# constructible Witt theory of Schemes

Onkar Kamlakar Kale and Girja S Tripathi

###### Abstract.

We study the constructible Witt theory of étale sheaves of $\Lambda$-modules
on a scheme $X$ for coefficient rings $\Lambda$ having finite characteristic
not equal to 2 and prime to the residue characteristics of the scheme $X$. Our
construction is based on the recent advances by Cisinski and Deglise on six-
functor formalism for derived categories of étale motives and offers a
background for the of study of constructible Witt theory as a cohomological
invariant for schemes.

##  1\. Introduction

Witt groups of sheaves of modules on topological spaces have been considered
as a generalised cohomology theory by Woolf in [Woo08] and by Woolf and
Schürmann in [SW20]. These provide signature-type invariants for topological
spaces taking values in Witt theories related to the coefficient ring and have
descriptions as symmetric forms on intersection cohomology complexes. This
work grew out of our interest in the analogous algebraic setting particularly
with an interest in algebraic counterpart for the interpretation of
$L$-classes as stable homology operations from (topological) constructible
Witt groups to the ordinary rational homology. We consider the derived
categories of constructible étale sheaves of modules over suitable coefficient
rings and use Balmer’s theory of triangular Witt groups to define
constructible Witt groups: Given a scheme $X$ and a ring $\Lambda$ having
finite characteristic not equal to 2 and prime to residue characteristics of
$X$, we define the constructible Witt groups
$W^{i}_{c}(X_{\acute{e}t},\Lambda)$ as the Witt groups of triangulated
category with duality $(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),D_{X}(T))$
consisting of étale sheaves of $\Lambda$-modules having finite Tor-dimension
and constructible cohomology sheaves.

As shown in the last section of the paper, with the identification of
$W^{i}_{c}((\mathrm{Spec\hskip 1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)$
with a $\mathbb{Z}/2\mathbb{Z}$-equivariant Witt theory of finitely generated
projective $\Lambda$-modules, an incentive to consider constructible Witt
theory is its identification with certain equivariant Witt theories related to
the coefficient rings. The algebraic analog of signature in [Woo08] can be
constructed by using the proper pushforwards for constructible Witt theory in
Theorem 5.6. Let $X$ be a projective real algebraic variety with the structure
morphism $f:X\rightarrow\mathrm{Spec\hskip 1.42262pt}\mathbb{R}$ and $\Lambda$
a ring of finite characteristic not equal to 2. Then we have the induced
proper pushforward for constructible Witt theory

$W^{i}(f_{*}):W^{i}_{c}(X_{\acute{e}t},\Lambda)\rightarrow
W^{i}_{c}((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)=W^{i}_{lf}(\Lambda[\mathbb{Z}/2\mathbb{Z}])$

where $W^{i}_{lf}(\Lambda[\mathbb{Z}/2\mathbb{Z}])$ denotes a
$\mathbb{Z}/2\mathbb{Z}$-equivariant Witt theory of finitely generated
projective $\Lambda$-modules.

It can be observed that the theory of étale sheaves on $\mathrm{Spec\hskip
1.42262pt}\mathbb{R}$ is the only one that can be related to an equivariant
theory frequently studied, namely, the $\mathbb{Z}/2\mathbb{Z}$-equivariant
theory since this is the only case in which the absolute Galois group is a
familiar one (the cyclic group of order 2). In this paper we identify the
constructible Witt theory of sheaves of $\Lambda$-modules for
$\mathrm{Spec\hskip 1.42262pt}\mathbb{R}$ with the Witt theory of the group
ring $\Lambda[\mathbb{Z}/2\mathbb{Z}]$. In general, over a field $k$ having
characteristic prime to the characteristic of the ring $\Lambda$ the groups
$W^{i}_{c}(\mathrm{Spec\hskip 1.42262pt}k,\Lambda)$ should be related to a
Witt theory of finitely generated projective $\Lambda$-modules with an action
of the absolute Galois group $Gal(k^{sep}/k)$ (a profinite group) of the
separable closure of $k$. Even with widening interest in equivariant theories
we are not aware of any studies in Witt theory for actions of pro-finite
groups and the work in this paper should encourage more research on this.

Now we describe the contents of the paper in details. In section 2 we quickly
recall the Witt theory of triangulated categories with duality. In section 3
the derived category $D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$ of étale sheaves of
$\Lambda$-modules having finite Tor-dimensions and constructible cohomology
sheaves is described as a triangulated category with duality. Under
appropriate restrictions these categories have been widely studied for a long
time in étale cohomology theory. In this paper we use the recent advances made
by Cisinski and Déglise on étale motives to understand the category
$D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$ by using an equivalent model
$DM_{h,lc}(X,\Lambda)$ of locally constructible $h$-motives. The categories
$DM_{h,lc}(X,\Lambda)$ offer a greater generality for the six-functor
formalism and also a roadmap for studying the relations between constructible
Witt groups and the rational motivic cohomology. In section 3 we use the work
of Cisinski and Déglise to recast the relevant results for the bounded derived
category $D^{b}_{ctf}((-)_{\acute{e}t},\Lambda)$ and using these in section 4
we develop the constructible Witt theory.

######  Theorem 1.1.

Let $B$ be an excellent noetherian scheme of dimension $\leq 2$ and $\Lambda$
a noetherian ring of positive characteristic prime to the residue
characteristics of $B$. Let $\phi:S\rightarrow B$ be a regular separated
finite type $B$-scheme and $f:X\rightarrow S$ a separated morphism of finite
type. The categories $D^{b}_{ctf}(X_{\acute{e}t},\Lambda)\subset
D^{b}(X_{\acute{e}t},\Lambda)$ are closed under the six-functors of
Grothendieck in $D^{b}(X_{\acute{e}t},\Lambda)$. For a $\otimes$-invertible
object $T\in D^{b}_{ctf}(S_{\acute{e}t},\Lambda)$ the functor

$D_{X}(T)=R\mathcal{H}om(-,f^{!}(T)):D^{b}_{ctf}(X,\Lambda)^{op}\rightarrow
D^{b}_{ctf}(X,\Lambda)$

is a duality functor on $D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$. In particular,
we have triangulated categories with duality
$(D^{b}_{ctf}(B_{\acute{e}t},\Lambda),R\mathcal{H}om(-,\Lambda))$ and
$(D^{b}_{ctf}(S_{\acute{e}t},\Lambda),R\mathcal{H}om(-,\Lambda))$

######  Theorem 1.2 (Pullbacks for étale morphisms and proper pushforwards).

Under the same assumptions on $B$, $\Lambda$ and $S$ as in the theorem above,
let $f:X\rightarrow S$ and $g:Y\rightarrow S$ be separated $S$-schemes of
finite type, and $T$ be a $\otimes$-invertible object in
$D^{b}_{ctf}(S_{\acute{e}t},\Lambda)$. Then an ëtale morphism $h:X\rightarrow
Y$ of $S$-schemes induces a morphism of triangulated categories with duality

$h^{*}:(D^{b}_{ctf}(Y_{\acute{e}t},\Lambda),g^{!}T)\rightarrow(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),f^{!}T)$

and homomorphisms $h^{*}:W^{i}_{c}(Y_{\acute{e}t},\Lambda)\rightarrow
W^{i}_{c}(X_{\acute{e}t},\Lambda)$ of constructible Witt groups. If the
morphism $h:X\rightarrow Y$ is proper, then the pushforward morphism

$h_{*}:(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),f^{!}T)\rightarrow(D^{b}_{ctf}(Y_{\acute{e}t},\Lambda),g^{!}T)$

is a morphism of triangulated categories with duality and induces pushforward
morphisms on constructible Witt groups
$h_{*}:W^{i}_{c}(X_{\acute{e}t},\Lambda)\rightarrow
W^{i}_{c}(Y_{\acute{e}t},\Lambda)$.

In the last section 5 we provide some details about the description of
constructible Witt theory of fields in terms of an equivariant Witt theory of
finitely generated $\Lambda$-modules with Galois action.

######  Theorem 1.3.

For a ring $\Lambda$ of finite characteristic not equal to $2$ the bounded
derived category $D^{b}_{ctf}((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)$ is equivalent to bounded derived
category $D^{b}(Proj(\Lambda[\mathbb{Z}/2\mathbb{Z}]))$ of finitely generated
projective $\Lambda$-modules with an action of the absolute Galois group
$Gal(\mathbb{C}/\mathbb{R})=\mathbb{Z}/2\mathbb{Z}$, and have an
identification of constructible Witt groups $W^{i}_{c}((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)$ with a
$\mathbb{Z}/2\mathbb{Z}$-equivariant Witt groups of finitely generated
projective $\Lambda$-modules.

Conventions. In the paper the coefficient ring $\Lambda$ will be assumed to be
of finite characteristic not equal to $2$ and all the schemes considered will
have the property that their residue characteristics are prime to the
characteristic of $\Lambda$. All the schemes will be assumed to be noetherian
and of finite Krull dimension.

Acknowledgements. We are thankful to K. Arun Kumar for helpful discussions
during the preparation of this manuscript.

##  2\. Witt theory of triangulated categories with duality

In this section we recall the basic theory of Witt groups for triangulated
categories with duality developed by Balmer in [Bal99], [Bal00], [Bal01].

###  2.1. Triangulated categories with duality

Let $\delta=\pm 1$. For triangulated categories $(K_{1},T_{1})$ and
$(K_{2},T_{2})$, an additive functor $F:K_{1}\rightarrow K_{2}$ is said to be
$\delta$-exact if $T_{2}^{-1}\circ F=F\circ T_{1}$ and if for any exact
triangle

$A\xrightarrow{u}B\xrightarrow{v}C\xrightarrow{w}T_{1}(A)$

the following triangle is exact:

$F(C)\xrightarrow{F(v)}F(B)\xrightarrow{F(u)}F(A)\xrightarrow{\delta\cdot
T_{2}(F(w))}T_{2}(F(C))$

######  Definition 2.1.

Let $(K,T)$ be a triangulated category and $\delta=\pm 1$. We will always
assume that $1/2\in K$. A _$\delta$ -duality_ is a $\delta$-exact
contravariant functor $\\#:K\rightarrow K$ such that there exist an
isomorphism $\omega:Id\xrightarrow{\simeq}\\#\circ\\#$ satisfying the
conditions:

$\omega_{T(M)}=T(\omega_{M})\hskip 14.22636pt\text{and}\hskip
14.22636pt(\omega_{M})^{\\#}\circ\omega_{M^{\\#}}=Id_{M^{\\#}}$

for any object $M$ of $K$. Then the triple $(K,\>\\#,\>\omega)$ is called a
$triangulated$ $category$ $with$ $\delta$-$duality$.

In case $\delta=1$, we shall talk about a $duality$ and in case $\delta=-1$,
we shall use the word $skew$-$duality$.

######  Example 2.2.

Let $(K,\>\\#,\>\omega)$ be a tringulated category with $\delta$-duality
$(\delta=\pm 1)$.

  1. (1)

Let $n\in\mathbb{Z}$. Then $(K,\>T^{n}\circ\\#,\>\omega)$ is again a
triangulated category with $((-1)^{n}\cdot\delta)$-duality.

  2. (2)

Also, $(K,\>\\#,\>-\omega)$ is again a tringulated category with
$\delta$-duality.

Consider a triangulated category $(K,\>\\#,\>\omega)$ with $\delta$-duality
$(\delta=\pm 1)$. A _symmetric space_ is a pair $(P,\phi)$ such that $P$ is an
object in $K$ and $\phi:P\xrightarrow{\simeq}P^{\\#}$ is an isomorphism such
that $\phi^{\\#}\circ\omega_{P}=\phi$. A _skew-symmetric_ $form$ is a pair
$(P,\phi)$ such that $P$ is an object in $K$ and
$\phi:P\xrightarrow{\simeq}P^{\\#}$ is an isomorphism such that
$\phi^{\\#}\circ\omega_{P}=-\phi$. A skew-symmetric form in $K$ is a symmetric
form in $(K,\\#,-\omega)$. Orthogonal sum and isometries are defined as usual.

######  Remark 2.3.

Let $(K,\>\\#,\>\omega)$ be a triangulated category with $\delta$-duality
$(\delta=\pm 1)$. Then as suggested by the above example, the _translated (or
shifted)_ structure of triangulated category with $(-\delta)$-duality is

$T(K,\>\\#,\>\omega)\coloneqq(K,\>T\circ\\#,\>(-\delta)\cdot\omega).$

Also, if $(K,\>\\#,\>\omega)$ is a triangulated category with duality (i.e.
$\delta=+1$) then

$T^{n}(K,\>\\#,\>\omega)=(K,\>T^{n}\circ\\#,\>(-1)^{n(n+1)/2}\cdot\omega)$

is a triangulated category with $(-1)^{n}$-duality, for all $n\in\mathbb{Z}$.

###  2.2. Witt groups of triangulated categories with duality

In his 1937 paper [Wit37], Ernst Witt introduced a group structure and even a
ring structure on the set of isometry classes of anisotropic quadratic forms,
over an arbitrary field k. This object is now called the Witt group W(k) of k.
Since then, Witt’s construction has been generalized and extended in many
ways. In this paper we work in the setting of triangulated categories with
duality developed by Balmer.

######  Definition 2.4.

Let $(K,\>\\#,\>\omega)$ be a triangulated category with $\delta$-duality (for
$\delta=\pm 1$) and $1/2\in K$. Let $(P,\phi)$ be a symmetric space. A pair
$(L,\alpha)$ where $L$ is an object of $K$ and $\alpha:L\rightarrow P$ is a
morphism, is called _sublagrangian_ of $(P,\phi)$ if
$\alpha^{\\#}\phi\alpha=0$. A triple $(L,\alpha,w)$ is called _Lagrangian_ if
the following triangle is exact:

$T^{-1}(L^{\\#})\xrightarrow{w}L\xrightarrow{\alpha}P\xrightarrow{\alpha^{\\#}\phi}L^{\\#}\hskip
14.22636pt\text{and}\hskip 14.22636pt\text{if $w$ is $\delta$-symmetric,
i.e.}\>T^{-1}(w^{\\#})=\delta\cdot w.$

######  Definition 2.5.

Let $(K,\>\\#,\>\omega)$ be a triangulated category with $\delta$-duality (for
$\delta=\pm 1$). A symmetric space $(P,\>\phi)$ is _neutral_ or _metabolic_ if
it possesses lagrangian.

######  Definition 2.6.

Let $(K,\>\\#,\>\omega)$ be a triangulated category with $\delta$-duality (for
$\delta=\pm 1$). We define the _Witt Monoid_ of $K$ to be the monoid of
isometry classes of symmetric spaces endowed with the diagonal sum and we
denote it by

$MW(K,\>\\#,\>\omega).$

The set of isometry classes of neutral spaces forms submonoid
$NW(K,\>\\#,\>\omega)\subset MW(K,\>\\#,\>\omega)$. The quotient monoid is a
group, called the _Witt Groups_ of $(K,\>\\#,\>\omega)$ and written as
$W(K,\>\\#,\>\omega)$. Thus,

$W(K,\>\\#,\>\omega)=\frac{MW(K,\>\\#,\>\omega)}{NW(K,\>\\#,\>\omega)}$

If $(P,\phi)$ is symmetric space, we write $[P,\phi]$ for its class in the
corresponding Witt group. We say that two symmetric spaces are _Witt-
equivalent_ if their classes in the Witt group are same. We also define

$W^{n}(K,\>\\#,\>\omega)\coloneqq W(T^{n}(K,\>\\#,\>\omega))$

for all $n\in\mathbb{Z}$. These are the $shifted$ Witt groups of
$(K,\>\\#,\>\omega)$.

###  2.3. Functoriality

Let $(K_{1},\\#_{1},\omega_{1})$ and $(K_{1},\\#_{2},\omega_{2})$ be
triangulated categories with duality. Assume that $\\#_{1}$ and $\\#_{2}$ are
both either exact or skew exact. A _morphism_ of triangulated categories with
duality refers to a covariant additive functor $F:K_{1}\rightarrow K_{2}$ that
preserves exactness or skew-exactness, and satisfies the following conditions:

$F\circ\\#_{1}=\\#_{2}\circ F\hskip 14.22636pt\text{and}\hskip
14.22636ptF(\omega_{1})=\omega_{2}.$

In this case, if $(P,\phi)$ is a symmetric space for $\\#_{1}$ then
$(F(X),F(\phi))$ is a symmetric space for $\\#_{2}$. If $(L,\alpha,w)$ is a
lagrangian of the starting space, then $(F(L),F(\alpha),\delta\cdot F(w))$ is
a lagrangian of its image, where $\delta=\pm 1$ comes from $\delta$-exactness
of $F$. This tells that neutral forms are mapped to neutral forms. Hence $F$
induces a group homomorphisms

$W^{n}(F):W^{n}(K_{1},\\#_{1},\omega_{1})\rightarrow
W^{n}(K_{1},\\#_{2},\omega_{2})$

for any $n\in\mathbb{Z}$.

##  3\. Derived category of constructible sheaves

In this section we recall some well-known results for the full triangulated
subcategory $D^{b}_{ctf}(X_{\acute{e}t},\ \Lambda)$ of the derived category
$D(X_{\acute{e}t},\ \Lambda)$ of étale sheaves of $\Lambda$-modules on the
small étale site $X_{\acute{e}t}$. We describe the category
$D^{b}_{ctf}(X_{\acute{e}t},\ \Lambda)$ as a triangulated category with
duality and discuss localization sequences. In the next section applying the
general machinery of Balmer the constructible Witt-theory of a scheme $X$ with
$\Lambda$-coefficients will be defined and studied via this triangulated
category with duality. For the description of duality and localization we use
the recent work of Cisinski and Déglise in [CD16]: The advantage of technical
foundations in terms of étale motives is a greater generality of the results
that otherwise will need more restrictive hypotheses within the framework of
[AGV71], [DGIV77] and [ILO14]. For instance, as compared to the restriction of
finiteness on coefficient rings one can develop the constructible Witt theory
for ‘good enough’ coefficient rings as defined in [CD16, Def. 6.3.6]\- this
includes $\mathbb{Z}$ and any noetherian ring of positive characteristics.
Another generality offered by the work of Cisinski and Déglise is extension of
the theory to a more general base scheme as recalled in Theorem 3.2 in terms
of the subcategory $DM_{h,lc}(X,R)$ of locally constructible $h$-motives of
the derived category $DM_{h}(X,R)$ of $h$-motives.

###  3.1. Constructible sheaves of $\Lambda$-modules

We begin by recalling the category to be used for defining constructible Witt
theory- the bounded derived category $D^{b}_{ctf}(X_{\acute{e}t},\ \Lambda)$
of étale sheaves having finite Tor-dimension and constructible cohomology
sheaves. For easy reference we recall the small étale site $X_{\acute{e}t}$ of
a scheme $X$ and the classically studied bounded derived category
$D^{b}_{ctf}(X_{\acute{e}t},\ \Lambda)$. As a category it is the category
$\acute{E}t/X$ of étale $X$-schemes, and covering in the site $X_{\acute{e}t}$
are surjective families $\\{X_{i}^{\prime}\xrightarrow{\phi_{i}}X^{\prime}\\}$
of morphisms in $\acute{E}t/X$. Thus,

$\begin{cases}\>\>cat(X_{\acute{e}t})=\acute{E}t/X\\\ \\\
\>\>cov(X_{\acute{e}t})=collection\>of\>surjective\>families\>of\>morpisms\>in\>\acute{E}t/X.\end{cases}$

Sheaves on $X_{\acute{e}t}$ are called _étale sheaves_ on $X$.

Let $X$ be a noetherian scheme and $\Lambda$ a noetherian ring. A sheaf
$\mathcal{F}$ of $\Lambda$-modules on $X_{\acute{e}t}$ is called
$constructible$ if there exists a decomposition $\bigcup_{i=1}^{n}X_{i}$ of
$X$ into finitely many locally closed subsets $X_{i}$ such that each
$\mathcal{F}|_{X_{i}}$ is locally constant and the stalks of $\mathcal{F}$ are
finitely generated $\Lambda$-modules. Along with constructibility the objects
in the category $D^{b}_{ctf}(X_{\acute{e}t},\ \Lambda)$ have additional
finiteness condition, namely that of having finite Tor-dimension. Let
$\mathcal{F}^{\bullet}$ be a bounded complex of sheaves of $\Lambda$-modules
on $X$. We say that $\mathcal{F}^{\bullet}$ has _finite Tor-dimension_ if
there exist an integer $n$ such that Tor${}_{i}(\mathcal{F}^{\bullet},M)=0$
for any $i>n$ and any constant sheaf of $\Lambda$-modules $M$ on $X$.

Let $X$ be a noetherian scheme and $\Lambda$ be a noetherian ring. Denote by
$D(X_{\acute{e}t},\Lambda)$ the derived category of sheaves of
$\Lambda$-modules over $X_{\acute{e}t}$. We get the full subcategories
$D^{\ast}(X_{\acute{e}t},\Lambda)$ by taking $\ast=+,-,b$.

######  Definition 3.1 (The category $D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$ of
sheaves having finite Tor-dimension and constructible cohomology sheaves).

The category $D^{b}_{ctf}(X_{\acute{e}t},\Lambda)\subset
D(X_{\acute{e}t},\Lambda)$ is the full subcategory consisting of bounded
complexes of sheaves of $\Lambda$-modules with finite Tor-dimension and having
constructible cohomology sheaves. The objects in the category
$D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$ are complexes that are quasi-isomorphic
in $D(X_{\acute{e}t},\Lambda)$ to bounded complexes whose components are
_flat_ and _constructible_ sheaves of $\Lambda$-modules [Fu11].

###  3.2. Duality on constructible sheaves

As mentioned in the beginning of this section we now use the six-functor
formalism for triangulated subcategories $DM_{h,lc}(X,\Lambda)$ of locally
constructible $h$-motives of the derived category $DM_{h}(X,\Lambda)$ of
$h$-motives from [CD16] to the case of classically studied bounded derived
categories $D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$ of étale sheaves having
finite Tor-dimension and constructible cohomology. For basic definitions and
related material about the derived category of étale motives the original
paper [CD16] by Cisinski and Déglise can be consulted.

In what follows recall that a scheme is said to be excellent if it has an open
affine cover given by excellent rings: Examples of such rings include the ring
$\mathbb{Z}$ of integers, fields, the ring $\mathbb{Z}_{p}$ of $p$-adic
integers, finitely generated rings of the form
$R[x_{1},\cdots,x_{r}]/<f_{1},\cdots,f_{n}>$ over an excellent ring $R$, and
localizations of excellent rings.

Compared to [AGV71] and [DGIV77] more general six-functor formalism for
$D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$ follows from the two results [CD16,
Theorem 6.3.11, Corollary 6.3.15] restated in the following.

######  Theorem 3.2.

Let $B$ be an excellent scheme of dimension $\leq 2$ and $\Lambda$ a
noetherian ring of positive characteristic prime to the residue
characteristics of $B$. Let $\phi:S\rightarrow B$ be a regular separated
finite type $B$-scheme and $f:X\rightarrow S$ a separated morphism of finite
type. The categories $D^{b}_{ctf}(X_{\acute{e}t},\Lambda)\subset
D^{b}(X_{\acute{e}t},\Lambda)$ are closed under the six-functors of
Grothendieck in $D^{b}(X_{\acute{e}t},\Lambda)$. For a $\otimes$-invertible
object $T\in D^{b}_{ctf}(S_{\acute{e}t},\Lambda)$ the functor

$D_{X}(T)=R\mathcal{H}om(-,f^{!}(T)):D^{b}_{ctf}(X_{\acute{e}t},\Lambda)^{op}\rightarrow
D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$

is a duality functor on $D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$. In particular,
we have triangulated categories with duality
$(D^{b}_{ctf}(B_{\acute{e}t},\Lambda),R\mathcal{H}om(-,\Lambda))$ and
$(D^{b}_{ctf}(S_{\acute{e}t},\Lambda),R\mathcal{H}om(-,\Lambda))$.

###### Proof.

Under the assumptions in the theorem the triangulated subcategory
$DM_{h,lc}(X,\Lambda)\subset DM_{h}(X,\Lambda)$ of locally constructible
$h$-motives is closed under the six-functors of Grothendieck in
$DM_{h}(X,\Lambda)$ and for a locally constructible $\otimes$-invertible
object $U\in DM_{h}(S,\Lambda)$, the functor

$D_{X}(U)=R\mathcal{H}om(-,f^{!}(U)):DM_{h,lc}(X,\Lambda)^{op}\rightarrow
DM_{h,lc}(X,\Lambda)$

is a duality functor on $DM_{h,lc}(X,\Lambda)$. Since the equivalence of
categories

$D^{b}_{ctf}(X_{\acute{e}t},\Lambda)\xrightarrow{\simeq}DM_{h,lc}(X,\Lambda)$

in [CD16, Theorem 6.3.11] is compatible with the six functor formalism in the
two settings, the proof of the theorem is complete. ∎

Notation. With the notations of the above theorem the triangulated category
$D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$ with the duality $D_{X}(T)$ will be
denoted by $(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),D_{X}(T))$ or
$(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),f^{!}T)$.

######  Proposition 3.3 (Pullbacks for étale morphisms and pushforwards for
proper morphisms).

Under the same assumptions on $B$, $\Lambda$ and $S$ as in the theorem above,
let $f:X\rightarrow S$ and $g:Y\rightarrow S$ be separated $S$-schemes of
finite type, and $T$ be a $\otimes$-invertible object in
$D^{b}_{ctf}(S_{\acute{e}t},\Lambda)$. Let $h:X\rightarrow Y$ be a morphism of
$S$-schemes. If $h$ is étale, then the _pullback_ morphism
$h^{*}:D^{b}_{ctf}(Y_{\acute{e}t},\Lambda)\rightarrow
D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$ is a morphism of triangulated categories
with duality

$h^{*}:(D^{b}_{ctf}(Y_{\acute{e}t},\Lambda),g^{!}T)\rightarrow(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),f^{!}T).$

If the morphism $h:X\rightarrow Y$ is proper, then the pushforward morphism
$h_{*}:D^{b}_{ctf}(X_{\acute{e}t},\Lambda)\rightarrow
D^{b}_{ctf}(Y_{\acute{e}t},\Lambda)$ is a morphism of triangulated categories
with duality

$h_{*}:(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),f^{!}T)\rightarrow(D^{b}_{ctf}(Y_{\acute{e}t},\Lambda),g^{!}T).$

###### Proof.

We explain the details for the pushforwards $h_{*}$ for proper morphisms of
schemes and leave the part on $h^{*}$ as an exercise. We show that in the
following diagram of triangulated categories

$\textstyle{D_{ctf}^{b}(X_{\acute{e}t},\Lambda)^{op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\ \ D_{X}(T)\ \ \
}$$\scriptstyle{h_{*}}$$\textstyle{D_{ctf}^{b}(X_{\acute{e}t},\Lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{*}}$$\textstyle{D_{ctf}^{b}(Y_{\acute{e}t},\Lambda)^{op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\ \ D_{Y}(T)\ \ \ }$$\textstyle{D_{ctf}^{b}(Y_{\acute{e}t},\Lambda)}$

the two compositions are naturally isomorphic by using the relations among the
functors in this diagram. We have $f=g\circ h$. Given an element $F\in
D_{ctf}^{b}(X_{\acute{e}t},\Lambda)$ we have the following natural
identifications

$\textstyle{h_{*}(D_{X}(T)(F))=h_{*}(R\mathcal{H}om(F,f^{!}(T)))\simeq
h_{*}(R\mathcal{H}om(F,h^{!}g^{!}(T)))}$$\textstyle{\simeq
R\mathcal{H}om_{X}(h_{!}F,f^{!}T)\simeq
R\mathcal{H}om_{X}(h_{*}F,f^{!}T)=D_{X}(T)(h_{*}(F)))}$

where in second to the last identification we use the fact that $h_{!}=h_{*}$
for the proper morphism $h$.∎

###  3.3. Localization

We know that for a closed immersion $i:Z\hookrightarrow X$ of schemes the
induced exact functor $i_{*}:Shv(Z_{\acute{e}t},\Lambda)\rightarrow
Shv(X_{\acute{e}t},\Lambda)$ is fully faithful and its essential image is the
subcategory of those sheaves whose support is contained in $Z$. The natural
transformation $Ri_{*}=i_{*}:D(Z_{\acute{e}t},\Lambda)\rightarrow
D(X_{\acute{e}t},\Lambda)$ of the derived categories induces an equivalence of
$D(Z_{\acute{e}t},\Lambda)$ with $D_{Z}(X_{\acute{e}t},\Lambda)$\- the
strictly full saturated subcategory of $D(X_{\acute{e}t},\Lambda)$ consisting
of complexes whose cohomology sheaves are supported in $Z$. Let
$j:U=X-Z\hookrightarrow X$ be the complementary open immersion and
$j^{*}:Shv(X_{\acute{e}t},\Lambda)\rightarrow Shv(U_{\acute{e}t},\Lambda)$ the
induced map of sheaves. Then we have an exact sequence of triangulated
categories

(3.1)
$D(Z_{\acute{e}t},\Lambda)\xrightarrow{i_{*}}D(X_{\acute{e}t},\Lambda)\xrightarrow{j^{*}}D(U_{\acute{e}t},\Lambda)$

in which the $D(Z_{\acute{e}t},\Lambda)$ can be identified with
$D_{Z}(X_{\acute{e}t},\Lambda)$. Let
$D^{b}_{ctf,Z}(X_{\acute{e}t},\Lambda)\subset
D_{ctf}^{b}(X_{\acute{e}t},\Lambda)$ be the strictly full saturated
triangulated subcategory consisting of bounded complexes having constructible
and finite Tor-dimension whose cohomology sheaves are supported in $Z$. Under
the assumptions in theorem 3.2 for the scheme $X$ and the ring $\Lambda$ the
six-functor formalism for étale motives in [CD16, Corollary 6.3.15] implies
that the natural equivalence of the categories $D(Z_{\acute{e}t},\Lambda)$
with $D_{Z}(X_{\acute{e}t},\Lambda)$ induces an equivalence of
$D_{ctf}^{b}(Z_{\acute{e}t},\Lambda)$ with
$D^{b}_{ctf,Z}(X_{\acute{e}t},\Lambda)$.

######  Lemma 3.4.

With the same assumptions as in Theorem 3.2 for the schemes $B,\ S,\ X$, the
ring $\Lambda$ and the map $f:X\rightarrow S$, let $i:Z\hookrightarrow X$ be a
closed immersion and $j:U=X-Z\hookrightarrow X$ the complementary open
immersion. The induced functors
$i_{*}:D_{ctf}^{b}(Z_{\acute{e}t},\Lambda)\rightarrow
D_{ctf}^{b}(X_{\acute{e}t},\Lambda)$ and
$j^{*}:D_{ctf}^{b}(X_{\acute{e}t},\Lambda)\rightarrow
D_{ctf}^{b}(U_{\acute{e}t},\Lambda)$ are natural transformations of
triangulated categories with duality

$\textstyle{(a.)\ \ \ \ \ \ \ \ \ \ \ \ \
i_{*}:(D_{ctf}^{b}(Z_{\acute{e}t},\Lambda),D_{Z}(T))\rightarrow(D_{ctf}^{b}(X_{\acute{e}t},\Lambda),D_{Z}(X))}$$\textstyle{(b.)\
\ \ \ \ \ \ \ \ \ \ \ \
j^{*}:(D_{ctf}^{b}(X_{\acute{e}t},\Lambda)),D_{X}(T)\rightarrow(D_{ctf}^{b}(U_{\acute{e}t},\Lambda),D_{U}(T))}$

for every $\otimes$-invertible object $T\in
D^{b}_{ctf}(S_{\acute{e}t},\Lambda)$.

###### Proof.

This is a special case of Proposition 3.3. ∎

Under the equivalence of $D_{ctf}^{b}(Z_{\acute{e}t},\Lambda)$ with
$D^{b}_{ctf,Z}(X_{\acute{e}t},\Lambda)$ we will consider
$D^{b}_{ctf,Z}(X_{\acute{e}t},\Lambda)$ as a triangulated category with
duality and denote the duality by $D_{Z}(T)$ for a $\otimes$-invertible object
$T\in D^{b}_{ctf}(S_{\acute{e}t},\Lambda)$. The localization property of the
categories $D^{b}_{ctf}((-)_{\acute{e}t},\Lambda)$ is the following exact
sequence of triangulated categories with the notations of this subsection.

######  Theorem 3.5.

Let $B$ be an excellent scheme of dimension $\leq 2$ and $\Lambda$ a
noetherian ring of positive characteristic prime to the residue
characteristics of $B$. Let $\phi:S\rightarrow B$ be a regular separated
finite type $B$-scheme and $f:X\rightarrow S$ a separated morphism of finite
type. The exact sequence 3.1 of triangulated categories induces a localization
sequence of triangulated categories with duality

(3.2)
$(D^{b}_{ctf,Z}(X_{\acute{e}t},\Lambda),D_{Z}(T))\xrightarrow{i_{*}}(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),D_{X}(T))\xrightarrow{j^{*}}D^{b}_{ctf}(U_{\acute{e}t},\Lambda),D_{U}(T))$

for a $\otimes$-invertible object $T\in D^{b}_{ctf}(S_{\acute{e}t},\Lambda)$.

###### Proof.

Using Grothendieck’s six-functor formalism for
$DM_{h,lc}((-)_{\acute{e}t},\Lambda)$ in [CD16, Corollary 6.3.15] and the
identification of $D^{b}_{ctf}((-)_{\acute{e}t},\Lambda)$ with
$DM_{h,lc}((-)_{\acute{e}t},\Lambda)$ in [CD16, Theorem 6.3.11] we observe
that the morphisms of triangulated categories in 3.2 define a localizing
sequence. In view of the lemma 3.4 the proof of the theorem is complete. ∎

##  4\. Witt theory of constructible sheaves

In this section we now define the constructible Witt groups of étale sheaves
of $\Lambda$-modules on a noetherian scheme $X$ as Balmer’s Witt groups of the
triangulated category with duality $D^{b}_{ctf}(X_{\acute{e}t},\Lambda)$
discussed in the section 3. We describe some of their basic properties. We
will work under the assumptions as in Theorem 3.2 for the schemes $B$, $S$,
$X$, the ring $\Lambda$ and the map $f:X\rightarrow S$.

######  Definition 4.1 (Constructible Witt groups).

The category $(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),D_{X}(T))$ is a
triangulated category with duality for a $\otimes$-invertible object $T\in
D^{b}_{ctf}(S_{\acute{e}t},\Lambda)$. The constructible Witt groups of sheaves
of $\Lambda$-modules on $X$ are Witt groups of this triangulated category with
duality. We will denote these by $W^{n}_{c}(X_{\acute{e}t},\Lambda,T)$. In
notations we may suppress the mention of the $\otimes$-invertible object $T$
and use the notation $W^{n}_{c}(X_{\acute{e}t},\Lambda)$.

###  4.1. Functoriality for constructible Witt groups

We can directly use Proposition 3.3 to get the functoriality of constructible
Witt groups. Consider the same assumptions on $B$, $X$, $\Lambda$ and $S$ as
in Proposition 3.3 and $T$ a $\otimes$-invertible object in
$D^{b}_{ctf}(S_{\acute{e}t},\Lambda)$.

For an étale morphism $h:X\rightarrow Y$ of $S$-schemes, we get a morphism of
triangulated categories with duality

$h^{*}:(D^{b}_{ctf}(Y_{\acute{e}t},\Lambda),g^{!}T)\rightarrow(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),f^{!}T).$

Then $h^{\ast}$ induces a group homomorphism of constructible Witt groups

$W^{n}(h^{\ast}):W^{n}_{c}(Y_{\acute{e}t},\Lambda)\rightarrow
W^{n}_{c}(X_{\acute{e}t},\Lambda)$

for all $n\in\mathbb{Z}$.

Also, when $h:X\rightarrow Y$ is a proper morphism of $S$-schemes, we get a
morphism of triangulated categories with duality

$h_{\ast}:(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),g^{!}T)\rightarrow(D^{b}_{ctf}(Y_{\acute{e}t},\Lambda),f^{!}T).$

Then $h_{\ast}$ induces a group homomorphism of constructible Witt groups

$W^{n}(h_{\ast}):W^{n}_{c}(X_{\acute{e}t},\Lambda)\rightarrow
W^{n}_{c}(Y_{\acute{e}t},\Lambda)$

for all $n\in\mathbb{Z}$.

###  4.2. Localization sequence for constructible Witt groups

Now we describe the localization sequence for the constructible Witt theory.
Recall that a _localization of triangulated categories_

(4.1) $J\xrightarrow{j}K\xrightarrow{q}L$

is an exact sequence of triangulated categories, that is, $L$ is a
localization of $K$ with respect to a saturated class of morphisms $S$, and
the triangulated category $J$ is the full subcategory of $K$ consisting of
objects $X\in K$ for which $q(X)=0$ in $L$.

If $(K,\\#,\omega)$ is a triangulated category with duality and the class of
morphisms $S$ is _compatible_ with duality on $K$ i.e. $\\#(S)=S$, then $J$
endowed with restriction of $\\#$ and $S^{-1}K$ endowed with the localization
of $\\#$ are triangulated categories with duality and the exact natural
transformations $j$ and $q$ are morphisms of triangulated categories with
duality. Such a localization sequence of triangulated categories with duality
gives a 12-term exact sequence of corresponding Witt groups.

The localization sequence for the constructible Witt theory is a consequence
of the localization sequence of triangulated categories with duality in
theorem 3.2. Given a closed immersion $i:Z\hookrightarrow X$ with the
complementry open immersion $j:U=X-Z\hookrightarrow X$, theorem 3.5 gives the
localization of triangulated categories with duality

$(D^{b}_{ctf}(Z_{\acute{e}t},\Lambda),D_{Z}(T))\xrightarrow{i_{*}}(D^{b}_{ctf}(X_{\acute{e}t},\Lambda),D_{X}(T))\xrightarrow{j^{*}}D^{b}_{ctf}(U_{\acute{e}t},\Lambda),D_{U}(T)).$

The corresponding 12-term exact sequence of constructible Witt groups can be
written as

$\cdots\rightarrow
W_{c}^{n-1}(U_{\acute{e}t},\Lambda)\xrightarrow{\partial^{n-1}}W^{n}_{c}(Z_{\acute{e}t},\Lambda)\xrightarrow{W^{n}(i_{\ast})}W^{n}_{c}(X_{\acute{e}t},\Lambda)$
$\xrightarrow{W^{n}(j^{\ast})}W^{n}_{c}(U_{\acute{e}t},\Lambda)\xrightarrow{\partial^{n}}W^{n+1}_{c}(Z_{\acute{e}t},\Lambda)\rightarrow\cdots$

using the 4-periodicity of triangular Witt groups.

###  4.3. Homotopy Invariance

With the techniques used in this paper we are not in position to prove
homotopy invariance of contructible Witt theory simply because duality is not
compatible with the six-functor formalism for
$D^{b}_{ctf}((-)_{\acute{e}t},\Lambda)$ in the same generality. To be
specific, under the running assumptions in this section, for a vector bundle
$p:V\rightarrow X$, the pullback functor
$p^{*}:D^{b}_{ctf}(X_{et},\Lambda)\rightarrow D^{b}_{ctf}(V_{et},\Lambda)$ is
fully faithful since the unit of adjunction $1\rightarrow p_{*}p^{*}$ is an
isomorphism but $p_{*}$ is not compatible with dualities. It is instinctive to
expect homotopy invariance for this theory, but then one should develop the
theory in a more flexible setting for six-functor formalism.

##  5\. Constructible Witt theory of $\mathrm{Spec\hskip 1.42262pt}\mathbb{R}$

In this section we will identify the constructible Witt theory of étale
sheaves of $\Lambda$-modules on $\mathrm{Spec\hskip 1.42262pt}\mathbb{R}$, for
a ring $\Lambda$ of finite characteristic not equal to 2, as a
$\mathbb{Z}/2\mathbb{Z}$-equivariant Witt theory of $\Lambda$. Also for a real
projective variety $f:X\rightarrow\mathrm{Spec\hskip 1.42262pt}\mathbb{R}$ we
construct natural homomorphisms

$W^{i}(f_{*}):W^{i}_{c}(X,\Lambda)\rightarrow
W^{i}_{c}(\mathbb{R},\Lambda)=W^{i}_{lf}(\Lambda[\mathbb{Z}/2\mathbb{Z}])$

which define an algebraic version of signature for $X$.

Let $X=\mathrm{Spec\hskip 1.42262pt}k$ be the affine scheme defined by a field
$k$. Let $k^{sep}$ be a separable closure of $k$ and let $Gal(k^{sep}/k)$ be
the Galois group of $k^{sep}/k$ equipped with the canonical structure of a
profinite group. For each $k$-scheme $X^{\prime}$ we denote by
$X^{\prime}(k^{sep})$ the set of $k^{sep}$-valued points on $X^{\prime}/k$,
i.e. the set of $k$-morphisms $\mathrm{Spec\hskip 1.42262pt}k^{sep}\rightarrow
X^{\prime}$. A $k^{sep}$-valued point of $X^{\prime}$ corresponds uniquely to
a point $x^{\prime}\in X^{\prime}$ together with a $k$-homomorphism
$k(x^{\prime})\rightarrow k^{sep}$. The following equivalence of the category
of étale sheaves of $\Lambda$-modules on $\mathrm{Spec\hskip 1.42262pt}k$ with
the category of $\Lambda$-modules with continuous $Gal(k^{sep}/k)$-action for
the profinite topology on the Galois group is well-known.

######  Theorem 5.1.

For the scheme $\mathrm{Spec\hskip 1.42262pt}k$ the functor defined by taking
stalks at the geometric point of $\mathrm{Spec\hskip 1.42262pt}k$ given by a
choice $k\subset k^{sep}$ of the separable closure

$\mathcal{F}\mapsto\varinjlim\ _{k^{\prime}}\ \mathcal{F}(\mathrm{Spec\hskip
1.42262pt}k^{\prime}),\ \ \ \ k\subset k^{\prime}\subset k^{sep}\ and\
k^{\prime}\ a\ finite\ extension\ of\ k$

is an equivalence between the category of sheaves of $\Lambda$-modules on
$(\mathrm{Spec\hskip 1.42262pt}k)_{\acute{e}t}$ and the category of continuous
$\Lambda[G]$-modules for a group ring $\Lambda[G]$. Thus, in the case of the
field $\mathbb{R}$ of real numbers we have an equivalence

$\phi:Shv((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)\rightarrow
Mod(\Lambda[\mathbb{Z}/2\mathbb{Z}])$

of the category of étale sheaves of $\Lambda$-modules on $\mathrm{Spec\hskip
1.42262pt}\mathbb{R}$ with the category of all
$\mathbb{Z}/2\mathbb{Z}$-equivariant $\Lambda$-modules.

###### Proof.

See [Tam12, Corollary 2.2] for details. ∎

###  5.1. Flat and constructible sheaves on $(\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t}$

Let $\mathcal{F}$ be a flat and constructible sheaf of $\Lambda$-modules on
$(\mathrm{Spec\hskip 1.42262pt}\mathbb{R})_{\acute{e}t}$. As $\mathcal{F}$ is
a flat sheaf, the stalk of $\mathcal{F}$ at the single geomtric point in
$\mathrm{Spec\hskip 1.42262pt}\mathbb{R}$ given by $\mathrm{Spec\hskip
1.42262pt}\mathbb{C}$ is a flat $\Lambda$-module. Also $\mathcal{F}$ being a
constructible sheaf of $\Lambda$-modules, it is locally constant and its stalk
$\mathcal{F}_{x}$ is finitely generated $\Lambda$-module. Thus, for a flat and
constructible sheaf $\mathcal{F}$ on $(\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t}$ the stalk $\mathcal{F}_{x}$ is a flat and
finitely generated $\Lambda$-module. In this paper the coefficient rings are
assumed to be noetherian and therefore for a flat and constructible sheaf
$\mathcal{F}$ the stalk $\mathcal{F}_{x}$ being flat and finitely generated is
a finitely generated projective $\Lambda$-module with a $G$ action.

Under the assumption that the characteristic of $\Lambda$ is different from 2
it so happens that the category of finitely generated projective
$\Lambda$-modules with $\mathbb{Z}/2\mathbb{Z}$-action is equivalent to the
category of finitely generated projective
$\Lambda[\mathbb{Z}/2\mathbb{Z}]$-modules. For a lack of reference known to us
we will recall this useful fact in terms of equivariant vector bundles on
schemes.

######  Lemma 5.2.

For an affine $G$-scheme $X$, under the assumption that $|G|$ is invertible in
$\Gamma(X,\ \mathcal{O}_{X})$, every short exact sequence of equivariant
vector bundles splits equivariantly.

###### Proof.

Given a short exact sequence
$0\rightarrow\mathcal{E}\xrightarrow{i}\mathcal{L}\xrightarrow{\phi}\mathcal{M}\rightarrow
0$ of $G$-equivariant vector bundles on $X$, choose a (non-equivariant)
splitting $s:\mathcal{M}\rightarrow\mathcal{L}$ and observe that the morphism

$\bar{s}:\mathcal{M}\rightarrow\mathcal{L},\ \ \
a\mapsto\frac{1}{|G|}\sum_{g\in G}g\cdot s(g^{-1}\cdot a),\ \
a\in\mathcal{M}(U),\ U\subset X\ \mathrm{open}$

of vector bundles is a $G$-equivariant splitting of $\phi$. ∎

For an equivariant affine $G$-scheme $(\mathrm{Spec\hskip 1.42262pt}R,\tau)$,
let $\rho_{{}_{R_{\tau}}}$ (or simply $\rho_{{}_{R}}$) be the skew group-ring
$R_{\tau}G$ viewed as the twisted regular representation of $G$ over $R$. The
$G$-action on $\rho_{{}_{R_{\tau}}}$ is given by

$g^{\prime}\cdot(\Sigma_{g\in G}x_{g}\cdot e_{g})=\Sigma_{g\in
G}\tau_{g^{\prime}}(x_{g})\cdot e_{g^{\prime}g},$

where $\\{e_{g}:g\in G\\}$ is the standard basis of $\rho_{{}_{R_{\tau}}}$

######  Lemma 5.3.

For an affine $G$-scheme $\mathrm{Spec\hskip 1.42262pt}R$, under the
assumption that $|G|$ is invertible in $R$, every $G$-equivariant finitely
generated projective $R$-module is a summand in a direct sum of regular
representations.

###### Proof.

Let $\tau:G\rightarrow\mathrm{Aut}_{Sch/\mathrm{Spec\hskip
1.42262pt}\mathbb{Z}}(\mathrm{Spec\hskip 1.42262pt}R)$ denote the $G$-action
on $\mathrm{Spec\hskip 1.42262pt}R$. Given an equivariant finitely generated
projective $R$-module $P$ choose an epimorphism $\alpha:R^{n}\rightarrow P$.
The free rank 1 module $R$ is a $G$-equivariant summand in
$\rho_{{}_{R_{\tau}}}$ via the inclusion

$R\longrightarrow\rho_{{}_{R_{\tau}}},\ \ \ x\mapsto\sum_{g\in G}\tau(x)e_{g}$

and hence $R^{n}\hookrightarrow
R^{n}\otimes_{R}\rho_{{}_{R_{\tau}}}=\rho_{{}_{R_{\tau}}}^{n}$ is a
$G$-equivariant direct summand. The induced surjective map

$\widetilde{\alpha}:\rho_{{}_{R_{\tau}}}^{n}\rightarrow P\ \ \ \ \ (x,\
\sum_{g\in G}x_{g}\cdot e_{g})\mapsto\sum_{g\in G}x_{g}\cdot\alpha(x)$

is $G$-equivariant and we have an exact sequence of $G$-equivariant finitely
generated projective $R$-modules $\mathrm{ker}\
\widetilde{\alpha}\xrightarrow{\ \ \ }\rho_{{}_{R_{\tau}}}^{n}\xrightarrow{\ \
\widetilde{\alpha}\ \ }P.$ Choosing an equivariant splitting by Lemma 5.2 we
see that $P$ is a direct summand of $\rho_{{R_{\tau}}}^{n}$. ∎

######  Corollary 5.4.

For the trivial action of the group $\mathbb{Z}/2\mathbb{Z}$ on
$\mathrm{Spec\hskip 1.42262pt}\Lambda$ the category of
$\mathbb{Z}/2\mathbb{Z}$-equivariant finitely generated projective
$\Lambda$-modules is equivalent to the category of finitely generated
projective $\Lambda[\mathbb{Z}/2\mathbb{Z}]$-modules. Thus, for a
constructible and flat sheaf $\mathcal{F}$ of $\Lambda$-modules on
$(\mathrm{Spec\hskip 1.42262pt}\mathbb{R})_{\acute{e}t}$, the stalk
$\mathcal{F}_{x}$ is a finitely generated projective
$\Lambda[\mathbb{Z}/2\mathbb{Z}]$-module.

###### Proof.

Follows from the Lemma 5.3 and discussion in the beginning of this subsection
in view of the assumption that the characteristic of $\Lambda$ is different
from 2. ∎

###  5.2. Equivalence of $D^{b}_{ctf}((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)$ with $D^{b}(Proj(\Lambda[G]))$
and Constructible Witt groups of $\mathrm{Spec\hskip 1.42262pt}\mathbb{R}$

Using the description of stalks of flat and constructible sheaves on
$(\mathrm{Spec\hskip 1.42262pt}\mathbb{R})_{\acute{e}t}$ as finitely generated
projective $\Lambda[\mathbb{Z}/2\mathbb{Z}]$-modules in Corollary 5.4 we now
describe the constructible Witt theory of $\Lambda$-modules on
$\mathrm{Spec\hskip 1.42262pt}\mathbb{R}$ as a
$\mathbb{Z}/2\mathbb{Z}$-equivariant Witt theory of $\Lambda$. The equivalence

$\phi:Shv((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)\rightarrow
Mod(\Lambda[\mathbb{Z}/2\mathbb{Z}])$

in Theorem 5.1 is an exact equivalence of categories and induces an exact
equivalence on the associated categories of chain complexes

$\phi:Ch(Shv((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda))\rightarrow
Ch(Mod(\Lambda[\mathbb{Z}/2\mathbb{Z}])).$

Let

(5.1) $\Phi:D((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)\rightarrow
D(Mod(\Lambda[\mathbb{Z}/2\mathbb{Z}]))$

be the induced functor on the derived categories. The category
$D^{b}_{ctf}((\mathrm{Spec\hskip 1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)$
of bounded complexes sheaves of $\Lambda$-modules having Tor-finite dimension
and constructible cohomology sheaves can now be identified with the more
familiar bounded derived category of
$D^{b}(Proj(\Lambda[\mathbb{Z}/2\mathbb{Z}]))$ of finitely generated
projective $\Lambda[\mathbb{Z}/2\mathbb{Z}]$-modules giving us the the
identification of the constructible Witt theory of $\mathrm{Spec\hskip
1.42262pt}\mathbb{R}$ with a Witt theory of finitely generated projective
$\Lambda[\mathbb{Z}/2\mathbb{Z}]$-modules.

######  Theorem 5.5.

Let $\Lambda$ a ring of finite characteristic not equal to $2$. Then the
functor in $\mathrm{(\ref{Phi})}$ induces an equivalence of triangulated
categories with duality

$\Phi:D^{b}_{ctf}((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda),\mathcal{H}om(-,\Lambda))\rightarrow(D^{b}(Proj(\Lambda[\mathbb{Z}/2\mathbb{Z}])),Hom_{\Lambda[\mathbb{Z}/2\mathbb{Z}]}(-,\Lambda))$

and induces an isomorphism of constructible Witt theory of $\mathrm{Spec\hskip
1.42262pt}\mathbb{R}$ with an equivariant Witt theory of finitely generated
projective $\Lambda$-modules for the action of the group
$\mathbb{Z}/2\mathbb{Z}$ (the absolute Galois group of $\mathbb{R}$).

###### Proof.

The category $D^{b}_{ctf}((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)\subset D((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)$ is the full triangulated
subcategory consisting of complexes that are quasi-isomorphic in
$D((\mathrm{Spec\hskip 1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)$ to bounded
complexes whose components are flat and constructible sheaves of
$\Lambda$-modules. By Corollary 5.4 we have the restriction

$\Phi:D^{b}_{ctf}((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)\rightarrow
D^{b}(Proj(\Lambda[\mathbb{Z}/2\mathbb{Z}]))$

of $\Phi$ is functor into the bounded derived category
$D^{b}(Proj(\Lambda[\mathbb{Z}/2\mathbb{Z}]))$ of finitely generated
projective $\Lambda[\mathbb{Z}/2\mathbb{Z}]$-modules. Recall that a quasi-
inverse

$\psi:Mod(\Lambda[\mathbb{Z}/2\mathbb{Z}])\rightarrow Shv((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)$

of the functor $\phi:Shv((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)\rightarrow
Mod(\Lambda[\mathbb{Z}/2\mathbb{Z}])$ is defined by associating to a
$\Lambda[\mathbb{Z}/2\mathbb{Z}]$-module $M$ it’s the
$\mathbb{Z}/2\mathbb{Z}$-fixed points $M^{\mathbb{Z}/2\mathbb{Z}}$ with the
base scheme $\mathrm{Spec\hskip 1.42262pt}\mathbb{R}$ and the module $M$ to
the scheme $\mathrm{Spec\hskip 1.42262pt}\mathbb{C}$. It is an exact functor
and induces a quasi-inverse

$\Psi:D^{b}(Proj(\Lambda,\mathbb{Z}/2\mathbb{Z}))\rightarrow
D^{b}_{ctf}((\mathrm{Spec\hskip 1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)$

of $\Phi$. The diagram of triangulated categories

$\textstyle{D_{ctf}^{b}((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)^{op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\ \ \mathcal{H}om(-,\Lambda)\ \ \
}$$\scriptstyle{\Phi}$$\textstyle{D_{ctf}^{b}((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{D^{b}(Proj(\Lambda[\mathbb{Z}/2\mathbb{Z}]))^{op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Hom_{\Lambda[\mathbb{Z}/2\mathbb{Z}]}(-,\Lambda)\
}$$\textstyle{D^{b}(Proj(\Lambda[\mathbb{Z}/2\mathbb{Z}]))}$

commutes since the sheaf $\Lambda$ defining the duality
$\mathcal{H}om(-,\Lambda)$ is the constant sheaf. Therefore the equivalence
$\Phi$ is an equivalence of triangulated categories with duality. This
completes the proof. ∎

###  5.3. Signatures for proper real algebraic varieties

For a Noetherian ring $\Lambda$ of positive characteristic not equal to 2 the
Theorem 5.5 provides an equivalence

$D^{b}_{ctf}((\mathrm{Spec\hskip
1.42262pt}\mathbb{R})_{\acute{e}t},\Lambda)\xrightarrow{\simeq}D^{b}(Proj(\Lambda[\mathbb{Z}/2\mathbb{Z}]))$

of the derived category of bounded complexes sheaves of $\Lambda$-modules
having Tor-finite dimension and constructible cohomology sheaves with the
bounded derived category of finitely generated projective
$\Lambda[\mathbb{Z}/2\mathbb{Z}]$-modules. This equivalence allows us to
identify the constructible Witt theory of $\mathrm{Spec\hskip
1.42262pt}\mathbb{R}$ with a $\mathbb{Z}/2\mathbb{Z}$-equivariant Witt theory
of finitely generated projective $\Lambda$-modules, and the algebraic analog
of signature defined in [Woo08] is given by the following.

######  Theorem 5.6.

Let $X$ be a projective real algebraic variety with the structure morphism
$f:X\rightarrow\mathrm{Spec\hskip 1.42262pt}\mathbb{R}$ and $\Lambda$ a ring
of finite characteristic not equal to $2$. Then we have the induced proper
pushforward for constructible Witt theory

$W^{i}(f_{*}):W^{i}_{c}(X,\Lambda)\rightarrow
W^{i}_{c}(\mathbb{R},\Lambda)=W^{i}_{lf}(\Lambda[\mathbb{Z}/2\mathbb{Z}])$

where $W^{i}_{lf}(\Lambda,\mathbb{Z}/2\mathbb{Z})$ denotes a
$\mathbb{Z}/2\mathbb{Z}$-equivariant Witt theory of finitely generated
projective $\Lambda$-modules.

###### Proof.

The proof follows from the Theorem 5.5 and the Proposition 3.3 for the duality
defined by the $\otimes$-invertible object $\Lambda$. ∎

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# Quantum correlation cost of the weak measurement

Jun Zhang  Shao-xiong Wu  Chang-shui Yu<EMAIL_ADDRESS>School of
Physics and Optoelectronic Technology, Dalian University of Technology, Dalian
116024, China

###### Abstract

Quantum correlation cost (QCC) characterizing how much quantum correlation is
used in a weak-measurement process is presented based on the trace norm. It is
shown that the QCC is related to the trace-norm-based quantum discord (TQD) by
only a factor that is determined by the strength of the weak measurement, so
it only catches partial quantumness of a quantum system compared with the TQD.
We also find that the residual quantumness can be ‘extracted’ not only by the
further von Neumann measurement, but also by a sequence of infinitesimal weak
measurements. As an example, we demonstrate our outcomes by the Bell-diagonal
state.

###### keywords:

weak measurement , trace norm , quantum correlation cost

##  1 Introduction

As one of the important quantum correlations, quantum entanglement has been
identified as an important physical resource in quantum information processing
tasks (QIPTs) [1]. But in some QIPTs such as the robust quantum algorithm
against the decoherence [2], the deterministic quantum computation with one
quantum bit (DQC1) [3] etc, is there not any quantum entanglement, but quantum
discord that was introduced in Ref. [4] and Ref. [5], respectively, has been
shown to be able to grasp more quantumness than entanglement. In recent years,
the research on quantum discord has been made great progress in various fields
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

Quantum discord was originally defined by the difference between the total
correlation and the classical correlation that is obtained by the optimal
local measurements, since classical information is locally accessible, and can
be obtained without perturbing the state of the system [4]. Later it was
generalized to various cases by using different distance measures between the
state taken into account and the post-measured state [24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38]. In usual, these mentioned measurements
are referred as to the von Neumann measurement which can cause collapse of the
wave function and hence has strong influences on (destroys) the initial state.
In 1988, Aharonov et al. introduced the weak measurement [39] which was
applied in many areas [40, 41, 42, 43]. It was later generalized by Oreshkov
and Brun [44] to the case with only preselection in terms of measurement
operator formalism and one of the important characteristics is that the
strength of the measurement process can be controlled to be very weak, so the
quantum state can be influenced weakly. Considering such weak measurements in
the quantification of quantum correlation, Singh and Pati proposed the super
quantum discord [45]. It has been shown that super discord is not less than
the quantum discord. Recently, the super quantum discord has attracted
increasing interests [46, 47, 48, 49, 50]. In particular, it is surprising
that the weak measurement can extract extra quantumness and the lost
quantumness could even be resurrected by weak measurement [50].

In this paper, we find very different phenomena from those in Refs. [45, 50].
Motivated by the super quantum discord, we present the quantum correlation
cost (QCC) by employing the trace norm as the distance measure between the
state of interests and post-weak-measured state. It is interesting that the
difference between the QCC and the quantum discord based on trace norm (TQD)
is in a factor that is determined by the strength of the employed weak
measurement. Instead of the extra quantumness [45], this factorization
relation shows that the QCC quantifies how much quantumness of a quantum
system is ‘extracted’ (used) by the employed weak measurements. In particular,
we also find that the residual quantumness which the weak measurement fails to
extract will be extracted further, if the post-weak-measured state is
succeeded by the von Neumann measurement or by a sequence of infinitesimal
weak measurements. In addition, all these conclusions can also be generalized
to $\left(2\otimes d\right)-$ dimensional systems and the case of multiple-
outcome weak measurements. As a demonstration, we study the QCC of the Bell-
diagonal state, which shows the consistency with our conclusions.

##  2 Quantum correlation cost and Residual quantumness

Quantum correlation cost.- To begin with, we would like to first introduce the
original super quantum discord $Q_{w}(\rho_{AB})$ which is defined as

$Q_{w}(\rho_{AB})=\min_{\Pi_{i}^{A}}S_{w}(B|\\{P^{A}(x)\\})+S(\rho_{A})-S(\rho_{AB}),$
(1)

where $S(\cdot)$ represents the von Neumann entropy and

$S_{w}(B|\\{P^{A}(x)\\})=p(+x)S(\rho_{B|P^{A}(+x)})+p(-x)S(\rho_{B|P^{A}(-x)}),$
(2)

with $\rho_{B|P^{A}(\pm x)}$ the post-weak-measured state given by

$\rho_{B|P^{A}(\pm x)}=\frac{Tr_{A}[(P^{A}(\pm x)\otimes
I_{B})\rho_{AB}(P^{A}(\pm x)\otimes I_{B})]}{Tr_{AB}[(P^{A}(\pm x)\otimes
I_{B})\rho_{AB}(P^{A}(\pm x)\otimes I_{B})]},$ (3)

and $p(\pm x)$ the corresponding probability given by

$p(\pm x)=Tr_{AB}[(P^{A}(\pm x)\otimes I_{B})\rho_{AB}(P^{A}(\pm x)\otimes
I_{B})].$ (4)

It is noted that

$\left\\{\begin{array}[]{c}P(+x)=\alpha\pi_{1}+\beta\pi_{2}\\\
P(-x)=\beta\pi_{1}+\alpha\pi_{2}\end{array}\right.,$ (5)

where $P^{\dagger}(+x)P(+x)+P^{\dagger}(-x)P(-x)=\mathbf{I}$ denotes the two-
outcome weak measurement with $\pi_{i}$ the normal projectors,

$\alpha=\sqrt{\frac{1-\tanh x}{2}},\beta=\sqrt{\frac{1+\tanh x}{2}}.$ (6)

and $x$$\in$ $\mathbb{R}$ the strength of measurement process. It is obvious
that the weak measurement operators will be reduced to orthogonal projectors
with $x\rightarrow\infty$.

The characteristic of the super quantum discord (SD) is that the SD can catch
‘extra’ quantumness compared with the initial quantum discord. However, it is
opposite to the original intention of quantum discord which requires to remove
the classical correlation from the total correlation as much as possible by
choosing the optimal measurements. It is obvious the weak measurement is the
bad choice which cannot extract enough classical correlation. That is, the
super quantum discord should include the residual classical correlation, so it
seems to be greater than the usual quantum discord. However, the super quantum
discord inspires us to study quantum correlation in a different way, that is,
to be effectively related to the weak-measurement process. To do so, we define
the quantum correlation cost (QCC) based on the trace norm as some kind of
quantum correlation measure which is given as follows.

Definition 1. The QCC $D_{w}(\rho_{AB})$ for a bipartite quantum state
$\rho_{AB}$ is defined as

$D_{w}(\rho_{AB}):=\min_{\pi}\left\|\rho_{AB}-\Pi_{2}(\rho_{AB})\right\|_{1},$
(7)

where $\left\|X\right\|_{1}=Tr\sqrt{XX^{\dagger}}$ and $\Pi_{2}(\cdot)$
denotes the operator of two-outcome weak measurement on subsystem $A$ with

$\Pi_{2}(\rho_{AB})=P(+x)\rho_{AB}P^{\dagger}(+x)+P(-x)\rho_{AB}P^{\dagger}(-x).$
(8)

It is implied that the QCC is defined in terms of the two-outcome weak
measurement, which holds throughout this paper if no particular statements
given. In addition, one can easily find that $D_{w}(\rho_{AB})$ inherits the
advantage of the trace norm such as the invariance under local unitary
operations and the contractivity under the local non-unitary evolution on
subsystem $A$, thus it is a reliable measure of quantum correlation. In
particular, one should note that the TQD $D(\rho_{AB})$ for $\rho_{AB}$ can be
directly obtained by requiring $x\rightarrow\infty$, that is,

$D(\rho_{AB})=\lim_{x\rightarrow\infty}D_{w}(\rho_{AB}).$ (9)

With Eq. (7) and Eq. (9), we can fortunately find the deeper relation between
the QCC and the TQD which is given by the following rigorous way.

Theorem 1. For a bipartite quantum system of qubits $\rho_{AB}$

$D_{w}(\rho_{AB})=(1-2\alpha\beta)D(\rho_{AB}).$ (10)

with $\alpha$, $\beta$ defined by Eq. (6) and only determined by the strength
of the measurement process.

Proof. Let’s consider an arbitrary two-qubit state $\rho_{AB}$. Suppose that
the final state after the weak measurement is $\Pi_{2}(\rho_{AB})$. Substitute
Eq. (8) into Eq. (7), we have

$\displaystyle D_{w}(\rho_{AB})$ $\displaystyle=$
$\displaystyle\min_{\pi}\left\|\rho_{AB}-\Pi_{2}(\rho_{AB})\right\|_{1}$ (11)
$\displaystyle=$
$\displaystyle\min_{\pi}\|\rho_{AB}-\left(\pi_{1}\rho_{AB}\pi_{1}+\pi_{2}\rho_{AB}\pi_{2}\right)$
$\displaystyle-2\alpha\beta\left(\pi_{1}\rho_{AB}\pi_{2}+\pi_{2}\rho_{AB}\pi_{1}\right)\|_{1}$
$\displaystyle=$
$\displaystyle(1-2\alpha\beta)\min_{\pi}\left\|\pi_{1}\rho_{AB}\pi_{2}+\pi_{2}\rho_{AB}\pi_{1}\right\|_{1},$

Insert Eq. (9) into Eq. (11), it follows that

$D_{w}(\rho_{AB})=(1-2\alpha\beta)D(\rho_{AB}).$ (12)

which completes the proof. $\blacksquare$

Theorem 1 shows us a very simple factorization relation between the QCC and
the TQD, which provides the important root for the next stories.

Residual quantumness.- Based on the factorization relation given above, one
will obviously see that $D_{w}(\rho_{AB})\leq D(\rho_{AB})$ due to the
reduction factor $(1-2\alpha\beta)\leq 1$ with ‘$=$’ satisfied for
$x\rightarrow\infty$. Since the weak measurement influences the system more
weakly than the normal projective measurement, the distance from the state of
interests to the post-weak-measured state is naturally less than that from the
state to the post-projective-measured state. Therefore, compared with the TQD
$D(\rho_{AB}),$ it is shown that the QCC $D_{w}(\rho_{AB})$ can only grasp the
partial quantumness instead of the extra quantumness. Thus the residual
quantumness can be written as

$\Delta=D(\rho_{AB})-D_{w}(\rho_{AB})=2\alpha\beta D(\rho_{AB}).$ (13)

Below we will show that the residual quantumness can be further extracted if
we continue performing a projective measurement on the post-weak-measured
state.

Theorem 2. Let $\tilde{\rho}_{AB}$ denote the final state of $\rho_{AB}$ after
the optimal weak measurement such that
$D_{w}(\rho_{AB})=\left\|\rho_{AB}-\tilde{\rho}_{AB}\right\|_{1}$, then we
have

$D(\tilde{\rho}_{AB})=D(\rho_{AB})-D_{w}(\rho_{AB}).$ (14)

Proof. At first, one should keep in mind that
$\tilde{\rho}_{AB}=\Pi_{2}(\rho_{AB})$ given by Eq. (8) can be further written
as that implied in Eq. (11) based on the Proof of Theorem 1. In addition, in
order to distinguish the projectors from those in the weak measurement, we
denote the projectors operated on subsystem $A$ in the normal projective
measurement by be $\pi_{1}^{\prime}$ and $\pi_{2}^{\prime}$. Therefore, in
terms of the definition of TQD, one can obtain $D(\tilde{\rho}_{AB})$ as

$D(\tilde{\rho}_{AB})=\min_{\pi^{\prime}}\left\|\tilde{\rho}_{AB}-\sum_{i=1}^{2}\pi_{i}^{\prime}\tilde{\rho}_{AB}\pi_{i}^{\prime}\right\|_{1}.$
(15)

Since for $x\rightarrow\infty$, the weak measurement operated on
$\tilde{\rho}_{AB}$ should become the normal projective measurement, one will
arrive at

$\displaystyle\lim_{x\rightarrow\infty}D(\tilde{\rho}_{AB})$ $\displaystyle=$
$\displaystyle\min_{\pi^{\prime}}\left\|\sum_{j=1}^{2}\pi_{j}\rho_{AB}\pi_{j}-\sum_{i,j=1}^{2}\pi_{i}^{\prime}\pi_{j}\rho_{AB}\pi_{j}\pi_{i}^{\prime}\right\|_{1}$
(16) $\displaystyle=$ $\displaystyle 0.$

where $\pi_{1}$ and $\pi_{2}$ are the projectors in the weak measurement. It
is obvious that the equality in Eq. (16) holds iff the set of the projectors
{$\pi_{i}^{\prime}\\}=\\{\pi_{j}\\}$. Thus the extremum operation in Eq. (16)
can be omitted, and then Eq. (16) can be rewritten as

$D(\tilde{\rho}_{AB})=\left\|\tilde{\rho}_{AB}-\sum_{i=1}^{2}\pi_{i}\tilde{\rho}_{AB}\pi_{i}\right\|_{1}.$
(17)

Substitute the expression of $\tilde{\rho}_{AB}$ into Eq. (17), we will arrive
at

$\displaystyle D(\tilde{\rho}_{AB})$ $\displaystyle=$
$\displaystyle\left\|\pi_{2}\tilde{\rho}_{AB}\pi_{1}+\pi_{1}\tilde{\rho}_{AB}\pi_{2}\right\|_{1}$
(18) $\displaystyle=$ $\displaystyle
2\alpha\beta\left\|\pi_{1}\rho_{AB}\pi_{2}+\pi_{2}\rho_{AB}\pi_{1}\right\|_{1}.$

where we use $\pi_{i}\pi_{j}=\delta_{ij}\pi_{i}$. The weak measurement in
$\tilde{\rho}_{AB}$ is required to be optimal in the sense of
$D_{w}(\rho_{AB})=\left\|\rho_{AB}-\tilde{\rho}_{AB}\right\|_{1}$, so the
projectors in the weak measurement will also be optimal for the corresponding
TQD for $x\rightarrow\infty$, because if it is not the case, it will lead to
two different TQDs for the same $\rho_{AB}$, which is also interpreted in Ref.
[50]. That is,

$\displaystyle D(\rho_{AB})$ $\displaystyle=$
$\displaystyle\left\|\rho_{AB}-\sum_{i=1}^{2}\pi_{i}\rho_{AB}\pi_{i}\right\|_{1}$
(19) $\displaystyle=$
$\displaystyle\left\|\pi_{1}\rho_{AB}\pi_{2}+\pi_{2}\rho_{AB}\pi_{1}\right\|_{1}.$

Eq. (18) and Eq. (19) show that

$D(\tilde{\rho}_{AB})=2\alpha\beta D(\rho_{AB}).$ (20)

which accompanied with Theorem 1 implies
$D(\tilde{\rho}_{AB})=D(\rho_{AB})-D_{w}(\rho_{AB})$. The proof is finished.
$\blacksquare$

In Theorem 2, we have shown that the residual quantum correlation that the
weak measurement fails to ‘extract’ can be ‘extracted’ by the latter
projective measurement. In addition, it is implied that no extra quantumness
is wasted compared with one optimal projective measurement, even though we use
weak measurements and projective measurements, respectively. That is, the
summation of the QCC and the residual TQD is completely consistent with the
TQD. Since any projective measurement can be implemented by a sequence of
continuous weak measurements, one could naturally ask if the quantum
correlation can be extracted little by little by these weak measurements. In
the following theorem, we will show that the weak measurement can do this job
indeed.

Corollary. Suppose $\rho_{n}$ to be the final state after $n$ optimal weak
measurements with the same infinitesimal measurement strength on the subsystem
$A$ of $\rho_{AB}$ such that
$D_{w}(\rho_{n})=\left\|\rho_{n}-\rho_{n+1}\right\|_{1}$ with
$\rho_{0}=\rho_{AB}$, then we will have

$D_{w}(\rho_{n})=(1-2\alpha\beta)\left(2\alpha\beta\right)^{n}D(\rho_{AB}),$
(21)

and

$D(\rho_{AB})=\lim_{x\rightarrow 0}\sum_{n=0}^{\infty}D_{w}(\rho_{n}).$ (22)

Proof. According to Theorem 1, for $\rho_{n}$ we have

$D_{w}(\rho_{n})=(1-2\alpha\beta)D(\rho_{n}).$ (23)

Using Theorem 2 and its proof, one can find

$D(\rho_{n})=D(\rho_{n-1})-D_{w}(\rho_{n-1})=2\alpha\beta D(\rho_{n-1}),$ (24)

which directly leads to

$D(\rho_{n})=\left(2\alpha\beta\right)^{n}D(\rho_{AB}).$ (25)

Insert Eq. (25) into Eq. (23), we will obtain

$D_{w}(\rho_{n})=(1-2\alpha\beta)\left(2\alpha\beta\right)^{n}D(\rho_{AB}).$
(26)

Since we consider the infinitesimal measurement strength which means
$x\rightarrow 0$, i.e., $\alpha\beta\rightarrow 0$, sum Eq. (26) over $n$, one
will get

$\displaystyle\lim_{x\rightarrow 0}\sum_{n=0}^{\infty}D_{w}(\rho_{n})$
$\displaystyle=$ $\displaystyle\lim_{x\rightarrow
0}(1-2\alpha\beta)D(\rho_{AB})\sum_{n=0}^{\infty}\left(2\alpha\beta\right)^{n}$
(27) $\displaystyle=$ $\displaystyle D(\rho_{AB}).$

Eq. (26) and Eq. (27) complete the proof. $\blacksquare$

Since only partial quantum correlation can be grasped by the weak measurement,
the above corollary first shows that the quantum correlation can be
‘extracted’ continuously by the infinitesimal weak measurements, which is
consistent with the fact that the continuous infinitesimal weak measurements
can realize the projective measurement. What is important is that the series
of weak measurements do not waste extra quantumness either. On the contrary,
if consider that the projective measurement will destroy all the quantum
correlation, in principle after the projective measurement, the latter weak
measurement will ‘extract’ no quantum correlation. In the following theorem,
we will prove that it is the case.

Theorem 3. Let the final state of $\rho_{AB}$ after any projective measurement
on subsystem $A$ is given by
$\tilde{\rho}_{AB}^{\prime}=\sum_{j=1}^{2}\tilde{\pi}_{j}^{\prime}\rho_{AB}\tilde{\pi}_{j}^{\prime}$
, then we have

$D_{w}(\tilde{\rho}_{AB}^{\prime})=0.$ (28)

Proof. Since
$\tilde{\rho}_{AB}^{\prime}=\sum_{j=1}^{2}\tilde{\pi}_{j}^{\prime}\rho_{AB}\tilde{\pi}_{j}^{\prime}$,
it is obvious that $D(\tilde{\rho}_{AB}^{\prime})=0$. According to Theorem 1,
one can easily find that
$D_{w}(\tilde{\rho}_{AB}^{\prime})=(1-2\alpha\beta)D(\tilde{\rho}_{AB}^{\prime})=0$.
The proof is completed. $\blacksquare$

The case of multiple-outcome weak measurement.-One can find that all the jobs
presented above only cover the two-outcome weak measurement. Next we will
extend the weak measurement to the case of $n$ outcomes which can be written
as

$P(i)=\alpha_{i}\pi_{1}+\beta_{i}\pi_{2},$ (29)

where the real $\alpha_{i},\beta_{i}$ with $\sum_{i=1}^{n}\alpha_{i}^{2}=1$
and $\sum_{i=1}^{n}\beta_{i}^{2}=1$ are determined by the strength vector
$\vec{x}$ of measurement process and similarly $\pi_{1}$ and $\pi_{2}$ are the
projectors satisfying $\pi_{1}+\pi_{2}=\mathbf{I}$. In particular, one should
note that in the limitation $\vec{x}\rightarrow\infty$, $P(i)$ will become the
normal projective measurement. Following the same procedure as we’ve done for
the two-outcome weak measurement, we will have the following theorem.

Theorem. 4. All the above conclusions hold for $n$-outcome weak measurement if
the factor $2\alpha\beta$ corresponding to two outcomes is replaced by
$\sum_{i=1}^{n}\alpha_{i}\beta_{i}$ corresponding to the $n$ outcomes.

Proof. The proof is completely analogous to those given for the two-outcome
case, so it is omitted. $\blacksquare$

$(2\otimes d)$-dimensional systems.- Finally, we would like to emphasize that
all the theorems as well as the corollary hold for $(2\otimes d)$-dimensional
systems, which can be confirmed if following the completely analogous proof
procedure for the two-outcome case.

To sum up, one can easily find that the QCC is quite different from the SD as
well as the TQD. That is, the QCC characterizes how much quantum correlation
is ’extracted’ (used) in the weak-measurement process. Even though the quantum
correlation in a system can be ’extracted’ by infinite weak measurements, the
QCC subject to the weak measurement per se does not describe how much
correlation is present in the system. As mentioned previously, the SD badly
characterizes the classical correlation in a system, which can be seen when
the measurement strength tends to zero ( the SD will reach its maximum, the
total correlation and the QCC will reach its minimum, zero). The TQD describes
the quantum correlation in the considered system.

##  3 The application

As a demonstration, let’s consider the Bell-diagonal states given by

$\rho_{AB}=\frac{1}{4}(\mathbf{I}_{AB}+\sum\limits_{k=1}^{3}c_{k}\sigma_{k}^{A}\otimes\sigma_{k}^{B}),$
(30)

with $\sigma_{k}$ the Pauli matrices and
$\left|c_{3}\right|>\left|c_{2}\right|>\left|c_{1}\right|$. Based on Refs.
[26, 28, 37], one can easily find that the QCC and TQD are given by

$D(\rho_{AB})=\left|c_{2}\right|,$ (31)

and

$D_{w}(\rho_{AB})=(1-2\alpha\beta)\left|c_{2}\right|.$ (32)

In particular, one can find that the optimal projectors that lead to the QCC
and TQD are $\hat{\pi}_{1}=\left|0\right\rangle\left\langle 0\right|$ and
$\hat{\pi}_{2}=\left|1\right\rangle\left\langle 1\right|$. Eq. (31) and Eq.
(32) show the consistency with Theorem 1. Substitute the optimal projectors
into the weak measurement, one can obtain the weak measurement as
$\hat{P}(x)=\alpha\hat{\pi}_{1}+\beta\hat{\pi}_{2}$ and
$\hat{P}(-x)=\beta\hat{\pi}_{1}+\alpha\hat{\pi}_{2}$. So we can write the
post-measured state as

$\displaystyle\tilde{\rho}_{AB}$ $\displaystyle=$
$\displaystyle\hat{P}(x)\rho_{AB}\hat{P}(x)+\hat{P}(-x)\rho_{AB}\hat{P}(-x)$
(33) $\displaystyle=$
$\displaystyle\frac{1}{4}(\mathbf{I}_{AB}+\sum\limits_{k=1}^{3}\tilde{c}_{k}\sigma_{k}^{A}\otimes\sigma_{k}^{B}),$

with $\tilde{c}_{k}=2\alpha\beta c_{k}$ for $k=1,2$ and $\tilde{c}_{3}=c_{3}$.
Use Eq. (18) again, one can easily find that

$D(\tilde{\rho}_{AB})=2\alpha\beta\left|c_{2}\right|,$ (34)

which associated with Eqs. (31,32) demonstrates the validity of Theorem 2. In
addition, if we continue performing a weak measurement on $\tilde{\rho}_{AB}$,
one will find that $\tilde{c}_{k}$ will be reduced further by a factor
$2\alpha\beta$ and $c_{3}$ will be still invariant. So one will find
$D(\tilde{\rho}_{2})=\left(2\alpha\beta\right)^{2}\left|c_{2}\right|$ and
$D_{w}(\tilde{\rho}_{2})=(1-2\alpha\beta)\left(2\alpha\beta\right)^{2}\left|c_{2}\right|$
with $\tilde{\rho}_{2}$ representing the twice weak measurements. This is
consistent with our corollary. Finally, let the projectors $\hat{\pi}_{1}$ and
$\hat{\pi}_{2}$ be performed on the subsystem $A$ of $\rho_{AB}$, one will get
the final state as

$\displaystyle\tilde{\rho}_{AB}^{\prime}=\hat{\pi}_{1}\rho_{AB}\hat{\pi}_{1}+\hat{\pi}_{2}\rho_{AB}\hat{\pi}_{2}$
$\displaystyle=\frac{1}{4}\left[\left|0\right\rangle\left\langle
0\right|\otimes\left(\mathbf{I}_{B}+c_{3}\sigma_{3}^{B}\right)+\left|1\right\rangle\left\langle
1\right|\otimes\left(\mathbf{I}_{B}-c_{3}\sigma_{3}^{B}\right)\right].$ (35)

which obviously has no quantum correlation. The similar conclusion for
$\tilde{\rho}_{AB}^{\prime}$ can also be found if one employs arbitrary
projectors instead of $\hat{\pi}_{1}$ and $\hat{\pi}_{2}$. Therefore, Theorem
3 in this Bell-diagonal state is also satisfied.

##  4 Conclusion and discussions

We have presented the QCC for the weak measurement based on the trace norm.
The QCC quantifies the quantumness ‘extracted’ by the employed weak
measurements during the measurement procedure. We have found a factorization
relation between the QCC and the TQD. This is easily understood since weak
measurement only influences the system weakly. It is especially interesting
that the residual quantumness after the weak measurement can be ‘extracted’
further by the latter projective measurement or by the latter sequence of
infinitesimal weak measurements, which shows the consistent nature with that
the sequence of infinitesimal weak measurements can realize the projective
measurements. It is important that these different measurements do not waste
extra quantumness. In contrast, it is shown that the weak measurement cannot
extract any quantum correlation from the state after projective measurements.
We believe that it provides a new point of view for us to understand the weak
measurement and the projective measurement. In addition, we also generalize
our conclusions to the cases of $2\otimes d$-dimensional systems and of the
multiple-outcome measurement. Finally, we demonstrate our theorems and
corollary by the Bell-diagonal states.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China,
under Grants No.11375036 and 11175033.

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