}E_{k-1,i}|=|E_{k-1}|$.

Let us firstly show that it suffices to construct for $i=1,\ldots,M_{k-1}$

  * •

a family of diffeomorphisms $\Phi_{ki}:P_{k-1,i}\to\Phi_{k-1}(P_{k-1,i})$,

  * •

compact sets $\hat{E}_{ki}\subset\mathring{P}_{k-1,i}$,

  * •

Borel sets $L_{ki}\subset E_{k-1,i}$,

  * •

a partition $\mathscr{P}_{ki}=\\{P_{kij}\\}_{j=1}^{2^{nN_{ki}}}$ of
${P}_{k-1,i}$

such that

  1. (1)

$\Phi_{ki}=\Phi_{k-1}$ near $\partial P_{k-1,i}$;

  2. (2)

$D\Phi_{ki}=T$ on $\hat{E}_{ki}$;

  3. (3)

$|\hat{E}_{ki}|>\beta|P_{k-1,i}|+(1-\beta)|E_{k-1,i}\setminus L_{ki}|$;

  4. (4)

$\Phi_{ki}=\Phi_{k-1}$ on $E_{k-1,i}\setminus L_{ki}$ and
$|L_{ki}|<\alpha^{k}\,M_{k-1}^{-1}$;

  5. (5)

partition $\mathscr{P}_{ki}$ is a diffeomorphic dyadic partition and satisfies
for $j=1,\ldots,2^{nN_{ki}}$

$|\Phi_{ki}(P_{kij})|=\int_{P_{kij}}\det T(x)\,dx$

and

$\operatorname{diam}P_{kij}<2^{-k},\quad\operatorname{diam}\Phi_{ki}(P_{kij})<2^{-k}.$

We set

$\Phi_{k}:=\Phi_{ki}\text{ on }P_{k-1,i}.$

By condition (1), $\Phi_{k}$ is indeed a diffeomorphism and
$\Phi_{k}=\mathrm{id}$ near $\partial\mathscr{Q}$, which is (a). Next, we set

$\hat{E}_{k}:=\bigcup_{i=1}^{M_{k-1}}\hat{E}_{ki}\quad\text{and}\quad
L_{k}:=\bigcup_{i=1}^{M_{k-1}}L_{ki}\subset E_{k-1}.$

Since $D\Phi_{k}=D\Phi_{ki}=T$ on $\hat{E}_{ki}$, we have that $D\Phi_{k}=T$
on $\hat{E}_{k}$.

Summing (4) over $i=1,\dots,M_{k-1}$, we readily see that
$\Phi_{k}=\Phi_{k-1}$ on $E_{k-1}\setminus L_{k}$ and $|L_{k}|<\alpha^{k}$, i.
e., (d) holds. Moreover, summing (3) over $i=1,\ldots,M_{k-1}$ and recalling
that $|E_{k-1}|>1-\alpha^{k-1}$ , we get

$|\hat{E}_{k}|>\beta|\mathscr{Q}|+(1-\beta)|E_{k-1}\setminus
L_{k}|>\beta+(1-\beta)(1-\alpha^{k-1}-\alpha^{k})>1-\alpha^{k},$

so $|\mathscr{Q}\setminus\hat{E}_{k}|<\alpha^{k}$. By discarding some points
from $\hat{E}_{k}$ if necessary, we can find another compact set
$E_{k}\subset\hat{E}_{k}$ such that $\alpha^{k+1}<|\mathscr{Q}\setminus
E_{k}|<\alpha^{k}$. Clearly, $D\Phi_{k}=T$ on $E_{k}$ i. e., conditions (b)
and (c) are satisfied. It remains to verify (e).

We define the partition444While $\mathscr{P}_{ki}$ are a diffeomorphic dyadic
partitions, $\mathscr{P}_{k}$ need not be, because we might divide each of the
diffeomorphic cubes $P_{k-1,i}$ into a different number of diffeomorphic
dyadic cubes. $\mathscr{P}_{k}$ as the union of partitions $\mathscr{P}_{ki}$

$\mathscr{P}_{k}=\bigcup_{i=1}^{M_{k-1}}\bigcup_{j=1}^{2^{nN_{ki}}}\\{P_{kij}\\}.$

After re-enumeration of the diffeomorphic cubes $P_{kij}$, we can write

$\mathscr{P}_{k}=\bigcup_{\ell=1}^{M_{k}}\\{P_{k\ell}\\},\quad\text{where}\quad
M_{k}=\sum_{i=1}^{M_{k-1}}2^{nN_{ki}}.$

Since $\Phi_{k}=\Phi_{ki}$ on $P_{k-1,i}$, and $\Phi_{ki}=\Phi_{k-1}$ near
$\partial P_{k-1,i}$, it follows that
$\Phi_{k}(P_{k-1,i})=\Phi_{k-1}(P_{k-1,i})$, which is (7.2). Since
$P_{k\ell}=P_{kij}$ for some $i,j$ and $\Phi_{k}=\Phi_{ki}$ on
$P_{k\ell}=P_{kij}\subset P_{k-1,i}$, condition (5) yields

$|\Phi_{k}(P_{k\ell})|=|\Phi_{ki}(P_{kij})|=\int_{P_{kij}}\det
T(x)\,dx=\int_{P_{k\ell}}\det T(x)\,dx,$

which is (7.3). Also

$\operatorname{diam}P_{k\ell}=\operatorname{diam}P_{kij}<2^{-k}\quad\text{and}\quad\operatorname{diam}\Phi_{k}(P_{k\ell})=\operatorname{diam}\Phi_{ki}(P_{kij})<2^{-k},$

which is (7.4). This completes the proof of (e) and hence that of (a)-(e).

Fix any $i=1,\ldots,M_{k-1}$. We shall now show that given $\Phi_{k-1}$, we
can construct $\Phi_{ki}$ as described above. As before, we begin by
correcting the way $\Phi_{k-1}$ distributes measure so that the subsequent
corrections from $\Phi_{k-1}$ to $\Phi_{k}$ are made at a smaller scale. To
this end, we use Proposition 5.11 for the diffeomorphic closed cube
$\Phi_{k-1}(P_{k-1,i})$, the compact set $\Phi_{k-1}(E_{k-1,i})$ and functions

$f(y)=\det T(\Phi_{k-1}^{-1}(y))\det D\Phi_{k-1}^{-1}(y)\quad\text{and}\quad
g(y)=1.$

We check that (5.20) is satisfied by change of variables and the inductive
assumption (7.3) for $\Phi_{k-1}$,

$\displaystyle\int_{\Phi_{k-1}(P_{k-1,i})}\det T(\Phi_{k-1}^{-1}(y))\det
D\Phi_{k-1}^{-1}(y)\,dy$ $\displaystyle=\int_{P_{k-1,i}}\det
T(x)\,dx\overset{\eqref{eq37}}{=}|\Phi_{k-1}(P_{k-1,i})|$
$\displaystyle=\int_{\Phi_{k-1}(P_{k-1,i})}1\,dx.$

Moreover, since $D\Phi_{k-1}=T$ on $E_{k-1,i}$, we have $f(y)=g(y)$ for all
$y\in\Phi_{k-1}(E_{k-1,i})$. Therefore, assumptions of Proposition 5.11 are
satisfied.

By Proposition 5.11, we get a diffeomorphism
$\Psi_{i}:\Phi_{k-1}(P_{k-1,i})\to\Phi_{k-1}(P_{k-1,i})$, a partition
$\mathscr{R}_{ki}=\\{R_{kij}\\}_{j=1}^{2^{nN_{ki}}}$ of
$\Phi_{k-1}(P_{k-1,i})$ and a compact set
$\widetilde{K}_{i}\subset\Phi_{k-1}(E_{k-1,i})$ with properties described
below.

Partition $\mathscr{R}_{ki}$ is a diffeomorphic dyadic partition and satisfies
an analogue of (7.7) and (7.8), namely

(7.19)
$\operatorname{diam}\Psi_{i}(R_{kij})<2^{-k},\quad\operatorname{diam}(\Phi_{k-1}^{-1}(R_{kij}))<2^{-k},$

and

(7.20) $\int_{\Phi^{-1}_{k-1}(R_{kij})}\det T(x)\,dx=|\Psi_{i}(R_{kij})|.$

We find $\eta>0$ such that for any measurable set $A\subset\mathscr{Q}$,

$|A|<\eta\quad\Rightarrow\quad\int_{A}\det
D\Phi_{k-1}^{-1}<\frac{\alpha^{k}}{2M_{k-1}}\,.$

We choose a compact set $\widetilde{K}_{i}\subset\Phi_{k-1}(E_{k-1,i})$ so
that

$|\Phi_{k-1}(E_{k-1,i})\setminus\widetilde{K}_{i}|<\eta.$

Moreover, $\Psi_{i}=\mathrm{id}$ near
$\widetilde{K}_{i}\cup\partial(\Phi_{k-1}(P_{k-1,i}))$. In view of the
inequality above, setting $K_{i}:=\Phi_{k-1}^{-1}(\widetilde{K}_{i})\subset
E_{k-1,i}$, we have

(7.21) $|E_{k-1,i}\setminus K_{i}|<\frac{\alpha^{k}}{2M_{k-1}}\,.$

Eventually, we set

$\widetilde{\Phi}_{ki}:=\Psi_{i}\circ\Phi_{k-1}\text{ in
}P_{k-1,i}\quad\text{and}\quad\mathscr{P}_{ki}:=\\{P_{kij}\\}_{j=1}^{2^{nN_{ki}}},\text{
where }P_{kij}:=\Phi^{-1}_{k-1}(R_{kij}).$

Then (7.19) and (7.20) become

(7.22)
$\operatorname{diam}{\widetilde{\Phi}_{ki}(P_{kij})}<2^{-k},\quad\operatorname{diam}({P_{kij}})<2^{-k},\quad\int_{P_{kij}}\det
T(x)\,dx=|\widetilde{\Phi}_{ki}(P_{kij})|,$

i. e., diffeomorphism $\widetilde{\Phi}_{ki}$ satisfies (5). Observe also that

(7.23) $\widetilde{\Phi}_{ki}=\Phi_{k-1}\text{ near }K_{i}\cup\partial
P_{k-1,i}\quad\text{so}\quad D\widetilde{\Phi}_{ki}=T\text{ on }K_{i}\subset
E_{k-1,i}.$

We will now replace $\widetilde{\Phi}_{ki}$ inside each $P_{kij}$ with a
diffeomorphism $\Phi_{kij}$ which has correct derivative on a larger set.

As explained earlier, above inequality (7.12), we need to use Lemma 5.12 to
get a compact set $\widetilde{E}_{kij}\subset\mathring{P}_{kij}\cap K_{i}$
such that $\mathring{P}_{kij}\setminus\widetilde{E}_{kij}$ is connected and

(7.24) $|(\mathring{P}_{kij}\cap
K_{i})\setminus\widetilde{E}_{kij}|<\alpha^{k}\,2^{-nN_{ki}-1}M_{k-1}^{-1}.$

Set

$\widetilde{E}_{ki}:=\bigcup_{j=1}^{2^{nN_{ki}}}\widetilde{E}_{kij}\subset
K_{i}\quad\text{and}\quad L_{ki}:=E_{k-1,i}\setminus\widetilde{E}_{ki}.$

Clearly, $\widetilde{E}_{ki}\subset\mathring{P}_{k-1,i}$. Summing (7.24) over
$j=1,\ldots,2^{nN_{ki}}$ yields

$|K_{i}\setminus\widetilde{E}_{ki}|<\frac{\alpha^{k}}{2M_{k-1}}\,.$

By (7.21), we arrive at

(7.25) $|L_{ki}|=|E_{k-1,i}\setminus
K_{i}|+|K_{i}\setminus\widetilde{E}_{ki}|<\alpha^{k}M_{k-1}^{-1}.$

Consequently,

(7.26)
$|\widetilde{E}_{ki}|=|E_{k-1,i}|-|L_{ki}|>|E_{k-1,i}|-\alpha^{k}\,M_{k-1}^{-1}.$

Let us stress that $\widetilde{E}_{ki}\subset K_{i}$ so (7.23) yields

(7.27) $D\widetilde{\Phi}_{ki}=T\text{ on }\widetilde{E}_{ki}.$

and

(7.28) $\widetilde{\Phi}_{ki}=\Phi_{k-1}\text{ on
}\widetilde{E}_{ki}=E_{k-1,i}\setminus L_{ki}.$

Although $D\Phi_{k-1}=T$ on $E_{k-1,i}$, the set $L_{ki}$ is the set on which
we will have spoiled the already prescribed derivative of $\Phi_{k-1}$. On the
other hand, the set $\widetilde{E}_{ki}$ consists of points in which the
prescribed derivative survives the transition from $\Phi_{k-1}$ to $\Phi_{ki}$
and, as we will soon see, to $\Phi_{k}$.

We now correct the derivative of $\widetilde{\Phi}_{ki}$. The set

$\Omega_{kij}:=\mathring{P}_{kij}\setminus\widetilde{E}_{kij}$

is open and connected. Observe that, setting
$\Omega_{ki}:=\bigcup_{j}\Omega_{kij}$,

$|\Omega_{ki}|=|P_{k-1,i}\setminus\widetilde{E}_{ki}|.$

Exactly as in (7.17), invoking (7.22) and (7.27) instead of (7.10) and (7.15),
we check that $\Omega_{kij}$ satisfies

$|\widetilde{\Phi}_{ki}(\Omega_{kij})|=\int_{\Omega_{kij}}\det T(x)\,dx.$

We are now in the position to use Theorem 1.2 for the domain $\Omega_{kij}$
and the diffeomorphism $\widetilde{\Phi}_{ki}$. We find a diffeomorphism
$\Phi_{kij}:\Omega_{kij}\to\widetilde{\Phi}_{ki}(\Omega_{kij})$ and a compact
set $E^{\prime}_{kij}\subset\Omega_{kij}$ such that

(7.29) $D\Phi_{kij}=T\text{ on
}E^{\prime}_{kij},\quad\Phi_{kij}=\widetilde{\Phi}_{ki}\text{ near
}\partial\Omega_{kij}\quad\text{and}\quad|E^{\prime}_{kij}|>\beta|\Omega_{kij}|.$

and thus $\Phi_{kij}(\Omega_{kij})=\widetilde{\Phi}_{ki}(\Omega_{kij})$. Let

$E^{\prime}_{ki}:=\bigcup_{j=1}^{2^{nN_{ki}}}E^{\prime}_{kij}\subset\mathring{P}_{k-1,i}.$

By the third condition in (7.29), we have

$|E^{\prime}_{ki}|>\beta|\Omega_{ki}|=\beta|P_{k-1,i}\setminus\widetilde{E}_{ki}|.$

We replace the diffeomorphism $\widetilde{\Phi}_{ki}$ with $\Phi_{kij}$ inside
each $\Omega_{kij}$, setting

$\Phi_{ki}:=\begin{cases}\Phi_{kij}&\text{on }\Omega_{kij},\\\
\widetilde{\Phi}_{ki}&\text{on }P_{k-1,i}\setminus\Omega_{ki}.\end{cases}$

Observe that thanks to the second condition in (7.29), $\Phi_{ki}$ is indeed a
diffeomorphism.

Moreover, $\Phi_{ki}=\widetilde{\Phi}_{ki}$ near $\partial
P_{k-1,i}\subset\bigcup_{j}\partial P_{kij}$. Since
$\widetilde{\Phi}_{ki}=\Phi_{k-1}$ near $\partial P_{k-1,i}$, property (1)
follows.

Since $\Phi_{ki}=\widetilde{\Phi}_{ki}$ near $\bigcup_{j}\partial P_{kij}$,
$\Phi_{ki}(P_{kij})=\widetilde{\Phi}_{ki}(P_{kij})$, so (7.22) proves property
(5).

Note that $\widetilde{E}_{ki}\subset\mathring{P}_{k-1,i}\setminus\Omega_{ki}$
and $\Phi_{ki}=\widetilde{\Phi}_{ki}$ in a neighborhood of
$\widetilde{E}_{ki}$, so the first condition in (7.29) together with (7.27)
imply that

$D\Phi_{ki}=T\text{ on
}\hat{E}_{ki}:=E^{\prime}_{ki}\cup\widetilde{E}_{ki}\subset\mathring{P}_{k-1,i}.$

Note that $\hat{E}_{ki}$ is compact as a finite sum of compact sets. This
proves property (2). We calculate

$|\hat{E}_{ki}|=|E^{\prime}_{ki}|+|\widetilde{E}_{ki}|>\beta|P_{k-1,i}\setminus\widetilde{E}_{ki}|+|\widetilde{E}_{ki}|=\beta|P_{k-1,i}|+(1-\beta)|E_{k-1,i}\setminus
L_{ki}|,$

where the last equality follows from $\widetilde{E}_{ki}=E_{k-1,i}\setminus
L_{ki}$. This proves (3).

Since $\Phi_{ki}=\widetilde{\Phi}_{ki}$ on
$\widetilde{E}_{ki}=E_{k-1,i}\setminus L_{ki}$, recalling (7.28) and (7.25),
we see that (4) also holds.

We verified conditions (1)-(5) and that completes the proof of the theorem. ∎

##  8\. Appendix

######  Lemma 8.1.

Let $E\subset\mathbb{R}^{n}$, $n\geq 1$, be a measurable set of finite
measure. Then the space of measurable functions $f:E\to\mathbb{R}$ is complete
with respect to the Lusin metric $d_{L}$.

###### Proof.

Let $\\{f_{k}\\}_{k}$ be a Cauchy sequence. It suffices to show that it has a
convergent subsequence in the metric $d_{L}$. To this end, we will show that a
subsequence $\\{f_{k_{\ell}}\\}_{\ell}$ such that

$d_{L}(f_{k_{\ell}},f_{k_{\ell+1}})=|\\{f_{k_{\ell}}\neq
f_{k_{\ell+1}}\\}|<2^{-(\ell+1)}$

is convergent. Let

$A_{\ell}:=\\{f_{k_{\ell}}=f_{k_{\ell+1}}\\}\quad\text{and}\quad
C:=\bigcup_{\ell=1}^{\infty}\bigcap_{i=\ell}^{\infty}A_{i}.$

Note that for any $\ell$,

$|E\setminus
C|\leq\Big{|}E\setminus\bigcap_{i=\ell}^{\infty}A_{i}\Big{|}\leq\sum_{i=\ell}^{\infty}|E\setminus
A_{i}|<2^{-\ell},\quad\text{so}\quad|E\setminus C|=0.$

If $x\in C$, then $x\in\bigcap_{i=\ell}^{\infty}A_{i}$ for some $\ell$ and
hence

$f_{k_{\ell}}(x)=f_{k_{\ell+1}}(x)=f_{k_{\ell+2}}(x)=\ldots\quad\text{so}\quad
f(x):=\lim_{\ell\to\infty}f_{k_{\ell}}(x)$

exists for all $x\in C$ and hence for almost all $x\in E$. In fact,
$f(x)=f_{k_{\ell}}(x)$ for $x\in\bigcap_{i=\ell}^{\infty}A_{i}$ and hence

$d_{L}(f,f_{k_{\ell}})\leq\Big{|}E\setminus\bigcap_{i=\ell}^{\infty}A_{i}\Big{|}<2^{-\ell},$

proving that $f_{k_{\ell}}\to f$ in the metric $d_{L}$. ∎

## References

  * [1] Alberti, G.: A Lusin type theorem for gradients. J. Funct. Anal. 100 (1991), 110–118.
  * [2] Alpern, S. Prasad, V. S.: Typical Dynamics of Volume Preserving Homeomorphisms, Cambridge Tracts in Mathematics, 139. Cambridge University Press, Cambridge, 2001.
  * [3] Ball, J.: Singularities and computation of minimizers for variational problems. In R. Devore, A. Iserles, E. Süli (Eds.), Foundations of Computational Mathematics, (London Mathematical Society Lecture Note Series, pp. 1-20). Cambridge University Press, Cambridge, 2001.
  * [4] Banyaga, A.: Formes-volume sur les variétés à bord. Enseignement Math. (2), 20 (1974), 127–131.
  * [5] Burago, D. Kleiner, B.: Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps. Geom. Funct. Anal. 8 (1998), 273–282.
  * [6] Campbell, D., D’Onofrio, L., Hencl, S.: A sense-preserving Sobolev homeomorphism with negative Jacobian almost everywhere. J. Lond. Math. Soc. 106 (2022), 235–310.
  * [7] Campbell, D., Hencl, S., Tengvall, V.: Approximation of $W^{1,p}$ Sobolev homeomorphism by diffeomorphisms and the signs of the Jacobian. Adv. Math. 331 (2018), 748–829.
  * [8] Csató, G., Dacorogna, B., Kneuss, O.: The pullback equation for differential forms. Progress in Nonlinear Differential Equations and their Applications, 83\. Birkhäuser/Springer, New York, 2012.
  * [9] Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences Vol. 78, Springer-Verlag, Berlin 1989.
  * [10] Dacorogna, B., Moser, J.: On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 1–26.
  * [11] D’Onofrio, L., Sbordone, C., Schiattarella, R.: On the approximate differentiability of inverse maps. J. Fixed Point Theory Appl., 15 (2014), 473–-499.
  * [12] Evans, L. C., Gariepy, R. F.: Measure theory and fine properties of functions. Revised edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015.
  * [13] Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York 1969.
  * [14] Federer, H.: Surface area. II. Trans. Amer. Math. Soc. 55 (1944), 438–456.
  * [15] Goffman, C., Nishiura, T., Waterman, D.: Homeomorphisms in analysis. Mathematical Surveys and Monographs, 54. American Mathematical Society, Providence, RI, 1997.
  * [16] Goldstein, P., Hajłasz, P.: Jacobians of $W^{1,p}$ homeomorphisms, case $p=[n/2]$. Calc. Var. Partial Differential Equations 58 (2019), no. 4, Paper No. 122, 28 pp.
  * [17] Goldstein, P., Hajłasz, P.: Modulus of continuity of orientation preserving approximately differentiable homeomorphisms with a.e. negative Jacobian. Ann. Acad. Sci. Fenn. Math. 43 (2018), 147–170.
  * [18] Goldstein, P., Hajłasz, P.: A measure and orientation preserving homeomorphism with approximate Jacobian equal $-1$ almost everywhere. Arch. Ration. Mech. Anal. 225 (2017), 65–88.
  * [19] Guerra, A., Koch, L., Lindberg, S.: The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces. Calc. Var. Partial Differential Equations 60 (2021), no. 1, Paper No. 55, 14 pp.
  * [20] Hajłasz, P.: Change of variables formula under minimal assumptions. Colloq. Math. 64 (1993), 93–101.
  * [21] Hencl, S., Koskela, P.: Lectures on mappings of finite distortion. Lecture Notes in Mathematics, 2096. Springer, Cham, 2014.
  * [22] Hencl, S., Malý, J.: Jacobians of Sobolev homeomorphisms. Calc. Var. Partial Differential Equations 38 (2010), 233–242.
  * [23] Hencl, S., Vejnar, B.: Sobolev homeomorphism that cannot be approximated by diffeomorphisms in $W^{1,1}$. Arch. Ration. Mech. Anal. 219 (2016), 183–202.
  * [24] Ho, C. W.: When are immersions diffeomorphisms? Canad. Math. Bull. 24 (1981), 491–492.
  * [25] Ho, C. W.: A note on proper maps. Proc. Amer. Math. Soc. 51 (1975), 237–241.
  * [26] Hurewicz, W., Wallman, H.: Dimension Theory. Princeton Mathematical Series, vol. 4. Princeton University Press, Princeton, NJ, 1941.
  * [27] Kauranen, A., Luisto, R., Tengvall, V.: On proper branched coverings and a question of Vuorinen. Bull. Lond. Math. Soc. 54 (2022), 145–160.
  * [28] McMullen, C. T.: Lipschitz maps and nets in Euclidean space. Geom. Funct. Anal. 8 (1998), 304–314.
  * [29] Milnor, J.W.: Topology from the differentiable viewpoint. The University Press of Virginia, Charlottesville, 1965.
  * [30] Moser, J.: On the volume elements on a manifold. Trans. Amer. Math. Soc., 120 (1965), 286–294.
  * [31] Munkres, J.: Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. of Math. 72 (1960), 521–554.
  * [32] Oxtoby, J. & Ulam, S.: Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math. 42 (1941), 874–920.
  * [33] Prasolov, V. V.: Elements of combinatorial and differential topology. Graduate Studies in Mathematics, 74. American Mathematical Society, Providence, RI, 2006.
  * [34] Rivière, T., Ye, D.,: Resolutions of the prescribed volume form equation. NoDEA Nonlinear Differential Equations Appl. 3 (1996), 323–369.
  * [35] Warner, F. W.: Foundations of differentiable manifolds and Lie groups. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
  * [36] Whitney, H.: On totally differentiable and smooth functions. Pacific J. Math. 1 (1951), 143–159.