# Multigraded Tor and local cohomology

Marc Chardin  Institut de Mathématiques de Jussieu, CNRS & Sorbonne
Université, France<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Rafael
Holanda  Departamento de Matemática, Universidade Federal da Paraíba -
58051-900, João Pessoa, PB, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Notions of Castelnuovo-Mumford regularity and of $a^{*}$ invariant were
extended from standard graded algebras to the toric setting.

We here focus our attention on the standard multigraded case, which
corresponds to a product of $k$ projective spaces. A natural notion for a
$\mathbb{Z}^{k}$-graded module is its support : degrees in which it is not
zero. A stabilized version of it is adding $-\mathbb{N}^{k}$, in order for the
complement (vanishing region) to be stable by addition of $\mathbb{N}^{k}$.

Cohomology of twists of a sheaf on a product of projective spaces, provided by
a graded module, are given by local cohomologies with respect the product $B$
of the ideals $B_{i}$ generated by the $k$ sets of variables.

Our results shed some light on a central issue, the relation between shifts in
graded free resolution and cohomology vanishing : it shows that stabilized
support of cohomology with respect to $B$ corresponds to the union of
stabilized supports for cohomologies in the $B_{i}$’s, while shifts in (some
of the) graded free resolutions are inside the intersection of these
stabilized supports. A one-to-one correspondence between stabilized supports
of Tor modules and of local cohomologies with respect to the sum of the
$B_{i}$’s is also established.

We then derive a consequence on linear resolutions for truncations of a graded
module.

###### Key words and phrases:

Graded free resolutions, Local cohomology, Multigraded regularity

###### 2020 Mathematics Subject Classification:

Primary 13D02, 13D07, 13D45 ; Secondary 14B15

##  1\. Introduction

Let $S$ be a commutative ring and $R$ a finitely generated standard
$\mathbb{Z}^{k}$-graded polynomial extension. In other words $R$ has $k\geq 1$
finite sets of variables, the $i$-th set having degree the $i$-th canonical
generator of $\mathbb{Z}^{k}$.

A graded $R$-module $M$ determines a quasi-coherent sheaf $\mathcal{F}$ on the
scheme $\operatorname{Proj}(R)$ which is the product (over the spectrum of
$S$) of projective spaces (again over the spectrum of $S$) corresponding to
the $k$ sets of variables. Any quasi-coherent sheaf arises this way similarly
as in the case of a single projective space. Sheaf cohomology of twists of
$\mathcal{F}$ could be given in terms of graded components of the local
cohomology of $M$ with respect to the product $B$ of the ideals
$B_{1},\ldots,B_{k}$ generated by the $k$ sets of variables.

In the case $k=1$, the notion of Castelnuovo-Mumford regularity provides a key
relation between shifts in graded free resolutions of $M$ and the degrees
where local (or sheaf) cohomologies vanish. This notion could also be defined
by considering truncations of the module (keeping only elements of degree at
least some given integer) and asking for generation in a single degree and
maximal shifts at each step in a graded resolution increasing at most by one.

In a series of works, first motivated by toric geometry (see [10]), the notion
of Castelnuovo-Mumford regularity was extended and studied for the multigraded
setting, in general requiring $S$ to be a field and $M$ to be finitely
generated. Although the properties do not extend perfectly, pretty sharp
results have been obtained and provide a more and more precise picture of the
situation; in the standard multigraded setting, see for instance [3] for a
recent refinement and references to previous works.

Besides the notion of Castelnuovo-Mumford regularity, a variation of this
notion sometimes called the $a^{*}$-invariant could as well be defined both in
terms of shifts in graded free resolutions or in terms of local cohomology. It
is bounded above by the regularity and the difference is at most the number of
variables; it gives a sharper estimate of the regularity of the Hilbert
function, but do not share all the nice features of regularity.

In this text, we consider the natural extension of the notion of
$a^{*}$-invariant by considering the following notion already present in the
work of Huy Tài Hà [8]:

$\mathbb{C}_{B}(M)^{*}:=\cup_{i}\operatorname{Supp}_{\mathbb{Z}^{k}}(H^{i}_{B}(M))^{*}$

where the support of a graded module is the set of degrees for which the
corresponding component is not $0$. The star of a set
$E\subseteq\mathbb{Z}^{k}$, $E^{*}:=\\{e-\mu\ |\ \mu\in\mathbb{N}^{k}\\}$, is
the smallest set containing $E$ such that its complement is stable under the
addition of any element in $\mathbb{N}^{k}$ (see Lemma 3.4).

In order to understand cohomology with support in $B$, which could be hard to
determine even in very simple examples (Remark 3.12 presents one such case),
one available tool is a spectral sequence (see [9] or our construction in 3.2)

$\bigoplus_{1\leq i_{1}<\cdots<i_{p}\leq
k}H^{p-q-1}_{B_{i_{1}}+\cdots+B_{i_{p}}}(M)\Rightarrow H^{q}_{B}(M).$

It shows that
$\mathbb{C}_{B}(M)\subseteq\bigcup_{i_{1},\dots,i_{p}}\mathbb{C}_{B_{i_{1}}+\cdots+B_{i_{p}}}(M)$.
The cohomology with respect to the sum of some $B_{j}$’s is much easier to
analyze, as it is cohomology with support in ideals generated by variables.

A first surprise was that in terms of supports, and after stabilization of the
complement by the star operation, the union of the supports on the right
equals $\mathbb{C}_{B}(M)^{*}$. More precisely, defining $\mathbb{C}_{I}(M)$
similarly as above for any graded ideal $I$:

######  Theorem 1.1.

(Theorem 3.11) Let $M$ be a graded $R$-module, then

$\bigcup_{i}\mathbb{C}_{B_{i}}(M)^{*}=\bigcup_{i_{1},\dots,i_{p}}\mathbb{C}_{B_{i_{1}}+\cdots+B_{i_{p}}}(M)^{*}=\mathbb{C}_{B}(M)^{*}.$

And also, explaining and completing the left equality:

$\mathbb{C}_{B_{i_{1}}+\dots+B_{i_{p}}}(M)^{*}\subseteq\mathbb{C}_{B_{i_{1}}}(M)^{*}\cap\cdots\cap\mathbb{C}_{B_{i_{p}}}(M)^{*}.$

Hence $\mathbb{C}_{B}(M)^{*}$ is fully determined by the supports of the
cohomology with respect to the $B_{i}$’s. This should help to determine
$\mathbb{C}_{B}(M)^{*}$ much more easily.

The second important, although less surprising, fact is that the sets
$\mathbb{C}_{B_{i_{1}}+\dots+B_{i_{p}}}(M)^{*}$ are fully determined by the
support of corresponding Tor modules, and vice-versa, as detailed in Theorem
3.8. This follows from the key case of $\mathfrak{m}:=B_{1}+\cdots+B_{k}$:

######  Theorem 1.2.

(Theorem 3.6) Let $M$ be a graded $R$-module, then

$\bigcup_{i}\operatorname{Supp}_{\mathbb{Z}^{k}}(\operatorname{Tor}_{i}^{R}(M,S))^{*}=(n_{1},\ldots,n_{k})+\mathbb{C}_{\mathfrak{m}}(M)^{*}.$

These two results give a quite precise picture of the difference between the
support of Tor modules, that reflects shifts in free resolutions (see Lemma
3.5 and following comments), and cohomology with respect to $B$, a central
question in many recent works on multigraded regularity :

$\bigcup_{i}\operatorname{Supp}_{\mathbb{Z}^{k}}(\operatorname{Tor}_{i}^{R}(M,S))^{*}\subseteq(n_{1},\ldots,n_{k})+\bigcap_{i}C_{B_{i}}(M)^{*}\
\hbox{ while }\mathbb{C}_{B}(M)^{*}=\bigcup_{i}\mathbb{C}_{B_{i}}(M)^{*}.$

This also supports the idea, already developed in a work of the first named
author with Nicolas Botbol [2] : one should not restrict the attention to only
the ideal $B$, but consider as well cohomologies supported on other graded
ideals that are naturally linked to the study.

In our text, we do not ask the ring to satisfy any property (besides
commutativity) nor finiteness hypotheses for the module, in order to give as
much flexibility in the use as it could be. However, it is of course crucial
to have in mind that the main tool to be able to assert that
$\mathbb{C}_{B}(M)^{*}\not=\mathbb{Z}^{k}$, or to show that the modules
$\operatorname{Tor}_{i}^{R}(M,S)$ have finite support, is to require $S$ to be
Noetherian and $M$ to be finitely generated.

Whenever $\mu\not\in\mathbb{C}_{B}(M)^{*}$, we show that, in degrees slightly
bigger then $\mu$, the truncation of the module is generated in a single
degree and shifts have total degree increasing at most by one at each step
(hence the truncated module has a linear resolution whenever the base ring is
a field). We state this result in terms of the regularity in $R$ for the total
degree grading (a $\mathbb{Z}^{k}$-graded module is also graded for the
$\mathbb{Z}$-grading given by total degree):

######  Proposition 1.3.

(Proposition 4.9) Let $M$ be a graded $R$-module and
$\mu\not\in\mathbb{C}_{B}(M)^{*}$.

Then for any $t\in\mu+(n_{1}-1,\ldots,n_{k}-1)+\mathbb{N}^{k}$,
$M_{t+\mathbb{N}^{k}}$ has a regularity $|t|$.

##  2\. Multigraded support

Let $R$ be a commutative unitary ring and $I=(f_{1},\ldots,f_{s})$ a finitely
generated ideal of $R$. Given an $R$-module $M$, the Čech cohomology modules
$H^{i}_{I}(M)$ of $M$ supported on $I$ are the cohomologies of the Čech
complex $\check{\mathcal{C}}^{\bullet}_{\mathbf{f}}(M)$ of $M$ with respect to
the finite tuple $\mathbf{f}=(f_{1},\ldots,f_{s})$. These modules only depend
upon $M$ and the radical of $I$, up to isomorphism; they coincide with the
derived functors of $H^{0}_{I}(-)$ applied to $M$ whenever $R$ is Noetherian,
or $I$ is generated by a regular sequence, or if $R$ is a polynomial ring over
another ring and $I$ is a monomial ideal – this will be the context in use in
most of this text.

We now setup notations for working in standard multigraded polynomial rings.
Let $S$ be a commutative unitary ring, $k\geq 1$ an integer and denote the
standard $\mathbb{Z}^{k}$-graded polynomial ring by
$R=S[X_{1,1},...,X_{1,n_{1}},...,X_{k,1},...,X_{k,n_{k}}]$. For each set of
variables $\mathbf{X}_{i}=\\{X_{i,1},\ldots,X_{i,n_{i}}\\}$ write $B_{i}$ for
the ideal generated by this $i$-th set of variables, $B=B_{1}\cap\ldots\cap
B_{k}=B_{1}\cdots B_{k}$ and $\mathfrak{m}=B_{1}+\ldots+B_{k}$. We write
$\mathbb{Z}^{k}=\oplus_{i=1}^{k}\mathbb{Z}e_{i}$, where the $i$-th canonical
generator of $\mathbb{Z}^{k}$ corresponds to the degree of the set of
variables generating $B_{i}$. Unless another grading is specified, a graded
$R$-module is a $\mathbb{Z}^{k}$-graded $R$-module. The degree will be written
$\deg$ and $\operatorname{tdeg}$ will denote the total degree, they
respectively belong to $\mathbb{Z}^{k}$ and $\mathbb{Z}$. Also
$\mathbb{N}=\mathbb{Z}_{\geq 0}$ and if $G$ is an additive group and $E,F$ two
subsets of $G$, $E+F=\\{e+f\ |\ e\in E,f\in F\\}$, $-E:=\\{-e\ |\ e\in E\\}$
and $E-F=E+(-F)$; to simplify notations if $e\in G$, $e+F:=\\{e\\}+F$. These
conventions and notations will be in use for all the text and we define

######  Definition 2.1.

The _support_ of a graded $R$-module $M$ is

$\operatorname{Supp}_{\mathbb{Z}^{k}}(M):=\\{\gamma\in\mathbb{Z}^{k}:M_{\gamma}\neq
0\\}.$

Also, given a homogeneous finitely generated ideal $I$, we set

$\mathbb{C}_{I}^{i}(M):=\operatorname{Supp}_{\mathbb{Z}^{k}}(H^{i}_{I}(M))\
\mbox{and}\ \mathbb{C}_{I}(M):=\bigcup_{i}\mathbb{C}_{I}^{i}(M).$

For a graded $R$-module $M$ and for $\mu\in\mathbb{Z}^{k}$ we define the
$R$-module $M(-\mu)$ as $M$ with grading defined by
$M(-\mu)_{\nu}:=M_{\mu-\nu}$.

######  Lemma 2.2.

If $M$ is a graded $R$-module, then
$\operatorname{Supp}_{\mathbb{Z}^{k}}(M(-\mu))=\operatorname{Supp}_{\mathbb{Z}^{k}}(M)+\mu$.

###### Proof.

Indeed $\nu\in\operatorname{Supp}_{\mathbb{Z}^{k}}(M(-\mu))$ if and only if
$\nu-\mu\in\operatorname{Supp}_{\mathbb{Z}^{k}}(M)$, equivalently
$\nu\in\operatorname{Supp}_{\mathbb{Z}^{k}}(M)+\mu$. ∎

Shifts appearing in local cohomology with respect to the sum of some of the
$B_{i}$’s will appear in many places, and we introduce the notation:

$a_{i_{1},\cdots,i_{p}}:=-\sum_{j=1}^{p}n_{i_{j}}e_{i_{j}}\in\mathbb{Z}^{k}\quad{\rm
and}\quad a:=a_{1,\ldots,k}=-(n_{1},\ldots,n_{k}).$

######  Example 2.3.

For $1\leq i_{1}<\cdots<i_{p}\leq k$,
$\mathbb{C}^{i}_{B_{i_{1}}+...+B_{i_{p}}}(R)=\emptyset$ unless
$i=n_{i_{1}}+\cdots+n_{i_{p}}$ and

$\mathbb{C}^{n_{i_{1}}+\cdots+n_{i_{p}}}_{B_{i_{1}}+...+B_{i_{p}}}(R)=a_{i_{1},\cdots,i_{p}}+\prod_{j=1}^{k}(-1)^{\varepsilon_{j}}\mathbb{N},$

with $\varepsilon_{j}:=1$ if $j\in\\{i_{1},\ldots,i_{p}\\}$ and
$\varepsilon_{j}:=0$ else.

In particular, $\mathbb{C}^{i}_{\mathfrak{m}}(R)=\emptyset$ for
$i\not=n_{1}+\cdots+n_{k}$ and

$\mathbb{C}_{\mathfrak{m}}(R)=\mathbb{C}_{\mathfrak{m}}^{n_{1}+\cdots+n_{k}}(R)=a-\mathbb{N}^{k}.$

The following example illustrates supports.

######  Example 2.4.

By taking $k=2$ and $n_{1}=3$ and $n_{2}=5$ in Example 2.3, we have the
following regions
$\mathbb{C}_{\mathfrak{m}}(R)=(-3,-5)-\mathbb{N}^{2},\mathbb{C}_{B_{1}}(R)=(-3,0)+(-\mathbb{N})\times\mathbb{N}$
and $\mathbb{C}_{B_{2}}(R)=(0,-5)+\mathbb{N}\times(-\mathbb{N})$.

As detailed in the work of Eisenbud, Mustata and Stillman [7] whenever the
base ring is a field, cohomology with respect to $B$ of a graded $R$-module
$M$ is providing the sheaf cohomology of the corresponding sheaf $\mathcal{F}$
on $\mathbb{P}:=\mathbb{P}^{n_{1}-1}\times\cdots\times\mathbb{P}^{n_{k}-1}$
(where projective spaces and products are taken over the spectrum of $S$):

$H^{i}(\mathbb{P},\mathcal{F}(\mu))\simeq H^{i+1}_{B}(M)_{\mu},\quad\forall
i>0,\quad\forall\mu\in\mathbb{Z}^{k},$

and there is a natural short exact sequence of graded modules

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{B}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\oplus_{\mu}H^{0}(\mathbb{P},\mathcal{F}(\mu))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{B}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The following result is essential as it shows that the regions where
cohomology vanishes that we will investigate are non trivial in many cases;
recall also that, for a same module, local cohomology is invariant under
arbitrary base change.

######  Theorem 2.5.

Let $M$ be a finitely generated graded $R$-module. If $S$ is Noetherian,

(i) $\operatorname{Supp}(\operatorname{Tor}_{i}^{R}(M,S))$ is a finite set,
for every $i$,

(ii) there exists $\mu$ in $\mathbb{Z}^{k}$ such that
$\mathbb{C}_{\mathfrak{m}}(M)\subseteq\mu-\mathbb{N}^{k}$,

(iii) there exists $\nu$ in $\mathbb{Z}^{k}$ such that
$\mathbb{C}_{B}(M)\cap(\nu+\mathbb{N}^{k})=\emptyset$.

###### Proof.

For (i), recall that these Tor modules are finitely generated $S$-modules.
Items (ii) and (iii) follow from [2, Theorem 4.14] : for (ii) by Examples 2.3
and because any finite union of shifts of $-\mathbb{N}^{k}$ is contained in
$\mu-\mathbb{N}^{k}$ for some $\mu$, and for (ii) as there exists
$\nu\in\operatorname{reg}_{B}(M)$ and, for any such $\nu$,
$\mathbb{C}_{B}(M)\cap(\nu+\mathbb{N}^{k})=\emptyset$. ∎

##  3\. Tor and local cohomology modules

###  3.1. An auxiliary result

We first recall the following classical result that is a key ingredient in our
first main result.

######  Theorem 3.1.

Let $I$ be a homogeneous ideal generated by a regular sequence ${\mathbf{f}}$
of length $r$ and $M$ a graded $R$-module. If $F_{\bullet}$ is a graded free
$R$-resolution of $M$, for any $p$, there exists a degree zero graded
isomorphism

$H^{p}_{I}(M)\simeq H_{r-p}(H^{r}_{I}(F_{\bullet})).$

###### Proof.

The second pages of the two spectral sequences from the double complex
$\check{\mathcal{C}}^{\bullet}_{\mathbf{f}}(F_{\bullet})$ are concentrated
respectively on a line (as $H^{i}_{I}(F_{\bullet})=0$ unless $i=r$) and on a
column (as localization is flat) and are isomorphic to the right and left hand
sides of the claimed isomorphism. ∎

As consequence of Theorem 3.1 and [1, Lemma 6.4.7] we obtain the following
spectral sequence.

######  Corollary 3.2.

Let $M$ be a graded $R$-module. There exists a spectral sequence of graded
modules,

$\bigoplus_{1\leq i_{1}<\cdots<i_{p}\leq
k}H^{p-q-1}_{B_{i_{1}}+\cdots+B_{i_{p}}}(M)\Rightarrow H^{q}_{B}(M).$

###### Proof.

Let $F_{\bullet}$ be a graded free resolution of $M$ and
$B=(f_{1},\ldots,f_{s})$. The double complex
$\check{\mathcal{C}}^{\bullet}_{\mathbf{f}}(F_{\bullet})$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}^{0}_{\mathbf{f}}(F_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}^{0}_{\mathbf{f}}(F_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}^{0}_{\mathbf{f}}(F_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}^{1}_{\mathbf{f}}(F_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}^{1}_{\mathbf{f}}(F_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}^{1}_{\mathbf{f}}(F_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$

yields two spectral sequences that converge to filtrations of the same graded
module $H$. Taking homologies in the horizontal direction first, we get a
spectral sequence ${}^{\prime}E$ such that
${}^{\prime}E_{2}^{0,-j}=H^{j}_{B}(M)$ and ${}^{\prime}E_{2}^{-i,-j}=0$
whenever $i\neq 0$; hence $H^{j}_{B}(M)\simeq H^{j}$ for all $j\geq 0$. On the
other hand, by [1, Lemma 6.4.7], the other spectral sequence $E$ is such that

$E_{2}^{-i,-j}=H_{i}\left(\bigoplus_{\begin{subarray}{c}1\leq
i_{1}<...<i_{p}\leq k\\\
n_{i_{1}}+...+n_{i_{p}}=j+p-1\end{subarray}}H^{n_{i_{1}}+...+n_{i_{p}}}_{B_{i_{1}}+...+B_{i_{p}}}(F_{\bullet})\right)\simeq\bigoplus_{1\leq
i_{1}<...<i_{p}\leq k}H^{p+(j-i)-1}_{B_{i_{1}}+...+B_{i_{p}}}(M)$

and $E_{2}^{-i,-j}\Rightarrow_{i}H^{i-j}\simeq H^{i-j}_{B}(M)$, whence the
result. ∎

It is worth mentioning that the spectral sequence above has similarities with
the one of Lyubeznik’s [9], although it has second page terms on a diagonal
equal to the ones on a diagonal in the first page in Lyubeznik’s approach.

###  3.2. Stable sets

It is convenient to introduce a notation for multigraded support of Tor
modules, and for some shifts of these that are tightly connected to support of
local cohomology, as we will show.

######  Definition 3.3.

Given a graded $R$-module $M$ and $i_{1},\dots,i_{p}$ distinct elements in
$\\{1,\dots,k\\}$

$\mathbb{T}_{j}^{i_{1},\dots,i_{p}}(M):=\operatorname{Supp}_{\mathbb{Z}^{k}}(\operatorname{Tor}_{j}^{R}(M,R/(B_{i_{1}}+\dots+B_{i_{p}}))$

and
${\mathbb{T}}^{i_{1},\dots,i_{p}}(M):=\cup_{j}\mathbb{T}_{j}^{i_{1},\dots,i_{p}}(M)$.
Also,

$\hat{\mathbb{T}}_{j}^{i_{1},\dots,i_{p}}(M):=\mathbb{T}_{j}^{i_{1},\dots,i_{p}}(M)+a_{i_{1},\ldots,i_{p}}\
{\rm and}\
\hat{\mathbb{T}}^{i_{1},\dots,i_{p}}(M):={\mathbb{T}}^{i_{1},\dots,i_{p}}(M)+a_{i_{1},\ldots,i_{p}}.$

For simplicity,

$\mathbb{T}_{j}(M):=\mathbb{T}_{j}^{1,\dots,k}(M)=\operatorname{Supp}_{\mathbb{Z}^{k}}(\operatorname{Tor}_{j}^{R}(M,S))\quad{\rm
and}\quad\mathbb{T}(M):=\cup_{j}\mathbb{T}_{j}(M),$

and similarly $\hat{\mathbb{T}}_{j}(M):=\mathbb{T}_{j}(M)+a$ and
$\hat{\mathbb{T}}(M):=\mathbb{T}(M)+a$.

######  Lemma 3.4.

Let $E\subset\mathbb{Z}^{k}$. The smallest set containing $E$ such that its
complement is stable under the addition of $\mathbb{N}^{k}$ is
$E-\mathbb{N}^{k}$.

###### Proof.

Let $E^{*}:=E-\mathbb{N}^{k}$. If $e\not\in E^{*}$, $e+n$ is not in $E^{*}$
for all $n\in\mathbb{N}^{k}$ as otherwise $e+n=e^{\prime}-n^{\prime}$ with
$e^{\prime}\in E$ and $n^{\prime}\in\mathbb{N}^{k}$, the equality
$e=e^{\prime}-n-n^{\prime}$ then contradicting the fact that $e\not\in E^{*}$.
Now $E^{*}$ is minimal as for $E\subseteq E^{\prime}\subseteq E^{*}$, with the
complement of $E^{\prime}$ stable, if $e\in E^{*}\setminus E^{\prime}$, then
$e+n\in E\subseteq E^{\prime}$ for some $n\in\mathbb{N}^{k}$, but also
$e+n\notin E^{\prime}$ due to the stability of the complement of $E^{\prime}$,
thus a contradiction. ∎

From now on, we denote by $E^{*}$ the set $E-\mathbb{N}^{k}$ as in the proof
above.

Recall from [2, Lemma 3.12 i)] a link between Tor modules and graded free
resolutions :

######  Lemma 3.5.

Let $M$ be a graded $R$ module such that $\operatorname{tdeg}(m)\geq u$ for
some $u\in\mathbb{Z}$ and every $m\in M$. Then $M$ admits a graded free
$R$-resolution $F_{\bullet}$ such that $F_{i}$ is generated by elements with
degrees sitting in $\cup_{p\leq i}\mathbb{T}_{p}(M)$.

Any minimal graded free resolution (no syzygy module has a superfluous
generator) satisfy this property. If $S$ is Noetherian and $M$ is finitely
generated, minimal graded free resolutions exist. Any element in
$\mathbb{T}_{i}$ appear as a shift in $F_{i}$ for any graded free
$R$-resolution $F_{\bullet}$ of $M$. One can replace the union by the only set
$\mathbb{T}_{i}$ if $S$ is a field.

We first show that support of cohomology and support of Tor modules are in one
to one correspondence, if one considers the union of all in both cases and
stabilize the complement by addition of $-\mathbb{N}^{k}$. The first and main
case is the one relative to the ideal $\mathfrak{m}$.

######  Theorem 3.6.

Let $M$ be a graded module. Then

$\mathbb{C}_{\mathfrak{m}}(M)^{*}=\mathbb{T}(M)+\mathbb{C}_{\mathfrak{m}}(R)=\hat{\mathbb{T}}(M)^{*}.$

###### Proof.

Recall that $\mathbb{C}_{\mathfrak{m}}(R)=a-\mathbb{N}^{k}$, by Example 2.3.
Thus
$\mathbb{T}(M)+\mathbb{C}_{\mathfrak{m}}(R)=\hat{\mathbb{T}}(M)-\mathbb{N}^{k}=\hat{\mathbb{T}}(M)^{*}$.

We first assume that there exists $u\in\mathbb{Z}$ such that
$\operatorname{tdeg}(m)\geq u$ for any $m\in M$ and let $F_{\bullet}$ be a
graded free $R$-resolution of $M$ as in Lemma 3.5. Then,

$H^{i}_{\mathfrak{m}}(M)\simeq H_{d-i}(H^{d}_{\mathfrak{m}}(F_{\bullet})),$

for all $i\geq 0$ by Theorem 3.1, where $d:=n_{1}+\ldots+n_{k}$ is the number
of variables of $R$. This shows that
$\mathbb{C}_{\mathfrak{m}}^{i}(M)\subseteq\bigcup_{p\leq
d-i}\mathbb{T}_{p}(M)+\mathbb{C}_{\mathfrak{m}}(R)$, hence
$\mathbb{C}_{\mathfrak{m}}(M)\subseteq\mathbb{T}(M)+\mathbb{C}_{\mathfrak{m}}(R)$
and therefore
$\mathbb{C}_{\mathfrak{m}}(M)^{*}\subseteq\mathbb{T}(M)+\mathbb{C}_{\mathfrak{m}}(R)$.

To show the inverse inclusion, notice that for all
$\mu\not\in\mathbb{C}_{\mathfrak{m}}(M)^{*}$,
$(\mu+\mathbb{N}^{k})\cap\mathbb{C}_{\mathfrak{m}}(M)=\emptyset$. Write
$\mathbf{X}$ for the tuple of all $d$ variables and consider the first
quadrant double complex
$\check{\mathcal{C}}_{\mathbf{X}}^{\bullet}(K_{\bullet}(\mathbf{X};M))$. It
gives rise to a spectral sequence with first terms
$K_{i}(\mathbf{X};H^{j}_{\mathfrak{m}}(M))$ that abuts to
$\operatorname{Tor}_{i-j}^{R}(M,S)$. As shifts in
$K_{i}(\mathbf{X};H^{j}_{\mathfrak{m}}(M))$ have all $p$-th coordinates at
most $n_{p}$ for any $p=1,...,k$, in degree $\mu-a$ all terms are zero because
$K_{i}(\mathbf{X};H^{j}_{\mathfrak{m}}(M))_{\mu-a}$ is a sum of copies of
$H^{j}_{\mathfrak{m}}(M)$ sitting in degrees $\mu+\delta$ for
$\delta\in\mathbb{N}^{k}$. It follows that $\mu-a\not\in\mathbb{T}(M)$, as
claimed.

We now turn to the general case. Notice that for $u\in\mathbb{Z}$, the
submodule $M[u]:=\\{m\in M,\ \operatorname{tdeg}(m)\geq u\\}$ is such that
$M/M[u]=H^{0}_{\mathfrak{m}}(M/M[u])$. It follows that
$H^{i}_{\mathfrak{m}}(M[u])_{\mu}$ coincides with
$H^{i}_{\mathfrak{m}}(M)_{\mu}$ for every $i$ and $\mu$ of total degree at
least $u$. On the other hand, $K_{\bullet}(\mathbf{X};M)$ and
$K_{\bullet}(\mathbf{X};M[u])$ coincide in total degree at least $u+d$.

Hence $\mathbb{C}_{\mathfrak{m}}(M)$ coincides with
$\mathbb{C}_{\mathfrak{m}}(M[u])$ and $\hat{\mathbb{T}}(M)$ coincides with
$\hat{\mathbb{T}}(M[u])$ in total degree at least $u$.

Since the sets $E_{u}:=\\{t\in\mathbb{Z}^{k},\ t_{1}+\cdots+t_{k}<u\\}$ are
stable under the addition of $-\mathbb{N}^{k}$, $(A\cup E_{u})^{*}=A^{*}\cup
E_{u}$ for any set $A\subseteq\mathbb{Z}^{k}$.

It follows that

$\begin{array}[]{rlr}\hat{\mathbb{T}}(M)^{*}&=\cap_{u\in\mathbb{Z}}(\hat{\mathbb{T}}(M)^{*}\cup
E_{u})&(\hbox{since }\cap_{u\in\mathbb{Z}}E_{u}=\emptyset)\\\
&=\cap_{u\in\mathbb{Z}}(\hat{\mathbb{T}}(M[u])^{*}\cup E_{u})&(\hbox{as
}\hat{\mathbb{T}}(M[u])=\hat{\mathbb{T}}(M)\hbox{ off }E_{u})\\\
&=\cap_{u\in\mathbb{Z}}(\mathbb{C}_{\mathfrak{m}}(M[u])^{*}\cup
E_{u})&(\hbox{by the first part})\\\
&=\cap_{u\in\mathbb{Z}}(\mathbb{C}_{\mathfrak{m}}(M)^{*}\cup E_{u})&(\hbox{as
}\mathbb{C}_{\mathfrak{m}}(M[u])=\mathbb{C}_{\mathfrak{m}}(M)\hbox{ off
}E_{u})\\\ &=\mathbb{C}_{\mathfrak{m}}(M)^{*}&(\hbox{since
}\cap_{u\in\mathbb{Z}}E_{u}=\emptyset).\\\ \end{array}$

∎

######  Lemma 3.7.

Let $M$ be a graded $R$-module and $i_{1},\ldots,i_{p}$ be distinct elements
in $\\{1,\ldots,k\\}$. Set
$T:=S[\mathbf{X}_{i_{1}},\ldots,\mathbf{X}_{i_{p}}]$ and let
$\nu\in\oplus_{q\not\in\\{i_{1},\ldots,i_{p}\\}}\mathbb{Z}e_{q}$. Then, for
any $j\geq 0$,

  * i)

$H^{j}_{B_{i_{1}}+\dots+B_{i_{p}}}(M)_{*,\nu}\simeq
H^{j}_{B_{i_{1}}+\dots+B_{i_{p}}}(M_{*,\nu})$ as graded $T$-modules;

  * ii)

$\operatorname{Tor}^{R}_{j}(M,R/B_{i_{1}}+\dots+B_{i_{p}})_{*,\nu}\simeq\operatorname{Tor}^{T}_{j}(M_{*,\nu},S)$
as graded $T$-modules.

###### Proof.

The variables in $\mathbf{X}_{i_{1}},\ldots,\mathbf{X}_{i_{p}}$ have degree
zero in $\oplus_{q\not\in\\{i_{1},\ldots,i_{p}\\}}\mathbb{Z}e_{q}$. ∎

A more general version of our first main result is directly derived from this
lemma.

######  Theorem 3.8.

Let $M$ be a graded $R$-module and $i_{1},\dots,i_{p}$ be distinct elements in
$\\{1,\dots,k\\}$. Then

$\hat{\mathbb{T}}^{i_{1},\dots,i_{p}}(M)-\sum_{j=1}^{p}e_{i_{j}}\mathbb{N}=\mathbb{C}_{B_{i_{1}}+\dots+B_{i_{p}}}(M)-\sum_{j=1}^{p}e_{i_{j}}\mathbb{N}.$

As a consequence,

$\mathbb{C}_{B_{i_{1}}+\cdots+B_{i_{p}}}(M)^{*}=\hat{\mathbb{T}}^{i_{1},\ldots,i_{p}}(M)^{*}.$

###### Proof.

According to Lemma 3.7, the proof follows along the same lines as in the one
of Theorem 3.6. ∎

In order to compare support with respect to several cohomologies, the
following result is useful. Let
$\mathcal{E}_{j}^{i_{1},\ldots,i_{p}}\subset\mathbb{N}^{k}$ be the set of
shifts in $K_{j}(\mathbf{X}_{i_{1}},\dots,\mathbf{X}_{i_{p}};R)$.

######  Proposition 3.9.

Let $M$ be a graded $R$-module and $i_{1},\dots,i_{p}$ be distinct elements in
$\\{1,\dots,k\\}$. For any finitely generated graded ideal
$I\subseteq\sqrt{B_{i_{1}}+\cdots+B_{i_{p}}+\operatorname{ann}_{R}(M)}$,

$\hat{\mathbb{T}}^{i_{1},\dots,i_{p}}_{j}(M)\subseteq\bigcup_{r\leq(\sum_{l=1}^{p}n_{i_{l}})-j}\mathbb{C}^{r}_{I}(M)+\mathcal{E}^{i_{1},...,i_{p}}_{j+r}+a_{i_{1},\ldots,i_{p}}$

for all $j\geq 0$.

###### Proof.

Let $\check{\mathcal{C}}^{\bullet}$ be the Čech complex of $R$ with respect to
a finite generating set of $I$ and consider the double complex
$\check{\mathcal{C}}^{\bullet}(K_{\bullet}(\mathbf{X}_{i_{1}},\dots,\mathbf{X}_{i_{p}};M))$.
It gives rise to a spectral sequence with first terms
$K_{i}(\mathbf{X}_{i_{1}},\dots,\mathbf{X}_{i_{p}};H^{j}_{I}(M))$ that abuts
to $\operatorname{Tor}_{i-j}^{R}(M,R/B_{i_{1}}+\dots+B_{i_{p}})$ since
$I\subseteq\sqrt{B_{i_{1}}+\cdots+B_{i_{s}}+\operatorname{ann}_{R}(M)}$.

Now,

${\mathbb{T}}^{i_{1},\dots,i_{p}}_{j}(M)\subseteq\bigcup_{l-r=j}\mathbb{C}^{r}_{I}(M)+\mathcal{E}^{i_{1},...,i_{p}}_{l}=\bigcup_{r\leq(\sum_{l=1}^{p}n_{i_{l}})-j}\mathbb{C}^{r}_{I}(M)+\mathcal{E}^{i_{1},...,i_{p}}_{j+r}$

whence the result. ∎

######  Corollary 3.10.

Let $M$ be a graded $R$-module and $i_{1},\dots,i_{p}$ be distinct elements in
$\\{1,\dots,k\\}$. For any finitely generated graded ideal $I\subseteq
B_{i_{1}}+\dots+B_{i_{p}}$,

$\mathbb{C}_{B_{i_{1}}+\dots+B_{i_{p}}}(M)\subseteq\mathbb{C}_{I}(M)-\sum_{j=1}^{p}e_{i_{j}}\mathbb{N}.$

In particular, if $p\geq 1$,

$\mathbb{C}_{B_{i_{1}}+\dots+B_{i_{p}}}(M)^{*}\subseteq\mathbb{C}_{B_{i_{1}}+\cdots+B_{i_{p-1}}}(M)^{*}.$

###### Proof.

By taking union over $j$ in Proposition 3.9 we obtain

$\hat{\mathbb{T}}^{i_{1},\dots,i_{p}}(M)\subseteq\mathbb{C}_{I}(M)-\sum_{j=1}^{p}e_{i_{j}}\mathbb{N}.$

The result follows from Theorem 3.8. ∎

Notice that this corollary implies that
$\mathbb{C}_{B_{i}}(M)^{*}\subseteq\mathbb{C}_{B}(M)^{*}$ for any $i$ and

$\mathbb{C}_{B_{i_{1}}+\dots+B_{i_{p}}}(M)^{*}\subseteq\mathbb{C}_{B_{i_{1}}}(M)^{*}\cap\cdots\cap\mathbb{C}_{B_{i_{p}}}(M)^{*}.$

As a consequence, we deduce the following description of
$\mathbb{C}_{B}(M)^{*}$. Notice that it is needed to stabilize the complement
by adding $-\mathbb{N}^{k}$, see Example 2.4.

######  Theorem 3.11.

Let $M$ be a graded $R$-module. Then,

$\bigcup_{i}\mathbb{C}_{B_{i}}(M)^{*}=\bigcup_{i_{1},\dots,i_{p}}\mathbb{C}_{B_{i_{1}}+\cdots+B_{i_{p}}}(M)^{*}=\mathbb{C}_{B}(M)^{*}.$

###### Proof.

First by Corollary 3.10,

$\bigcup_{i}\mathbb{C}_{B_{i}}(M)^{*}=\bigcup_{i_{1},\dots,i_{p}}\mathbb{C}_{B_{i_{1}}+\cdots+B_{i_{p}}}(M)^{*}\subseteq\mathbb{C}_{B}(M)^{*}.$

On the other hand,
$\mathbb{C}_{B}(M)\subseteq\bigcup_{i_{1},\dots,i_{p}}\mathbb{C}_{B_{i_{1}}+\cdots+B_{i_{p}}}(M)$
by Corollary 3.2 or [2, Lemma 2.1]. ∎

######  Remark 3.12.

In the example of $R:=S[a,b,c,x,y,z]$ with $S$ a field ($k=2$,
$n_{1}=n_{2}=3$) and $M:=R/(ax+by+cz)$, the sets $\mathbb{C}_{B_{i}}(M)$ and
$\mathbb{C}_{\mathfrak{m}}(M)$ are stable under the addition of
$-\mathbb{N}^{2}$ and independent of $S$, while $\mathbb{C}_{B}(M)$ is not
stable under the addition of $-\mathbb{N}^{2}$ and depends upon the
characteristic of $S$ (two distinct characteristics give rise to two distinct
supports). It shows that the determination of $\mathbb{C}_{B}(M)$ could
reserve some delicate points (for this example, we could only fully determine
$\mathbb{C}_{B}(M)$ in characteristic zero or two).

By Theorem 3.8, the first equality in Theorem 3.11 follows from an inclusion
of support of Tor modules that could be proved directly, as we now show.

######  Proposition 3.13.

Let $M$ be a graded $R$-module and $i_{1},\dots,i_{p},j_{1},\dots,j_{q}$ be
distinct elements in $\\{1,\ldots,k\\}$, then

${\mathbb{T}}^{i_{1},\dots,i_{p},j_{1},\dots,j_{q}}_{j}(M)\subseteq\bigcup_{l\leq
j}{\mathbb{T}}^{i_{1},\dots,i_{p}}_{j-l}(M)+\mathcal{E}_{l}^{j_{1},\dots,j_{q}}$

for all $j\geq 0$.

###### Proof.

The double complex
$K_{\bullet}(\mathbf{X}_{j_{1}},\dots,\mathbf{X}_{j_{q}};K_{\bullet}(\mathbf{X}_{i_{1}},\dots,\mathbf{X}_{i_{p}};M))$
gives rise to a spectral sequence with first terms

$K_{\bullet}(\mathbf{X}_{j_{1}},\dots,\mathbf{X}_{j_{q}};\operatorname{Tor}_{\bullet}^{R}(M,R/B_{i_{1}}+\ldots+B_{i_{p}}))$

that abuts to a filtration of
$\operatorname{Tor}_{\bullet}^{R}(M,R/B_{i_{1}}+\ldots+B_{i_{p}}+B_{j_{1}}+\ldots+B_{j_{q}})$.
∎

######  Corollary 3.14.

Let $M$ be a graded $R$-module and $i_{1},\dots,i_{p},j_{1},\dots,j_{q}$ be
distinct elements in $\\{1,\ldots,k\\}$, then

$\hat{\mathbb{T}}^{i_{1},\dots,i_{p},j_{1},\dots,j_{q}}(M)^{*}\subseteq\hat{\mathbb{T}}^{i_{1},\dots,i_{p}}(M)^{*}.$

##  4\. Some consequences on the truncation of modules

We first recall two results in the classical case where
$R=S[X_{1},\ldots,X_{n}]$ is a standard $\mathbb{Z}$-graded polynomial ring
and $\mathfrak{m}=(X_{1},\ldots,X_{n})$.

######  Proposition 4.1.

Let $C_{\bullet}$ be a graded complex of $R$-modules such that
$H^{p}_{\mathfrak{m}}(H_{i}(C_{\bullet}))=0$ for any $i\not=0$ and $p>i$.
Then,

$\operatorname{reg}(H_{0}(C_{\bullet}))\leq\max_{i}\\{\operatorname{reg}(C_{i})-i\\}.$

###### Proof.

It follows along the same lines as for instance in the proof of [4, Lemma 2.2]
by consideration of the two spectral sequences from
$\check{\mathcal{C}}^{\bullet}_{\mathbf{X}}(C_{\bullet})$ (the hypothesis
$C_{i}=0$ for $i<0$ there is not used except to restrict the $\max$ to
positive values of $i$). ∎

######  Lemma 4.2.

Let $M$ be a graded $R$ module and $t$ an integer.Then,

  * (i)

$\mathbb{C}^{0}_{\mathfrak{m}}(M_{t+\mathbb{N}})=\mathbb{C}^{0}_{\mathfrak{m}}(M)\cap(t+\mathbb{N})$;

  * (ii)

$\mathbb{C}^{1}_{\mathfrak{m}}(M_{t+\mathbb{N}})=\mathbb{C}_{\mathfrak{m}}^{1}(M)\cup\\{\operatorname{Supp}_{\mathbb{Z}}(M/H^{0}_{\mathfrak{m}}(M))\cap(t-1-\mathbb{N})\\}$;

  * (iii)

$\mathbb{C}^{j}_{\mathfrak{m}}(M_{t+\mathbb{N}})=\mathbb{C}^{j}_{\mathfrak{m}}(M)$
for all $j\geq 2$.

###### Proof.

It directly derives from the exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\mathfrak{m}}(M_{t+\mathbb{N}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\mathfrak{m}}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M/M_{t+\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\mathfrak{m}}(M_{t+\mathbb{N}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\mathfrak{m}}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and the isomorphisms $H^{j}_{\mathfrak{m}}(M_{t+\mathbb{N}})\simeq
H^{j}_{\mathfrak{m}}(M)$ for all $j\geq 2$, that are in turn provided by the
short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{t+\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M/M_{t+\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and the vanishing of $H^{i}_{\mathfrak{m}}(M/M_{t+\mathbb{N}})$ for $i>0$. ∎

We now return to our general setting of a standard multigraded ring $R$.

######  Definition 4.3.

Let $\mathbb{E}_{i}:=\\{(q_{1},\ldots,q_{k})\in\mathbb{Z}^{k}\ |\ q_{i}\geq
0\\}$ and

$\mathbb{E}_{i_{1},\ldots,i_{p}}:=\mathbb{E}_{i_{1}}\cap\cdots\cap E_{i_{p}}.$

Notice that $\mathbb{E}_{1,\ldots,k}=\mathbb{N}^{k}$. With these notations,

######  Proposition 4.4.

Let $M$ be a graded $R$-module, $i\in\\{1,\ldots,k\\}$ and
$t\in\mathbb{Z}^{k}$. Then

  * (i)

For $\mu\in t+\mathbb{N}^{k}$,

$H^{j}_{B_{i}}(M_{t+\mathbb{N}^{k}})_{\mu}=H^{j}_{B_{i}}(M)_{\mu},\quad\forall
j.$

  * (ii)

For $\mu\not\in t+\mathbb{N}^{k}$,

    * (1)

If $\mu\not\in t+\mathbb{E}_{1,\ldots,\widehat{i,}\ldots,k}$,
$H^{j}_{B_{i}}(M_{t+\mathbb{N}^{k}})_{\mu}=0$ for all $j$;

    * (2)

If $\mu\in t+\mathbb{E}_{1,\ldots,\widehat{i,}\ldots,k}$,
$H^{0}_{B_{i}}(M_{t+\mathbb{N}^{k}})_{\mu}=0$, the sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(M/H^{0}_{B_{i}}(M))_{\mu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{B_{i}}(M_{t+\mathbb{N}})_{\mu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{B_{i}}(M)_{\mu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is exact and $H^{j}_{B_{i}}(M_{t+\mathbb{N}^{k}})=H^{j}_{B_{i}}(M)$ for all
$j\geq 2$.

###### Proof.

According to Lemma 3.7 i), this follows from Lemma 4.2 applied to the ring
$T=S[\mathbf{X}_{i}]$. ∎

Set $\mathbf{1}:=(1,\ldots,1)=\sum_{i=1}^{k}e_{i}\in\mathbb{N}^{k}$.

######  Corollary 4.5.

Let $M$ be a graded $R$-module and $t\in\mathbb{Z}^{k}$. Then,

$\mathbb{C}_{\mathfrak{m}}(M_{t+\mathbb{N}^{k}})^{*}\subseteq\bigcap_{i}(\mathbb{C}_{B_{i}}(M)\cap(t+\mathbb{N}^{k}))^{*}\cup(t-\mathbf{1}-\mathbb{E}_{i}).$

###### Proof.

By Proposition 4.4,

$\mathbb{C}_{B_{i}}(M_{t+\mathbb{N}^{k}})\subseteq(\mathbb{C}_{B_{i}}(M)\cap(t+\mathbb{N}^{k}))\cup(t+(\mathbb{E}_{1,\ldots,\widehat{i,}\ldots,k}\setminus\mathbb{N}^{k})).$

As
$(\mathbb{E}_{1,\ldots,\widehat{i,}\ldots,k}\setminus\mathbb{N}^{k})^{*}=-\mathbb{E}_{i}-\mathbf{1}$,
the conclusion follows from Corollary 3.10.∎

######  Proposition 4.6.

Let $M$ be a graded $R$-module
$\Delta:=[0,n_{1}-1]\times\cdots\times[0,n_{k}-1]\subseteq\mathbb{N}^{k}$ and
$t\in\mathbb{Z}^{k}$. Then

$\mathbb{T}(M_{t+\mathbb{N}^{k}})\subseteq(-a+(\mathbb{C}_{B}(M)^{*}\cap(t+\mathbb{N}^{k})))\cup(t+\Delta).$

In particular, $\mathbb{T}(M_{t+\mathbb{N}^{k}})\subseteq t+\Delta$ if
$t\not\in\mathbb{C}_{B}(M)^{*}$.

###### Proof.

First,
$\hat{\mathbb{T}}(M_{t+\mathbb{N}^{k}})\subseteq\hat{\mathbb{T}}(M_{t+\mathbb{N}^{k}})^{*}=\mathbb{C}_{\mathfrak{m}}(M_{t+\mathbb{N}^{k}})^{*}$
by Theorem 3.6. By Theorem 3.11,
$\mathbb{C}_{B}(M)^{*}=\cup_{i}\mathbb{C}_{B_{i}}(M)^{*}$, hence

$\mathbb{T}(M_{t+\mathbb{N}^{k}})+a\subseteq(\mathbb{C}_{B}(M)\cap(t+\mathbb{N}^{k}))^{*}\cup(t-\mathbf{1}-\mathbb{N}^{k}).$

by Corollary 4.5. But
$\mathbb{T}(M_{t+\mathbb{N}^{k}})\subseteq\operatorname{Supp}(M_{t+\mathbb{N}^{k}})\subseteq
t+\mathbb{N}^{k}$ and

$(-a-\mathbf{1}-\mathbb{N}^{k})\cap\mathbb{N}^{k}=\Delta.$

If $t\not\in\mathbb{C}_{B}(M)^{*}$, then
$\mathbb{C}_{B}(M)^{*}\cap(t+\mathbb{N}^{k})=\emptyset$ by Lemma 3.4. ∎

Every $\mathbb{Z}^{k}$-graded module $M$ is also a $\mathbb{Z}$-graded module,
for the total degree grading, and we set $\operatorname{reg}(M)$ for its
Castelnuovo-Mumford regularity. For
$\mu=(\mu_{1},\ldots,\mu_{k})\in\mathbb{Z}^{k}$, write
$\mu^{+}:=(\max\\{\mu_{1},0\\},\ldots,\max\\{\mu_{k},0\\})$ and
$|\mu|:=\mu_{1}+\cdots+\mu_{k}$.

The next lemma follows from [6, the proof of 1.7]; it provides a criterion for
the truncation of a module $M_{t+\mathbb{N}^{k}}$ with $t\in\mathbb{Z}^{k}$ to
satisfy $\operatorname{reg}(M)=|t|$ (if the base ring $S$ is a field, usual
terminology is that $M$ is generated in degree $|t|$ and has a linear
resolution).

######  Lemma 4.7.

Let $M$ be a graded $R$-module such that $\operatorname{tdeg}(m)\geq u$ for
some $u\in\mathbb{Z}$ and every $m\in M$. For $t\in\mathbb{Z}^{k}$, let

$\delta_{i}:=\sup_{\mu\in\mathbb{T}_{i}(M)}\\{|(t-\mu)^{+}|-|t-\mu|\\}.$

Then
$\operatorname{reg}(M_{t+\mathbb{N}^{k}})\leq|t|+\max_{i}\\{\delta_{i}-i\\}$.

In particular, unless $M_{t+\mathbb{N}^{k}}=0$, if
$|(t-\mu)^{+}|\leq|t-\mu|+i$ for any $i$ and $\mu\in\mathbb{T}_{i}(M)$,

$\operatorname{reg}(M_{t+\mathbb{N}^{k}})=|t|.$

###### Proof.

As noticed by Eisenbud, Erman and Schreyer, any truncation of $R$ has a linear
resolution (given by the tensor product of some Eagon-Northcott complexes). As
a consequence, considering $R$ with its standard $\mathbb{Z}$-grading

$\operatorname{reg}(R_{t+\mathbb{N}^{k}})=\operatorname{indeg}(R_{t+\mathbb{N}^{k}})=|t^{+}|.$

Hence $\operatorname{reg}(R(-\mu)_{t+\mathbb{N}^{k}})=|\mu|+|(t-\mu)^{+}|$. As
$\operatorname{tdeg}(m)\geq u$ for every $m\in M$, $M$ admits a graded free
$R$-resolution $F_{\bullet}$ with $F_{i}=\oplus_{j}R(-t_{i,j})$ satisfying
$t_{i,j}\in\cup_{p\leq i}\mathbb{T}_{p}(M)$, by Lemma 3.5. The complex
$(F_{\bullet})_{t+\mathbb{N}^{k}}$ resolves $M_{t+\mathbb{N}^{k}}$ and for the
standard grading of $R$,

$\operatorname{reg}((F_{i})_{t+\mathbb{N}^{k}})-|t|=\sup_{j}\\{|t_{i,j}|+|(t-t_{i,j})^{+}|\\}-|t|\leq\max_{p\leq
i}\\{\delta_{p}\\}.$

The conclusion follows by Proposition 4.1. ∎

######  Remark 4.8.

To apply the above results in order to find $t$ such that
$\operatorname{reg}(M_{t+\mathbb{N}^{k}})=|t|$, it is useful to have in mind
that :

(i) $\delta_{i}=0$ if and only if $t\geq\mu$ (i.e. $t_{j}\geq\mu_{j}$,
$\forall j$) for every $\mu\in\mathbb{T}_{i}$,

(ii) one may replace $M$ by $M^{\prime}$ such that
$M_{\mu+\mathbb{N}^{k}}=M^{\prime}_{\mu+\mathbb{N}^{k}}$ and choose
$t\in\mu+\mathbb{N}^{k}$,

(iii) for any $\mu\in\mathbb{Z}^{k}$, $N:=M/M_{\mu+\mathbb{N}^{k}}$ satisfies
$N=H^{0}_{B}(N)$ and therefore

$H^{i}_{B}(M)_{\nu}=H^{i}_{B}(M_{\mu+\mathbb{N}^{k}})_{\nu},\quad\forall\nu\in\mu+\mathbb{N}^{k}.$

######  Proposition 4.9.

Let $M$ be a graded $R$-module and $\mu\not\in\mathbb{C}_{B}(M)^{*}$.

Then for any $t\in\mu+(n_{1}-1,\ldots,n_{k}-1)+\mathbb{N}^{k}$,
$M_{t+\mathbb{N}^{k}}$ has a regularity $|t|$.

###### Proof.

It follows from Proposition 4.6 and Lemma 4.7, according to Remark 4.8. ∎

For $M=R$, $\operatorname{reg}(R_{t+\mathbb{N}^{k}})=|t|$ if and only if
$t\in\mathbb{N}^{k}$ and
$\mathbb{C}_{B}(R)^{*}+(n_{1}-1,\ldots,n_{k}-1)=\mathbb{Z}^{k}\setminus\mathbb{N}^{k}$.

Acknowledgements. The second-named author was supported by a CAPES Doctoral
Scholarship.

## References

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# Symmetries and monotones in Markovian quantum dynamics

Georgios Styliaris  Paolo Zanardi  Department of Physics and Astronomy, and
Center for Quantum Information Science and Technology, University of Southern
California, Los Angeles, California 90089-0484, USA

###### Abstract

What can one infer about the dynamical evolution of quantum systems just by
symmetry considerations? For Markovian dynamics in finite dimensions, we
present a simple construction that assigns to each symmetry of the generator a
family of scalar functions over quantum states that are monotonic under the
time evolution. The aforementioned monotones can be utilized to identify
states that are non-reachable from an initial state by the time evolution and
include all constraints imposed by conserved quantities, providing a
generalization of Noether’s theorem for this class of dynamics. As a special
case, the generator itself can be considered a symmetry, resulting in non-
trivial constraints over the time evolution, even if all conserved quantities
trivialize. The construction utilizes tools from quantum information-geometry,
mainly the theory of monotone Riemannian metrics. We analyze the prototypical
cases of dephasing and Davies generators.

##  I Introduction

One of the main tasks in the study of non-relativistic quantum dynamical
systems is predicting how quantum states and observables evolve over time
given some dynamical law, for instance, a Hamiltonian operator and the
associated equations of motion. It is often the case, however, that in
practice the trajectory either does not admit an explicit closed form, or even
if it does, it can be complicated to draw conclusions to physical questions of
interest from it. It is therefore important to have accessible methods and
tools that allow one to extract the essential features of the evolution
directly from the dynamical laws.

In the heart of such approaches lies the far-reaching idea of symmetry. The
most prominent example is perhaps provided by Noether’s theorem which, in the
context of Lagrangian mechanics, yields a conserved quantity for each
differentiable symmetry of the generator [1]. In the quantum realm, the
theorem implies that all moments of an observable that is a symmetry of the
time evolution are conserved [2].

Connections between symmetries and conserved quantities in Markovian dynamics
have been discussed by Albert and Jiang [3]. Consequences of symmetries in
quantum dynamics have been systematically considered in the framework of
quantum resource theories [4], specifically in the theory of reference frames
and asymmetry [5, 6]. There, one investigates the allowed state transitions
under quantum dynamics that respects a specified symmetry111More specifically,
given a group $G$, one investigates the possibility of state transitions under
the class of quantum operations that are symmetric with respect to a unitary
representation of $G$, namely the operations such that
$\left[\mathcal{E},\mathcal{U}_{g}\right]=0$ $\forall g\in G$.. In particular,
Marvian and Spekkens in Ref. [7] established the fact that, for the case of
Hamiltonian dynamics, asymmetry monotones yield conserved quantities that can
be independent of the Noether ones [7]. In the case of symmetric dynamics that
is Markovian, asymmetry monotones have been utilized to put constraints on the
evolution of quantum coherences by Lostaglio et al. [8].

In this paper we consider open quantum systems in finite dimensions evolving
under Markovian dynamics [9]. We address the question: given the generator of
the dynamics and an initial state, what constraints can symmetry
considerations impose on the set of states that are reachable under the time
evolution?

Open quantum dynamics is, in general, dissipate and so the role of conserved
quantities can be rather limited. For instance, Markovian dynamics with a
unique steady state does not admit any non-trivial conserved quantities [3].
For this reason, we approach the problem by instead seeking for monotones,
i.e., functions of the state that are monotonic (in our case, non-increasing)
under the time evolution. Similarly to conserved quantities, monotones can be
utilized to exclude state transitions that are impossible under the dynamics,
i.e., states that do not belong in the trajectory of a given initial state.

Monotones generalize the concept of a conserved quantity and, as we show in
our construction, in fact one can deduce from them the full set of conserved
quantities. Our main result consists of a method to assign to each pair of
symmetries of the generator a one-parameter family of monotones for the time
evolution. As a special case, the generator itself can be considered a
symmetry of the dynamics, resulting in non-trivial constraints over the time
evolution, even if all conserved quantities trivialize.

The basic idea we invoke to obtain monotones of the time evolution relies on
the “infinitesimal” version of quantum data-processing inequalities.
Distinguishability measures $D(\rho,\sigma)\geq 0$, defined over pairs of
quantum states such that $D(\rho,\rho)=0$, are said to obey the data-
processing inequality if they are non-increasing under the joint action of a
quantum operation on both arguments and play a central role in quantum
information theory [10, 11]. Since Markovian dynamics is fully characterized
by its generator, we invoke an infinitesimal version of distinguishability
measures, connecting with monotone Riemannian metrics in the space of quantum
states. Such metrics have proven useful for the study of Markovian quantum
dynamics, as for instance for the study of the mixing time and convergence
rates [12, 13, 14].

The paper is organized as follows. In section II we begin by first non-
rigorously presenting the basic idea for establishing the correspondence
between symmetries and monotones of the evolution. We then present the general
construction based on monotone Riemannian metrics and, after giving a simple
example, we establish a connection with the Noether conserved quantities in
open systems. We also discuss how one can construct symmetries for the
generator of the dynamics and the monotones. In section III, we apply the
method to dephasing generators and in section IV for generators of the Davies
form, extracting in both cases the qualitative features of the evolution with
symmetry considerations. We conclude in section V.

##  II Monotones of the evolution from symmetries of the dynamics

###  II.1 Setting the stage

We consider quantum systems described by a state
$\rho\in\mathcal{S}(\mathcal{H})\subseteq\mathcal{B}(\mathcal{H})$, where the
Hilbert space $\mathcal{H}\cong\mathbb{C}^{d}$ is finite dimensional. We
assume that the dynamics is Markovian and time-homogeneous, i.e.,

$\displaystyle\frac{d}{dt}\rho_{t}=\mathcal{L}(\rho_{t})$ (1)

where the generator of the dynamics $\mathcal{L}$ (also known as the
Lindbladian) is time independent and hence can be expressed in the standard
form (see, e.g., Ref. [9] for more details)

$\displaystyle\mathcal{L}(X)=-i\left[H,X\right]+\sum_{i}\big{(}L_{i}XL_{i}^{\dagger}-\frac{1}{2}\big{\\{}L_{i}^{\dagger}L_{i},X\big{\\}}\big{)}\,\;.$
(2)

Equation (1) gives rise to a one-parameter family of time-evolution
superoperators

$\displaystyle\mathcal{E}_{t}=\exp\left(t\mathcal{L}\right)\,\;,\quad t\geq
0\,\;,$ (3)

belonging in the space of Completely Positive and Trace Preserving (CPTP)
maps, also known as quantum channels. We define as symmetries of a Lindbladian
those superoperators that commute with $\mathcal{L}$, i.e.,

$\displaystyle\operatorname*{Sym}(\mathcal{L})\coloneqq\left\\{\mathcal{M}\in\mathcal{B}\left(\mathcal{B}\left(\mathcal{H}\right)\right)\mid\left[\mathcal{M},\mathcal{L}\right]=0\right\\}\,\;.$
(4)

We first informally present the basic idea of how one can obtain monotones of
the evolution from symmetries of the generator. Let us consider a
distinguishability measure, i.e., a function $D(\rho,\sigma)$ over pairs of
states with

$\displaystyle D(\rho,\sigma)\geq 0\quad\text{and}\quad D(\rho,\rho)=0\,,$
(5a) such that it respects the data processing inequality $\displaystyle
D(\rho,\sigma)\geq D\left(\mathcal{E}(\rho),\mathcal{E}(\sigma)\right)$ (5b)

for all states $\rho$, $\sigma$ and quantum channels $\mathcal{E}$. Let us
also define

$\displaystyle f_{s}(\rho)=D(\rho,e^{s\mathcal{M}}\rho)\,\;,\quad\text{with
}\mathcal{M}\in\operatorname*{Sym}\left(\mathcal{L}\right)\,\;,$ (6)

which is a measure of the difference between a state $\rho$ and its
variation222Notice that for $\exp(s\mathcal{M})(\rho)$ to be a valid state for
$s\in I\subseteq\mathbb{R}_{\geq 0}$, the superoperator $\mathcal{M}$ should
obey certain constraints. We ignore this technical point for now, since we
later generalize this construction to include any
$\mathcal{M}\in\operatorname*{Sym}\left(\mathcal{L}\right)$. generated by the
symmetry $\mathcal{M}$.

The data processing inequality, together with the fact that $\mathcal{M}$ is a
symmetry of the time evolution, implies that

$\displaystyle f_{s}(\rho)\geq f_{s}(e^{t\mathcal{L}}\rho)\,\;,\quad\forall
t\geq 0\,\;,$ (7)

i.e., that $f_{s}$ is a monotone of the dynamics. However, to calculate
$f_{s}$ for a finite value of the parameter $s$ one needs to first calculate
the action of the exponential $\exp(s\mathcal{M})$ on $\rho$, which in general
is impractical. Instead, if the function $f_{s}$ is suitably well-behaved, one
can examine its expansion

$\displaystyle f_{s}(\rho)=s^{k}g_{\mathcal{M}}(\rho)+O(s^{k+1})\,,\quad
k\in\mathbb{N}\,\;,$ (8)

where the form of $g_{\mathcal{M}}$ depends on $D$ and $\mathcal{M}$. Since
the inequality (7) is valid for all $s\geq 0$, no matter how small, it follows
that also $g_{\mathcal{M}}(\rho)$ is a monotone of the evolution

$\displaystyle g_{\mathcal{M}}(\rho)\geq
g_{\mathcal{M}}(\mathcal{E}_{t}\left(\rho\right))\,\;,\quad t\geq 0\,\;.$ (9)

Before proceeding to formalize the previous observation via basic tools from
quantum information-geometry, let us consider an example. If the
distinguishability measure is taken to be the relative $\alpha$-Rényi entropy
with $\alpha=1/2$, i.e.,

$\displaystyle
D(\rho,\sigma)=S_{\frac{1}{2}}(\rho,\sigma)=-2\log\left(\operatorname*{Tr}\left[\sqrt{\rho}\sqrt{\sigma}\right]\right)$
(10)

then it is not hard to show by a straightforward calculation that the
expansion (8) yields $k=2$ and

$\displaystyle
g_{\mathcal{M}}(\rho)=\sum_{ij}\frac{\left|\braket{i}{\mathcal{M}(\rho)}{j}\right|^{2}}{\left(\sqrt{p_{i}}+\sqrt{p_{j}}\right)^{2}}\,\;,$
(11)

where we invoked the spectral decomposition of the argument
$\rho=\sum_{i=1}^{d}p_{i}\ket{i}\\!\bra{i}$. The above quantity is well-known
in quantum information-geometry as the Wigner-Yanase metric [15], here
evaluated on the operator $\mathcal{M}\left(\rho\right)$.

###  II.2 Monotones of the evolution and monotone Riemannian metrics

Let us specialize to the case where $\sqrt{D(\rho,\sigma)}$ is an “honest”
contractive distance function, i.e., in addition to properties (5) it is also
symmetric, non-degenerate, and obeys the triangle inequality. We will assume
that $\sqrt{D}$ arises from a Riemannian metric and that is defined over pairs
in $\mathcal{S}_{>0}(\mathcal{H})$ (the space of positive definite states). In
that case, the expansion (8) yields $k=2$ and the function
$g_{\mathcal{M}}(\rho)$ is nothing else that the square of the length of the
tangent vector $\mathcal{M}(\rho)$ over the tangent space
$T_{\rho}\left(\mathcal{S}_{>0}(\mathcal{H})\right)$.

The aforementioned class of Riemannian metrics, called monotone metrics [16,
17, 18, 19], constitute the quantum analog of the Fisher metric for
probability distributions (see also Ref. [20] for an accessible introduction
to the subject). While the latter is uniquely determined from the monotonicity
property under classical stochastic maps (up to a normalization constant), the
same does not hold for its quantum counterparts, which show a rich variety.
Monotone metrics are also closely related to generalized relative entropies
[21], as every monotone Riemannian metric arises from a generalized relative
entropy [22].

For the purposes of this work, the crucial property of such quantities is that
they satisfy the data processing inequality. This monotonicity property can be
understood as a consequence of a key operator inequality, first proved in Ref.
[22], which we repeat here due to its central importance for what follows.

######  Theorem (Lesniewski and Ruskai [22]).

$\displaystyle\operatorname*{Tr}\left(A^{\dagger}\frac{1}{\mathsf{R}_{\sigma}+\lambda\mathsf{L}_{\tau}}A\right)\geq\operatorname*{Tr}\left(\mathcal{E}(A)^{\dagger}\frac{1}{\mathsf{R}_{\mathcal{E}(\sigma)}+\lambda\mathsf{L}_{\mathcal{E}(\tau)}}\mathcal{E}(A)\right),$
(12)

where $A\in\mathcal{B}(\mathcal{H})$ is a linear operator,
$\mathsf{L}_{\sigma}\coloneqq\sigma X$ $(\mathsf{R}_{\tau}(X)\coloneqq X\tau)$
is the superoperator representing left (right) multiplication, $\mathcal{E}$
is a quantum channel and $\lambda\in\mathbb{R}_{\geq 0}$. The operators
$\sigma,\tau\in\mathcal{B}_{>0}(\mathcal{H})$ are positive definite, assuring
that the superoperator inverses entering the inequality are well-defined, as
well as that the resulting traces are non-negative.

Let us now return to considering quantum Markovian dynamics generated by some
time-independent Lindbladian $\mathcal{L}$. For our purposes, the quantum
channel in the inequality (12) is specialized to the time evolution
superoperator $\mathcal{E}_{t}\coloneqq\exp\left(t\mathcal{L}\right)$ for
$t\geq 0$. Letting $\rho\in\mathcal{S}_{>0}(\mathcal{H})$ be a full-rank
state, we take

$\displaystyle A=\mathcal{M}(\rho)$ $\displaystyle\,,\quad\text{where
}\mathcal{M}\in\operatorname*{Sym}\left(\mathcal{L}\right)$ (13)
$\displaystyle\sigma=\tau=\mathcal{N}(\rho)$ $\displaystyle\,,\quad\text{where
}\mathcal{N}\in\operatorname*{Sym}\left(\mathcal{L}\right)$ (14)

such that $\mathcal{N}(\rho)\in\mathcal{B}_{>0}(\mathcal{H})$ is positive
definite for all $\rho\in\mathcal{S}_{>0}(\mathcal{H})$. We will refer to any
such superoperators $\mathcal{M}$ and $\mathcal{N}$ as symmetries of the
dynamics.

The resulting inequality, due to the commutation relations
$\left[\mathcal{M},\mathcal{E}_{t}\right]=\left[\mathcal{N},\mathcal{E}_{t}\right]=0$,
expresses the fact that the quantity

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}\coloneqq\operatorname*{Tr}\left(\mathcal{M}(\rho)^{\dagger}\frac{1}{\mathsf{R}_{\mathcal{N}(\rho)}+\lambda\mathsf{L}_{\mathcal{N}(\rho)}}\mathcal{M}(\rho)\right)\,\;,$
(15)

satisfies

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}\geq\llbracket\mathcal{E}_{t}(\rho)\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}\,,\quad
t\geq 0.$ (16)

We have shown the following.

######  Proposition 1.

If $\mathcal{M},\mathcal{N}$ are symmetries of the dynamics and $\lambda\geq
0$, then the function

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}:\mathcal{S}_{>0}(\mathcal{H})\to\mathbb{R}_{\geq
0}$

defined in Eq. (15) is non-increasing under the time evolution generated by
$\mathcal{L}$.

In the rest of this paper, we will mainly focus on two cases.

  1. (i)

$\mathcal{N}=\mathcal{I}$, in which case for simplicity we denote the
resulting family of monotones as

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{M}}^{(\lambda)}=\operatorname*{Tr}\left(\mathcal{M}(\rho)^{\dagger}\frac{1}{\mathsf{R}_{\mathcal{\rho}}+\lambda\mathsf{L}_{\mathcal{\rho}}}\mathcal{M}(\rho)\right)\,\;.$
(17)

  2. (ii)

If the dynamics admits a full-rank stationary state $\omega$, then
$\mathcal{N}(X)=\operatorname*{Tr}\left(X\right)\omega$ is a symmetry. In that
case, it is convenient to denote directly the fixed state $\omega$ instead of
the symmetry, i.e., write

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{M},\omega}^{(\lambda)}=\operatorname*{Tr}\left(\mathcal{M}(\rho)^{\dagger}\frac{1}{\mathsf{R}_{\mathcal{\omega}}+\lambda\mathsf{L}_{\mathcal{\omega}}}\mathcal{M}(\rho)\right)\,\;.$
(18)

The monotones $\llbracket\rho\rrbracket_{\mathcal{M}}^{(\lambda)}$ can be also
expressed in coordinates via spectrally decomposing the argument
$\rho=\sum_{i}p_{i}\ket{i}\\!\bra{i}$. Substituting, one gets

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{M}}^{(\lambda)}=\sum_{ij}\frac{1}{\lambda
p_{i}+p_{j}}\left|\braket{i}{\mathcal{M}(\rho)}{j}\right|^{2}\,\;.$ (19)

Similarly, for the case of
$\llbracket\rho\rrbracket_{\mathcal{M},\omega}^{(\lambda)}$ one can decompose
$\omega=\sum_{i}q_{i}\ket{i_{\omega}}\\!\bra{i_{\omega}}$ resulting in

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{M},\omega}^{(\lambda)}=\sum_{ij}\frac{1}{\lambda
q_{i}+q_{j}}\left|\braket{i_{\omega}}{\mathcal{M}(\rho)}{j_{\omega}}\right|^{2}\,\;.$
(20)

For Hamiltonian dynamics,
$\mathcal{L}(X)=\mathcal{K}_{H}(X)\coloneqq-i\left[H,X\right]$, the monotones
(15) are, in fact, conserved. This follows because unitary channels are
invertible, hence for this case Eq. (16) holds true for $t\in\mathbb{R}$. This
forces monotones to maintain a constant value along the orbit.

For every family
$\\{\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}\\}_{\lambda}$,
one can consider convex combinations according to the measure $\mu(\lambda)$,
namely

$\displaystyle\\{\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}\\}_{\lambda}\mapsto\int
d\mu(\lambda)\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}\,\;.$
(21)

From this construction one obtains valid monotones, possibly admitting a
convenient mathematical form for an appropriate choice of the measure.
However, notice that the resulting functions do not impose any additional
constraints compared to the ones from the parent functions333I.e., if the
possibility of a transition $\rho\mapsto\sigma$ is ruled out by the inequality
$\int
d\mu(\lambda)\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}<\int
d\mu(\lambda)\llbracket\sigma\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}$,
then there exists a (non-zero measure) set of $\lambda$’s for which also
$\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}<\llbracket\sigma\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}$..

###  II.3 A first example: Dephasing of a qubit

It might be useful at this stage to consider a simple example to illustrate
the formalism. Let us analyze a two-level system with dephasing dynamics
described by the Lindbladian

$\displaystyle\mathcal{L}(X)=-ig\left[\sigma_{z},X\right]+\sigma_{z}X\sigma_{z}-X\,,\quad
g\in\mathbb{R}\,\;.$ (22)

The time evolution in terms of the Bloch vector representation of the state
$\rho(t)=\frac{1}{2}\left(I+\boldsymbol{v}\cdot\boldsymbol{\sigma}\right)$ is,
in cylindrical coordinates,

$\displaystyle r_{t}=r_{0}e^{-2t}\,,\quad\phi_{t}=\phi_{0}+2gt\,,\quad
z_{t}=z_{0}\,\;.$ (23)

That is, the Bloch vector lies onto a horizontal plane evolving inwards in a
spiral motion.

Let us now illustrate how one can deduce the qualitative features of the
evolution just by symmetry considerations, by use of Eq. (17). Since the
Hamiltonian $H=g\sigma_{z}$ and the single Lindblad operator $L=\sigma_{z}$
are both diagonal in the
$\sigma_{z}\coloneqq\ket{0}\\!\bra{0}-\ket{1}\\!\bra{1}$ eigenbasis, clearly
the left multiplication superoperators $\mathsf{L}_{\ket{0}\\!\bra{0}}$ and
$\mathsf{L}_{\ket{1}\\!\bra{1}}$ are symmetries of the Lindbladian. One
immediately gets that the populations

$\displaystyle\llbracket\rho\rrbracket_{\mathsf{L}_{\ket{i}\\!\bra{i}}}^{(0)}=\operatorname*{Tr}\left(\rho\ket{i}\\!\bra{i}\right)\,,\quad
i=0,1$ (24)

are non-increasing. However, since the evolution is trace preserving, each of
the populations is separately conserved. Notice that these are exactly the two
linearly independent conserved quantities of the evolution predicted by
Noether’s theorem for Lindbladians, which we discuss momentarily.

Now we consider again Eq. (17) but for the symmetry
$\mathcal{M}(X)=\mathcal{K}_{\sigma_{z}}$ and $\lambda=1$. Spectrally
decomposing $\rho=p_{+}P_{+}+p_{-}P_{-}$ and invoking Eq. (19), we have

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{K}_{\sigma_{z}}}^{(1)}$
$\displaystyle=2\operatorname*{Tr}\left(P_{+}\sigma_{z}P_{-}\sigma_{z}\right)(p_{+}-p_{-})^{2}=2\left|\braket{0}{\rho}{1}\right|^{2}\,\;,$
(25)

that is, coherences are non-increasing under the time evolution.

The monotones from Eqs. (24) and (25) jointly limit the set of states $R$ that
are (possibly) reachable under time evolution. The aforementioned set consists
of all Bloch vectors lying on horizontal disk with radius equal to $r_{0}$
(the radial distance of the initial Bloch vector from the $z$-axis.)

In fact, the set $R$ of our example is as constrained as possible, in the
sense that additional symmetry arguments (relying on the set
$\operatorname*{Sym}(\mathcal{L})$) cannot restrict it more. To see this,
first notice that the set of symmetries of our dephasing Lindbladian does not
depend on the value of the parameter $g$, as long as $g\neq 0$. On the other
hand, by varying $g$, all points in the bulk of $R$ can be reached by the
trajectories of Eq. (23) (for any fixed initial condition). This is because
$g$ controls the frequency of the rotational motion, which can be made
arbitrarily high. Therefore no more points in $R$ can be excluded by symmetry
arguments.

###  II.4 Monotones imply Noether conserved quantities

We will say that an operator $Y\in\mathcal{B}(\mathcal{H})$ is conserved if

$\displaystyle\operatorname*{Tr}\left[Y^{\dagger}\exp(t\mathcal{L})(\rho)\right]=\text{const}\quad\forall\rho\text{
and }t\geq 0\,\;.$ (26)

We will also refer to such functions of the state as Noether conserved
quantities, since they generalize the corresponding Hamiltonian construction.
It is well-known that

$\displaystyle Y\text{ is conserved\quad}\Longleftrightarrow\quad
Y\in\operatorname*{Ker}\mathcal{L}^{*}\,\;.$ (27)

The claim follows by noticing that the above trace can be expressed via the
Hilbert-Schmidt inner product
($\braket{A,B}\coloneqq\operatorname*{Tr}\left(A^{\dagger}B\right)$ for
$A,B\in\mathcal{B}(\mathcal{H})$) as

$\displaystyle\operatorname*{Tr}\left[Y^{\dagger}\exp(t\mathcal{L})(\rho)\right]=\braket{Y,\exp(t\mathcal{L})(\rho)}=\braket{\exp(t\mathcal{L}^{*})(Y),\rho}.$

A consequence is that the number of linearly independent conserved quantities
equals the dimension of the subspace spanned by the steady states of the
evolution, a fact also noted in [3]. This observation follows from the
identity
$\dim\operatorname*{Ker}(\mathcal{L})=\dim\operatorname*{Ker}(\mathcal{L}^{*})$.
In particular, if there is a unique fixed point of the evolution then all
conserved quantities trivialize, in the sense that they are necessarily
proportional to the trace of the (time-evolved) state, since always
$I\in\operatorname*{Ker}{\mathcal{L}^{*}}$.

On the other hand, the monotones
$\llbracket\rho\rrbracket_{\mathcal{M}}^{(\lambda)}$ impose non-trivial, in
general, constraints on the reachable states of the evolution, for instance by
taking $\mathcal{M}=\mathcal{L}$ (a specific example is given later in Figure
1). What is more, if a steady state $\omega$ is known (regardless whether it
is unique or not), then also monotones of the form
$\llbracket\rho\rrbracket_{\mathcal{M},\omega}^{(\lambda)}$ can be utilized,
imposing additional constraints.

A natural question to be asked is whether the constraints imposed by the
Noether conserved quantities are included in the monotones arising from 1 or
not. The answer is affirmative.

######  Proposition 2.

If the Lindbladian admits a full-rank stationary state, then for each
conserved operator $Y$ there is an appropriate choice of the symmetries
$\mathcal{M},\mathcal{N}$ in the family monotones
$\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(0)}$ that implies the
conservation.

Notice, however, the converse is not true; constraints imposed by the
monotones $\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(0)}$ cannot
necessarily be inferred from conservation laws.

###### Proof.

Let us consider some conserved $Y>0$. The superoperator
$\mathcal{M}(\rho)=\operatorname*{Tr}\left(Y\rho\right)\omega$, for $\omega$ a
full-rank stationary state of the evolution, is a symmetry of the dynamics.
Therefore
$\llbracket\rho\rrbracket_{\mathcal{M},\omega}^{(0)}=\operatorname*{Tr}\left(Y\rho\right)^{2}$
is non-increasing under the time evolution, hence also
$\operatorname*{Tr}\left(Y\rho\right)$ is non-increasing.

On the other hand, since $\omega$ is a full-rank steady state and
$\operatorname*{Tr}\left(Y\rho\right)>0$, also
$\llbracket\rho\rrbracket_{\omega,\mathcal{M}}^{(0)}=\left[\operatorname*{Tr}\left(Y\rho\right)\right]^{-1}$
is a valid non-increasing monotone. As a result,
$\operatorname*{Tr}\left(Y\rho\right)$ has to be a constant of the evolution.

To complete the proof, we need to show that the constraints imposed by
conserved $Y\in\mathcal{B}_{>0}(\mathcal{H})$ are the same as the ones imposed
by arbitrary $Y\in\mathcal{B}(\mathcal{H})$. This is true because
$\mathcal{L}^{*}$ preserves hermiticity and therefore
$Y\in\operatorname*{Ker}(\mathcal{L}^{*})$ also implies
$Y^{\dagger}\in\operatorname*{Ker}(\mathcal{L}^{*})$, allowing to trade a
possibly non-hermitian $Y$ for its (separately conserved) hermitian and anti-
hermitian parts. Finally, since $\mathcal{L}^{*}$ is unital, if $Y$ is
conserved then also $Y+aI$ is conserved, which can always be made positive for
large enough $a$. Notice that this amounts to just adding the constant $a$ to
the value of the Noether conserved quantity.

The above demonstrate that the monotone family
$\llbracket\rho\rrbracket_{\mathcal{M},\mathcal{N}}^{(0)}$ imposes at least as
many constraints as conserved quantities. ∎

###  II.5 Symmetries of the generator and the monotones

We now very briefly address the question of how one can find symmetries of a
Lindbladian, beyond those guessed by inspection. Determining explicitly the
entire set of symmetries $\operatorname*{Sym}\left(\mathcal{L}\right)$ of a
Lindbladian is usually too complicated to be of practical importance. Instead,
when a representation of the Lindbladian in Lindblad operators in known, it
might be convenient to look at operators simultaneously commuting with all
elements of the set
$S=\\{H\\}\cup\big{\\{}L_{i},L_{i}^{\dagger}\big{\\}}_{i}$. If $\mathcal{A}$
is the algebra generated by $S$, then the element of the algebra generated by
$\\{\mathsf{L}_{X},\mathsf{R}_{X}\\}_{X\in\mathcal{A}^{\prime}}$ belong to
$\operatorname*{Sym}\left(\mathcal{L}\right)$ (here $\mathcal{A}^{\prime}$
denotes the commutant of the algebra). We will use this fact to analyze
dephasing generators later in the paper.

Fixed points $\omega$ also give raise to symmetries, for instance via maps
$\mathcal{N}(X)=\operatorname*{Tr}(X)\omega$, as in Eq. (18). Although
$\operatorname*{Ker}\left(\mathcal{L}\right)$ is not in general closed with
respect to operator multiplication, it was shown in Ref. [23] that it forms an
Euclidean Jordan Algebra, i.e., if
$F\coloneqq\lim_{t\to\infty}\exp\left(t\mathcal{L}\right)I$ is the projection
of the identity onto $\operatorname*{Ker}\left(\mathcal{L}\right)$ and $A,B$
are fixed points of the evolution, then

$\displaystyle
A\bullet_{\\!{}_{F}}\\!\\!B\coloneqq\frac{1}{2}\left(A_{F}B_{F}+B_{F}A_{F}\right)$
(28)

is also a fixed point, where $X_{F}\coloneqq F^{-1/2}XF^{-1/2}$. This fact can
be of use to generate fixed points from a set of known ones.

Finally, we address the question of how to construct monotones that themselves
possess some symmetry. Let us consider a unitary representation $U_{g}$ of a
group $G$ such that $\mathcal{U}_{g}=U_{g}(\cdot)U_{g}^{\dagger}$ is a
symmetry of the Lindbladian for all $g\in G$. Then one can define a (left)
group action on the space of monotones via

$\displaystyle\llbracket\cdot\rrbracket_{\mathcal{M},\mathcal{N}}^{(\lambda)}\xmapsto{\,g\,}\llbracket\cdot\rrbracket_{\mathcal{U}_{g}\mathcal{M},\mathcal{N}}^{(\lambda)}=\llbracket\cdot\rrbracket_{\mathcal{M},\mathcal{U}_{g^{-1}}\mathcal{N}}^{(\lambda)}\,\;.$
(29)

From the latter, it is straightforward to construct monotones that are
invariant under the action by invoking the (Haar) group average, i.e.,
consider

$\displaystyle Q_{\mathcal{M},\mathcal{N}}^{(\lambda)}(\rho)\coloneqq\int
d\mu_{\text{Haar}}(g)\llbracket\rho\rrbracket_{\mathcal{U}_{g}\mathcal{M},\mathcal{N}}^{(\lambda)}\,\;.$
(30)

For instance,

$\displaystyle
Q_{\mathcal{M},\omega}^{(0)}(\rho)=\operatorname*{Tr}\left[\mathcal{P}\left(\mathcal{M}(\rho)^{\dagger}\mathcal{M}(\rho)\right)\omega^{-1}\right]\,\;,$
(31)

where $\mathcal{P}(X)\coloneqq\int d\mu_{\text{Haar}}(g)\mathcal{U}_{g}(X)$ is
the projector superoperator onto the $G$-invariant subspace, i.e., for any
operator $X$ it holds $U_{g}\mathcal{P}(X)=\mathcal{P}(X)U_{g}$ $\forall g$.
The projector $\mathcal{P}$ can be constructed explicitly in terms of the
irreducible representations of $\\{U_{g}\\}_{g}$ using standard techniques
from theory of noiseless subsystems [24, 25].

##  III Dephasing generators

Here we consider Lindblad generators of the general form (2) such that all
elements of the set $S=\\{H\\}\cup\big{\\{}L_{i},L_{i}^{\dagger}\big{\\}}_{i}$
mutually commute with each other. We will refer to this class as dephasing
generators. Notice that this family is a generalization of the qubit example
of subsection II.3.

Dephasing Lindblians are both unital and Hilbert-Schmidt normal, as can be
inferred with a direct calculation invoking the standard form (2) of the
generator. Hence by the spectral theorem they admit a complete family of
eigenoperators. The corresponding time evolution can be formally expressed via
resorting to the (maximal) projectors onto the joint eigenspaces of all
elements in $S$. Let us denote as $\\{P_{i}\\}_{i=1}^{n}$ ($n\leq d$) this
complete family of orthogonal projectors, and also as
$\mathcal{P}_{ij}(X)\coloneqq P_{i}XP_{j}$ the corresponding family of
(Hilbert-Schmidt orthogonal) projector superoperators. Then, for dephasing
generators, the time evolution can be written as

$\displaystyle\exp\left(t\mathcal{L}\right)=\sum_{ij}e^{\lambda_{ij}t}\mathcal{P}_{ij}\,\;.$
(32)

Invoking once again the standard form of the generator (2), it follows that
the eigenvalues satisfy $\lambda_{ij}=\lambda_{ji}^{*}$ and $\lambda_{ii}=0$
$\forall i$, while $\operatorname*{Re}\left(\lambda_{ij}\right)<0$ for $i\neq
j$.

The last inequality expresses that fact that there are no steady states with
support over any of the eigenspaces $\mathcal{P}_{ij}$ with $i\neq j$. This is
true since, for unital Lindbladians,
$\operatorname*{Ker}\left(\mathcal{L}\right)$ coincides with
$\mathcal{A}^{\prime}$ [26], where $\mathcal{A}$ is the (abelian) algebra
generated by the set $S$. In our case, this commutant is exactly given by
operators in the range of $\sum_{i}\mathcal{P}_{ii}$.

Qualitatively, the evolution dictates that the diagonal elements
$\sum_{i}\mathcal{P}_{ii}(\rho)$ are conserved while the off-diagonal parts
$\sum_{i}\mathcal{P}_{ij}(\rho)$ ($i\neq j$) decay, possibly with
oscillations, which is the defining characteristic of dephasing.

Let us now extract the same qualitative information from a symmetry analysis.
Clearly, $\left[\mathsf{L}_{P_{i}},\mathcal{L}\right]=0$ $\forall i$ hence

$\displaystyle\llbracket\rho\rrbracket_{\mathsf{L}_{P_{i}}}^{(0)}=\operatorname*{Tr}\left(\rho
P_{i}\right)\,,\quad\forall i$ (33)

are non-increasing. On the other hand, the evolution is trace-preserving,
therefore all $P_{i}$ are, in fact, conserved. These are all the linearly
independent Noether conserved quantities.

In addition, $\mathsf{L}_{P_{i}}\mathsf{R}_{P_{j}}$ is a symmetry of the
evolution since all elements in $S$ mutually commute. Combining this with the
fact that the evolution is unital, hence $\omega=I$ is a fixed point, we
directly get from Eq. (18) that

$\displaystyle\llbracket\rho\rrbracket_{\mathsf{L}_{P_{i}}\mathsf{R}_{P_{j}},I}^{(0)}=\operatorname*{Tr}\left(P_{i}\rho
P_{j}\rho\right)$ (34)

is non-increasing. This corresponds to the decay of the $(i,j)$-coherences of
the state $\rho$ and is a generalization of Eq. (25).

We have shown the following general fact.

######  Proposition 3.

Let $\mathcal{L}$ be a Lindbladian.

  1. (i)

If $\mathsf{L}_{P}$ is a symmetry of $\mathcal{L}$, then
$\operatorname*{Tr}\left(\rho_{t}P^{\dagger}P\right)$ is a non-increasing
function of time.

  2. (ii)

If $\mathcal{L}$ is unital and $\mathsf{L}_{P}\mathsf{R}_{Q}$ is a symmetry,
then $\operatorname*{Tr}\left(P^{\dagger}P\rho_{t}Q^{\dagger}Q\rho_{t}\right)$
is a non-increasing function of time.

For $P,Q$ orthogonal projectors, then $(i)$ corresponds to decaying population
and $(ii)$ to decaying coherence.

##  IV Davies generators

###  IV.1 Preliminaries

Figure 1: Constraints imposed on the time evolution by considering as symmetry
the Lindbladian $\mathcal{L}$ itself. In this example $\mathcal{L}$ is a
single qubit Davies generator with $H=\sigma_{z}$ and jump operators
$L_{1}=\sqrt{2}\sigma^{+}$, $L_{2}=\sigma^{-}$. We depict in Bloch coordinates
contours separating the allowed (inside) from the forbidden (outside) regions
as predicted by the monotone $\llbracket\rho\rrbracket_{\mathcal{L}}^{(1/2)}$
for various $\rho$. The monotone is symmetric under rotations around the
$z$-axis, so we only plot an $x-z$ slice of the Bloch ball. The corresponding
states $\rho$ are marked with a point and are chosen to correspond to
instances of the actual trajectory (as projected onto the $x-z$ plane) of the
initially pure state $\boldsymbol{v}=(1/2,0,-\sqrt{3}/2)$.

A physically significant class of Lindbladians is provided by the Davies
generators [27] (for a more modern treatment, see also Refs. [28, 29]). This
family arises from a first-principles derivation under the main assumption
that the interaction between the system and its environment is weak and gives
rise to steady states that are thermal. Davies generators have the following
properties:

  1. (i)

The Hamiltonian part commutes with the dissipative part, i.e.,
$\mathcal{L}=\mathcal{K}_{H}+\mathcal{D}$ with

$\displaystyle\left[\mathcal{K}_{H},\mathcal{D}\right]=0\,\;.$ (35a)

  2. (ii)

It satisfies the quantum detailed balance condition, i.e., there exists a
state $\tau$ such that

$\displaystyle\mathcal{D}\mathsf{R}_{\tau}$
$\displaystyle=\mathsf{R}_{\tau}\mathcal{D}^{*}\,\;,$ (35b)
$\displaystyle\mathcal{K}_{H}\left(\tau\right)$ $\displaystyle=0\,\;.$ (35c)

For the scope of this paper, we will refer to any Lindbladian that satisfies
the conditions of Eqs. (35) as a Davies generator.

Let us recall a few useful mathematical facts regarding the detailed balance
condition [28, 30] . First of all, since $\mathcal{D}^{*}$ is unital, Eq.
(35b) implies that $\tau$ is a fixed point of the evolution. The detailed
balance condition can be equivalently understood as the requirement that
$\mathcal{D}^{*}$ is hermitian with respect to the (possibly degenerate)
scalar product

$\displaystyle\braket{A,B}_{\tau}\coloneqq\operatorname*{Tr}\left(\tau
A^{\dagger}B\right)\,\;.$ (36)

Assuming that $\tau$ is full rank, the detailed balance condition Eq. (35b)
can be recast in a yet another form, namely $\mathcal{D}$ being hermitian with
respect to the scalar product $\braket{\cdot,\cdot}_{\tau^{-1}}$. The last
claim follows by multiplying both from the left and the right Eq. (35b) with
$\mathsf{R}_{\tau}^{-1}$.

Let us now express the general form of the time evolution of a Davies
generators. Notice that Davies generators are always normal operators, hence
admit a complete family of eigenoperators. This is because, by assumption,
$\mathcal{D}$ is hermitian with respect to the scalar product (36) (for
$\tau^{-1}$) and it commutes with the Hamiltonian part. Let as formally denote
as $\\{\mathcal{P}_{i}\\}_{i}$ the complete family of hermitian (with respect
to $\braket{\cdot,\cdot}_{\tau^{-1}}$) superoperator projectors. Then the
general solution has the form

$\displaystyle\exp\left(t\mathcal{L}\right)=\sum_{i}e^{\lambda_{i}t}\mathcal{P}_{i}$
(37)

with $\mathcal{P}_{i}\mathcal{P}_{j}=\delta_{ij}\mathcal{P}_{i}$ and
$\sum_{i}\mathcal{P}_{i}=\mathcal{I}$.

###  IV.2 Davies generator: Single qubit

Let us consider a qubit system whose Lindbladian is described by
$H=\sigma_{z}$ together with two Lindblad operators
$\\{a\sigma^{+},b\sigma^{-}\\}$. The unique steady state
$\tau=\frac{1}{2}\left(I+\boldsymbol{w}\cdot\boldsymbol{\sigma}\right)$ has a
corresponding Bloch vector lying along the $z$-axis with
$w_{z}=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}$. One can easily check directly that
Eqs. (35) are satisfied, hence $\mathcal{L}$ is indeed a valid Davies
generator. Setting $g=|a|^{2}+|b|^{2}$, the dynamical evolution of the Bloch
vector is, in cylindrical coordinates,

$\displaystyle
r_{t}=r_{0}e^{-gt/2}\,,\;\,\phi_{t}=\phi_{0}+2t\,,\;\,z_{t}=(1-e^{-gt})w_{z}+e^{-gt}z_{0}\,.$
(38)

Let us analyze some of its symmetries now. By definition of the Davies
generator, $\mathcal{K}_{H}=\mathcal{K}_{\sigma_{z}}$ is a symmetry, therefore
Eq. (25) is a valid monotone, indicating that coherences can only decay. This
corresponds to the inward spiral motion of the trajectory.

Regarding the $z$-evolution of the Bloch vector towards the fixed point, let
us assume that $\tau$ is known to be a fixed state of the evolution (but we do
not explicitly assume that it is the unique fixed state). Then by inspection
one can verify that
$\mathcal{M}(X)=\braket{\sigma_{z},X}_{\tau^{-1}}\sigma_{z}$ is a symmetry of
the evolution. Therefore

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{M},\tau}^{(0)}=\left|\operatorname*{Tr}\left(\frac{1}{\tau}\sigma_{z}\rho\right)\right|^{2}\operatorname*{Tr}\left(\tau^{-1}\right)$
(39)

is non-increasing, implying that also
$\left|\operatorname*{Tr}\left(\tau^{-1}\sigma_{z}\rho\right)\right|$ is a
monotone. In the Bloch representation, the last quantity equivalently
expresses the fact that $\left|v_{z}-w_{z}\right|$ is non-increasing, i.e.,
the $z$-component of the Bloch vector monotonically approaches the steady
state.

Finally, we numerically investigate in Figure 1 a example for the constraints
imposed on the qubit time evolution just by considering as symmetry the
Lindbladian itself.

###  IV.3 Davies generators: General remarks

Let us make a few observations regarding monotones and Davies generators.
First of all, notice that monotones (18) for $\omega=\tau$ (the detailed
balance fixed point) and $\lambda=0$ are directly related to the Davies inner
product

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{M},\tau}^{(0)}=\braket{\mathcal{M}(\rho),\mathcal{M}(\rho)}_{\tau^{-1}}\,\;.$
(40)

In particular, the case $\mathcal{M}=\mathcal{I}$ expresses the fact that

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{I},\tau}^{(0)}=\operatorname*{Tr}\left(\frac{1}{\tau}\rho^{2}\right)$
(41)

is non-increasing under the time evolution, which is a generalization of the
well-known fact that the purity
$\operatorname*{Pur}(\rho)\coloneqq\operatorname*{Tr}(\rho^{2})$ is non-
increasing under unital dynamics. Equivalently, the distance

$\displaystyle\left\|\rho_{t}-\tau\right\|^{2}_{\tau^{-1}}=\braket{\rho_{t}-\tau,\rho_{t}-\tau}_{\tau^{-1}}$
(42)

is non-increasing.

One can easily generalize the construction of the monotone (39). Each of the
projectors in Eq. (37) can be written as
$\mathcal{P}_{i}(X)=\sum_{k=1}^{d_{i}}\braket{Y_{i,k},X}_{\tau^{-1}}Y_{i,k}$,
where $\\{Y_{i,k}\\}_{i,k}$ is a complete orthonormal family of
eigenoperators. Hence, for any such operator,
$\mathcal{M}(X)=\braket{Y,X}_{\tau^{-1}}Y$ is clearly a symmetry of the
dynamics, and because for these symmetries

$\displaystyle\llbracket\rho\rrbracket_{\mathcal{M},\tau}^{(0)}=\left|\operatorname*{Tr}\left(\frac{1}{\tau}Y^{\dagger}\rho\right)\right|^{2}$
(43)

the functions
$\left|\operatorname*{Tr}\left(\tau^{-1}Y^{\dagger}\rho\right)\right|$ are
monotones, generalizing Eq. (39).

##  V Discussion and outlook

Identifying symmetries and their consequences in physical systems is an
important task for various fields of physics. The present manuscript
constitutes an attempt to provide a simple correspondence between symmetries
in the generators of quantum Markovian dynamics and monotones of the
corresponding evolution. More specifically, we presented a construction that
assigns to every pair of symmetries of the generator a one-parameter family of
functions over quantum states that are non-increasing under the time
evolution. Such monotonic functions can be employed in order to identify
states that are non-reachable by the evolution. The construction utilizes
powerful tools from quantum information-geometry, mainly from the theory of
monotone Riemannian metrics. Finally, we have demonstrated how one can deduce
from symmetries the qualitative features of the evolution for the prototypical
cases of dephasing and Davies generators.

One can easily rephrase the question addressed in the present paper to fit in
the framework of quantum resource theories. Given a Lindbladian, one can
define as free operations those associated with the time evolution
$\\{\mathcal{E}_{t}\coloneqq\exp\left(t\mathcal{L}\right)\\}_{t\geq 0}$. Our
main result, 1, provides a family of monotones for each pair of symmetries of
the generator. However, even if all possible symmetries are considered, the
resulting families of monotones are not in general complete, i.e., they do not
exclude all states that are non-reachable by the evolution. It remains an open
question to find a general construction for a complete family of monotones.

We hope that the present work demonstrates yet another way in which ideas and
tools from quantum information theory can provide useful insights to the
analysis of quantum dynamics.

###### Acknowledgements.

G.S. is thankful to Á.M. Alhambra, N. Anand, L. Campos Venuti, A. Hamma, S.
Montangero, and N.A. Rodríguez Briones for the interesting discussions and to
Th. Apostolatos and P.J. Ioannou for their teachings. P.Z. acknowledges
partial support from the NSF award PHY-1819189.

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