Simplifying Schmidt number witnesses via higher-dimensional embeddings

0.25 MB·English

by  Florian Hulpke

SIMPLIFYING SCHMIDT NUMBER WITNESSES VIA
HIGHER-DIMENSIONAL EMBEDDINGS
FLORIANHULPKE,DAGMARBRUSS,MACIEJLEWENSTEIN andANNASANPERA
Institut fu¨r Theoretische Physik, Universit¨atHannover
D-30167 Hannover, Germany
4
0
0 We apply the generalised concept of witness operators to arbitrary convex sets, and
2 review the criteria for the optimisation of these general witnesses. We then define an
embeddingofstatevectors andoperatorsintoahigher-dimensionalHilbertspace. This
n embedding leads to a connection between any Schmidt number witness in the original
a Hilbert space and a witness for Schmidt number two (i.e. the most general entangle-
J
ment witness) in the appropriate enlarged Hilbert space. Using this relation we arrive

0 at a conceptually simple method for the construction of Schmidt number witnesses in
2 bipartitesystems.

1 Keywords: Witnessoperators,Schmidtnumber,Classificationofentanglement
v
8
1 1 Introduction
1
In spite of the tremendous effort devoted in the recent years to characterize (i.e. to classify,
1
0 quantify, detect and measure) entanglement [1], the description of entanglement remains
4 an eluding problem whose complexity grows very fast with the number of subsystems of
0
a given composite quantum system and with the dimension of the Hilbert space. Several
/
h operational separability criteria have been introduced to determine if a given state ρ acting
p
on H=H ⊗H ⊗...⊗H is entangled or not (i.e separable). Among them, the criterion of
- 1 2 n
t thePositivePartialTransposition(PPT)[2]andthe realignmentcriterion[3]areparticularly
n
a powerful. Recently Doherty et al. [4] have introduced a new family of separability criteria
u which gives a characterization of mixed bipartite entangled states with a finite number of
q
tests.
:
v
An apparently different approach to treat the same problem is based on entanglement
i
X witness operators [5, 6]. An entanglement witness W is a hermitian operator which has a
r positiveexpectationvalueonallseparablestates,butanegativeoneforatleastoneentangled
a
state. This state is said to be detected by the witness operator. The existence of these
operators is a direct consequence of the nested convex structure of the sets of (mixed) states
actingonthe Hilbert spaceHofacompositesystem. Since foranygiven(finite-dimensional)
Hilbert space the subset of separable states is convex and closed, it is always possible to find
entanglementwitnessoperatorsregardlessofthedimensionsand/orthenumberofsubsystems
of the composite system. Equivalently each entangled state can be detected by a witness
operator. Notice that by using this approach the problem of determining whether a given
state ρ is entangled or not is transformed into the problem of finding a suitable witness
operator that detects it. The most suitable witnesses will be those that detect more states
than any other ones, and for that reason they are called optimal witnesses.
Remarkably, entanglement witnesses have become a very powerful tool not only for de-
1
tecting entanglement, but also in the context of various other tasks in quantum information
theory. Forinstance,establishingasecretkeyinquantumcryptographyrequirestheexistence
ofquantum correlations,which canbe characterizedby optimalwitnesses [7]. The activation
anddistillationpropertiesofastateρ(thatis,thepossibilitytolocallydistillfromanensem-
ble of mixed states a subset of maximally entangled pure states) can also be recast in terms
of witness operators [8]. By far the best-known and most famous entanglement witnesses
are the so-called Bell inequalities [9]. It is easy to see from the definition of entanglement
witnesses that each Bell inequality corresponds to an entanglement witness. However, this
correspondencedoes notnecessarilyholdthe otherwayround,asthereexistentangledstates
that do not violate any Bell inequality but nevertheless are detected by a witness operator
[10]. Thus, for the detection of entanglement witness operators are, in this respect, stronger
than Bell inequalities. Furthermore, witness operators can be generalized to distinguish be-
tween different types of entanglement as long as the different entanglement types correspond
to nested convex subsets. This is indeed the case for bipartite systems and, at least, for the
simplest multipartite system, i.e. H = C2 ⊗C2 ⊗C2 [11]. For larger multipartite systems,
wherethestructureofentangledstatesismuchricherandmuchless-known,witnessoperators
are also a useful tool to explore the structure of the Hilbert space. Finally, let us point out
that since entanglement witnesses are observables (although not positive semidefinite) they
can be measured. The experimental implementation of a witness operator can be realisedby
meansoflocalmeasurements[12,13]andhasalreadybeenachievedinthelaboratory[14,15].
The paper is organized as follows. In section 2 we first review the concept of a general
witness as well as its optimization following the arguments given in [6, 16, 17]. Most of the
lemmas and theorems stated in this part of the paper are a straightforward generalization
of the formalism developed previously, but for completeness we have included them here. In
section 3 we restrict ourselves to bipartite systems, and review first the concepts of Schmidt
number and Schmidt number witnesses. We show then that by embedding the state vectors
andoperatorsoftheoriginalHilbertspaceintoahigher-dimensionalHilbertspaceitispossible
to connect any Schmidt number witness in the originalHilbert space to a witness of Schmidt
number two (i.e. the mostgeneralentanglementwitness) in the appropriateenlargedHilbert
space. Using this method one can simplify the construction and optimization of the desired
general witness. We close this section with an explicit example to illustrate our method.
Finally, we present our conclusions in section 4.
2 Optimisation of a general witness operator
Weconsiderquantumsystemsofarbitrary,finitedimensions. ByHwedenoteaHilbertspace
overthe fieldK, by B(H)the spaceofbounded operatorsacting onthis Hilbert spaceandby
P ⊂ B(H) the subset of positive semidefinite operators with trace one (the set of states on
H).
Consider the following situation, sketched in figure 1: for two given nested convex, closed
subsets S ⊂ S ⊂ P, does a given ρ ∈ S belong to S ? Without loss of generality we will
1 2 2 1
2
Fig.1. Thestructureofthenestedconvexsets. Fig.2. Thewitnesscorrespondingtothehyper-planeW1
is(S1,S2)-finer,butnot(S1,P)-finerthanW2. BothW1
andW2 are(S1,P)-finerthanW3.
assume that the identity belongs to S and that S is not of measure zero in S [18].
1 1 2
Definition 1: For any convex set X we denote the border of X by
∃b∈X, s.t. ∀λ>0 one has:
δX := a∈X (1)
(1+λ)a−λb6∈X
(cid:26) (cid:12) (cid:27)
(cid:12)
(cid:12)
Some of the operators on the borde(cid:12)r, that have special properties, are referred to as edge-
operators. Notice that a full characterization of the border-operators of S immediately
1
implies a full characterizationof the operatorsthat belong to S \S . In fact a full character-
2 1
ization of edge-operators is sufficient for this aim. Without loss of generality one can always
shift the set S such that 1l∈S \δS .
1 1 1
Beforeansweringthequestionwhetheragivenρ∈S belongstoS wefirstformallydefine
2 1
ageneralwitnessoperator. Tosimplifythenotation,byW weshalldenotean(S ,S )-witness
1 2
defined as follows:
Definition 2: A hermitian operator W is called an (S ,S )-witness iff:
1 2
(i) Tr (Wσ)≥0 ∀σ ∈S .
1
(ii) ∃ρ∈S such that Tr (Wρ)<0.
2
(iii) Tr W =1.
We noteinpassingthatW is notpositivesemidefinite andthatcondition(iii)correspondsto
a normalization of the operator W. This normalization is useful for the comparison between
different witnesses.
To prove that ρ∈S \S it is sufficient to find a witness operator W that detects ρ. For the
2 1
casesinwhichthe setS is alsoclosed(andthereforecompact, due to the boundedness ofP)
1
the existence of a witness that detects ρ is also necessary for ρ∈S \S [19].
2 1
Note that without the requirement 1l ∈ S there could be cases for which no witnesses
1
exist. For example, assume σ 6= 1l, S = {σ} and S = {ρ(λ) = λ1l+(1−λ)σ|1 ≥ λ > 0}.
1 2
Then no linear operator exists such that Tr (Wσ)≥0 for all σ ∈S and Tr (Wρ)<0 for a
1
ρ∈S and Tr W =1.
2
3
To proceed further, let us introduce now some basic concepts and notations related to
witness operators. We shall adopt here the notation developed in Ref. [6] and [17]. Our
notation is as follows:
1. DWS2 :={ρ∈S2|Tr(Wρ)<0}, i.e. the set of the states in S2 that are detected by W.
2. Q := {Z| Tr (Zρ)≥ 0 ∀ρ ∈S }, i.e. the set of operators which do not detect any state
2 2
S
in S , and, therefore, are non-negative on S .
2 2
3. PWS1 :={|ψihψ|∈S1|hψ|W|ψi=0},i.e. the setofone-dimensionalprojectors(purestates)
in S for which the expectation value of W vanishes.
1
4. Finer witness: W1 is (S1,S2)-finer than W2 iff DWS22 ⊂ DWS21, that is, if any state detected
by W is also detected by W .
2 1
5. Optimal witness: W is an (S ,S )-optimal witness iff there exists no other witness that is
1 2
(S ,S )-finerthanW. SoforallW thatare(S ,S )-finerthanW theequalityW =W holds.
1 2 1 1 2 1
The concept of a witness being (S ,S )-finer than another one depends on S . As illus-
1 2 2
trated in figure 2 it is possible that a witness W is (S ,S )-finer than W , but that there
1 1 2 2
exists an S with S ⊂S , such that W is not (S ,S )-finer.
2′ 2 2′ 1 1 2′
The above definitions provide the necessary tools to optimize a given general witness.
Lemma 1 (Lemma 1 in [6]): Let W be (S ,S )-finer than W and
1 1 2 2
Tr ρW
1
λ:= inf (2)
ρ∈DWS22 Tr ρW2
Then for any positive operator ρ we have:
(i) If Tr ρW =0 then Tr ρW ≤0.
2 1
(ii) If Tr ρW <0 then Tr ρW ≤ Tr ρW .
2 1 2
(iii) If Tr ρW >0 then Tr ρW ≤λ Tr ρW .
2 1 2
(iv) λ≥1. In particular if λ=1⇔W =W .
1 2
In complete analogy with the optimization of entanglement witnesses, one canalso construct
(S ,S )-finergeneralwitnessesbysimplysubtractinganyoperatorthatispositivedefiniteon
1 2
S (i.e. Z ∈ Q ) from the original witness in such a way that the remaining operator still
2 2
S
fulfills the necessary conditions for being a witness operator.
Lemma 2 (Lemma 2 in [6]): W is (S ,S )-finer than W ⇔ there exists a Z ∈ Q
1 1 2 2 2
S
(i.e. Tr (Zρ) ≥ 0 for all ρ ∈ S ), and there exists an ǫ with 1 > ǫ ≥ 0 such that W =
2 2
(1−ǫ)W +ǫZ (⇔ for all ρ∈S : Tr (W ρ)−(1−ǫ) Tr (W ρ)≥0).
1 1 2 1
Theorem 1 (Theorem 1 in [6]): A witness W is optimal ⇔ for all Z ∈ Q and ǫ > 0
1 2
S
the operator W :=(1+ǫ)W −ǫZ is not a witness, i.e. it does not fulfill Tr (W ρ)≥0 for
′ 1 ′
all ρ∈S .
1
Lemma 3( Lemma 3 in[6]): If ZPWS1 6= 0 (i.e. for all P ∈ PWS1 one has Z 6⊥ P), then Z
4
cannot be subtracted from W, that is, (1+ǫ)W −ǫZ is not a witness for any ǫ>0.
Corollary 1 (Corollary 2 in [6]): If PWS1 spans H then W is optimal.
Lemma4: If,foragiven(S ,S )-witnessW,thereexistsaρ∈δS \δS ,suchthat Tr Wρ=
1 2 1 2
0, then W is (S ,S )-optimal. Geometrically W can be interpreted as an S -edge witness
1 2 1
operator, namely it is tangent to the set S at a point which does not belong to the border
1
of S .
2
Proof: Assuming that W is not (S ,S )-optimal, there would exist an (S ,S )-finer general
1 2 1 2
witness W 6=W. Followingthe assumptionthere has to exist a state σ ∈S that is detected
′ 2
by W (i.e. Tr W σ < 0) but not by W (i.e. Tr Wσ ≥ 0). Since ρ ∈ S and W is
′ ′ 1 ′
(S ,S )-finer than W, we know according to Lemma 1 (i) that Tr W ρ = 0. Furthermore
1 2 ′
ρ 6∈ δS . This leads by definition to the fact that there is a λ > 0, with the property that
2
the state (1+λ)ρ−λσ ∈ S . Evaluating both witnesses W and W on this state we see
2 ′
that Tr W((1+λ)ρ−λσ)=−λ Tr Wσ ≤0, but Tr (W (1+λ)ρ−λσ)=−λ Tr W σ >0.
′ ′
This is in contradictionto Lemma 1 (ii). Therefore, we conclude that there exists no witness
W 6=W which is (S ,S )-finer than W, and since all witnesses which are (S ,S )-finer than
′ 1 2 1 2
W have to be equal to W, we conclude that W is optimal (cid:3).
Witness operators are often used to detect generic entanglement in bipartite systems. In
thiscasethetwoconvexsetsS ,S canbeidentifiedasS =S ={set of all separable states}
1 2 1
and S = P = {set of all positive (semidefinite) operators with trace one}. It is well known
2
thatforsystemsactingonaHilbertspaceofdimensiondimH>6(wheredimH ,dimH ≥
A B
2) there exists entanglement which cannot be detected by means of the partial transposi-
tion. The set of states that remain positive under partial transposition (PPT-set) also form
a convex set. We can distinguish PPT-entangled states from separable ones if we search for
witnessesassociatedtoS =S andS ={PPT}withS ⊂S . ThosewitnessesW whichare
1 2 1 2
capable to detect ρ ∈S \S are necessarily non-decomposable, since they cannot be written
2 1
as W = P +QT, where P and Q are positive semidefinite operators [6]. First examples of
non-decomposable entanglement witnesses were provided in [9], and the characterisation of
such witnesses was presented in [16]. Notice that the concept of witness operators only relies
on a nested subset structure, and thus is not restricted to bipartite systems. For multipar-
tite systems, there exist typically various classes of distinct multipartite entanglement. For
instance, for 2×2×2 systems, there are classes of separable S, biseparable B, W [20] and
GHZ mixed states, which are ordered in the nested structure S ⊂ B ⊂ W ⊂ GHZ [11]. For
this case and similar ones one can tailor the appropriate witness W to discriminate between
the different nested convex sets.
In the next section we will study so-calledSchmidt (number) witnesses [21], which detect
how many degrees of freedom of the subsystems of a bipartite system are entangled with
each other. Thus Schmidt witnesses can distinguish between entangled states from different
Schmidt classes. In this sense Schmidt number witnesses provide a refinement of general
entanglement witnesses.
5
3 From Schmidt number witnesses to entanglement witnesses
For bipartite systems it is possible to extend the useful concept of the Schmidt rank for pure
states [22] to the Schmidt number for mixed states [23]. The Schmidt number of a mixed
state ρ characterisesthe maximal Schmidt rank of the pure states which is at least needed to
construct ρ. It is defined as:
SN(ρ):= min max{SR(ψ )}, (3)
i
ρ= ri=1pi|ψiihψi| i
P
where SR(ψ ) is the Schmidt rank of |ψ i. By (SN) we denote the Schmidt class k, i.e. the
i i k
set of states that have Schmidt number k or less. This set is a convex, compact subset of P,
and there is a nested structure of the form(SN) ⊂(SN) ⊂...⊂(SN) ⊂...⊂P. Clearly,
1 2 k
(SN) = S corresponds to the set of separable states. As explained in 2 it is clear that one
1
can construct a Schmidt number witness operator S which is non-negative on all states in
k
(SN) , but detects at least one state belonging to (SN) . Using our previous notation
k 1 k
−
this would correspond to a ((SN) ,(SN) )-witness, but as a shorter notation and to be
k 1 k
−
consistentwiththenotationof[21]wedenoteS asak-Schmidt-witness(k-SW).Inthesame
k
way we define the terms k-finer and k-optimal as abbreviations.
In the previous part of this paper we have established tools to answer (a) under which
conditions a given k-SW is k-finer than another one, and (b) how to optimise a given k-SW
(namely one has to subtract operators Z which are positive definite on the set (SN) , such
k
that the operator W = S −Z has the property Tr (Wσ) ≥ 0 for all σ ∈ (SN) ). To
k k 1
−
verify positivity on all states from the set (SN) it suffices to restrict oneself to the set of
k 1
−
all pure states |ψ i, since those are the extremal points of (SN) . But nonetheless such
k 1 k 1
− −
verificationremainslaborious,sincethese statesdonotexhibitaparticularlyusefulstructure
that allows to check easily whether Tr (W|ψ ihψ |) ≥ 0 for all |ψ i. Notice, how-
k 1 k 1 k 1
− − −
ever,that for the case of a generalentanglementwitnesses (or more precisely,of2-SW’s) this
problem becomes much easier, since semi-positivity of an operator W on all product states
|e,fi ∈ H ⊗H is equivalent to the semi-positivity of the (dimH −1)-parameter family
A B A
of operators he|W|ei ∈ B(H ). Thus, rather than checking a (dimH −1)(dimH −1)-
B A B
dimensional parameter space corresponding to all product vectors, the task is greatly sim-
plified. Furthermore, positivity on a whole space is mathematically a simpler concept than
positivity on a given subset.
LetuspointoutforclaritythateverySchmidtnumberwitnessisanentanglementwitness
since itdetects somekind ofentanglement. Awitness forSchmidt numbertwo (2-SW)corre-
spondstoawitnessthatdetectsallkindsofentanglementwithoutdiscriminatingbetweenthe
differenttypes. Thereforewe sometimes use the name “entanglementwitness” synonymously
with “2-SW”.
In this section we will show that it is always possible to reformulate the problem of
findingawitnessofSchmidtclassk intofindinga2-SWdenotedbySinahigher-dimensional
Hilbert space. Our method relies on embedding the original Hilbert space H ⊗H into
A B
an enlarged Hilbert space, such that each pure state with Schmidt rank k or less in the
original space becomes a product state in the enlarged space. The enlarged Hilbert space
consists of the original one with two added local ancillas of dimension k, i.e. H =
enlarged
(H ⊗Kk)⊗(H ⊗Kk). The embedding isperformedbymeansofamapI . We shallstudy
A B k
6
theeffectofthismaponthesetofoperatorsactingonH ⊗H andshowthatthismapalso
A B
connectstheexpectationvaluesofoperatorsintheoriginalHilbertspacewiththeexpectation
values of operators in the enlarged Hilbert space.
3.1 Mapping from the original Hilbert space to the enlarged one
In this subsection we first define a map which transforms states in the original Hilbert space
H ⊗H into states in the enlarged one. We then study the effect of this map on the
A B
operators acting on H ⊗H and define for each operator S an operator S acting in the
A B
enlarged space.
Definition3: WedenotebyI : H ⊗H →(H ⊗Kk)⊗(H ⊗Kk)amapthattransforms
k A B A B
every pure state |ψi ∈ H ⊗H into a pure state |I (ψ)i ∈ (H ⊗Kk)⊗(H ⊗Kk). This
A B k A B
map is defined by:
n k k
|ψi= λ |a b i7→|I (ψ)i = |a i⊗|ii ⊗ λ |b i⊗|ji
i i i k i j j
!  
i=1 i=1 j=1
X X X
 
2k 2k
+ |a i⊗|i−ki ⊗ λ |b i⊗|j−ki +...
i j j
!  
i=k+1 j=k+1
X X
 
n n
+ |a i⊗|i−ui ⊗ λ |b i⊗|j−ui . (4)
i j j
!  
i=u+1 j=u+1
X X
 
where n ≤ min(dimH ,dimH ). Here we have fixed a basis {|1i,...,|ki} for both ancilla
A B
spaces. The ancilla states are orthogonal, i.e. hi|ji = δ . The vectors |a i,|b i denote the
ij i i
Schmidt bases of |ψi and u := floor (n/k)k where floor indicates the integer part [24]. We
denote the action of this map as “lifting up”. In the following we describe the properties of
this map and its action on states and operators.
Remark 2: For each pure state |ψi= n λ |a b i∈H ⊗H it holds that:
i=1 i i i A B
k P 2k n
I (|ψi)=I ( λ |a b i)+I ( λ |a b i)+...+I ( λ |a b i). (5)
k k i i i k i i i k i i i
i=1 i=k+1 i=u+1
X X X
In particular,I maps by definition pure states in H ⊗H with Schmidt rankk or less into
k A B
pure product states in (H ⊗Kk)⊗(H ⊗Kk).
A B
We now define for each operator acting on H ⊗H an operator acting on
A B
(H ⊗Kk)⊗(H ⊗Kk).
A B
Definition 4: For a given S = σ |iihl| ⊗|jihm| acting on H ⊗H , and
i,j,l,m i,j,l,m A B A B
k ∈N we define an operator S on (H ⊗Kk)⊗(H ⊗Kk) by:
k P A B
k
S := σ |i,sihl,t|⊗|j,sihm,t|. (6)
k i,j,l,m
s,t=1i,j,l,m
X X
7
If Tr (S)=1, then Tr (S )=k.
k
Remark 3: Reordering the tensor-product structure of the enlarged Hilbert space as
H ⊗H ⊗Kn⊗Kn leads to the following expression for S :
A B k
k
S = S⊗|ssihtt|. (7)
k
s,t=1
X
In the following the notation S (i.e. blackboard font) will be used only for those operators
k
actingon(H ⊗Kk)⊗(H ⊗Kk),whichcanbewrittenlikeineqn. (7),withacorresponding
A B
operator S ∈B(H ⊗H ).
A B
Remark 4: For every pure state |ψi ∈ H ⊗H with Schmidt rank less or equal to k, i.e.
A B
|ψi= k λ |a b i with λ ≥0 and every operator S it holds that:
j=1 i i i i
P k
hψ|S|ψi = λ λ ha b |S|a b i
∗s t s s t t
s,t=1
X
k
= λ λ ha |hb |S|a i|b ihij|ssihtt|lmi
∗j m i j l m
i,j,l,m,s,t=1
X
= hI (ψ)|S |I (ψ)i. (8)
k k k
Remark 5: Notice that the same construction holds for a pure state of Schmidt rank larger
thank,namely|ψi= n λ |a i|b i= u |ψ i,withu= floor (n/k)andtheabbreviation
i=1 i i i j=1 j
|ψ i= min(jk,n) λ |a b i. By using Remark 2 one immediately obtains:
j l=(j 1)k+1 lPl l P
−
P u u
hψ|S|ψi = hψ |S|ψ i= hI (ψ )|S |I (ψ )i
i j k i k k j
i,j=1 i,j=1
X X
= hI (ψ)|S |I (ψ)i. (9)
k k k
Remark 6: Given a mixed state ρ and an arbitrary decomposition ρ = l p |φ ihφ |, it
i=1 i i i
follows that
l l P
Tr (Sρ)= p hφ |S|φ i= p hI (φ )|S |I (φ )i. (10)
i i i i k i k k i
i=1 i=1
X X
Thus, by defining a (non-normalized) mixed state Γ := p |I (φ )ihI (φ )| ∈
k i i k i k i
B((H ⊗Kk)⊗(H ⊗Kk)) one arrives at
A B P
Tr (Sρ)= Tr (S Γ ). (11)
k k
3.2 Mapping from the enlarged Hilbert space to the original one
In this part we now define a map which transforms states in the enlarged Hilbert space into
states in the original one.
Remark 7: Every pure product state |Ai⊗|Bi = |A,Bi ∈ (H ⊗Kk)⊗(H ⊗Kk) can be
A B
8
expressed by using the Schmidt decomposition of each pure state |Ai,|Bi in the bipartite
splitting H ⊗Kk as:
A,B
k k
|Ai= λ |a i⊗|c i ; |Bi= µ |b i⊗|d i, (12)
i i i j j j
i=1 j=1
X X
where the Schmidt coefficients λ and µ are positive and λ2 =1, µ2 =1.
i j i i j j
P P
Definition 5: By J˜ : (H ⊗Kk)⊗(H ⊗Kk) → H ⊗H we denote the map that
k A B A B
transforms any pure product state |Ai⊗|Bi ∈ (H ⊗Kk)⊗(H ⊗Kk) into a pure state
A B
|ψi∈H ⊗H with Schmidt rank less or equal to k. This map is defined by:
A B
k k k
J˜ :|Ai⊗|Bi7→|J˜ (A⊗B)i = hii| λ |a i⊗|c i ⊗ µ |b i⊗|d i
k k l l l m m m
! !
i=1 l=1 m=1
X X X
k
= F λ µ |a b i (13)
lm l m l m
l,m=1
X
where F := k hii|c d i.
lm i=1 l m
Recalling that the Schmidt rank ofa pure state is equalto the rankof its reduceddensity
P
matrices we find:
n k
Tr B|J˜k(A⊗B)ihJ˜k(A⊗B)| = hbs| Fi∗jFlmλ∗iλlµ∗jµm|alihai|⊗|bmihbj||bsi
s=1 i,j,l,m=1
X X
k  k 
= λ λ F F |µ |2|a iha |. (14)
∗i l i∗s ls s l i
i,l=1 s=1
X X
The rank of this operator cannot exceed k, and therefore, the Schmidt rank of |J˜ (A⊗B)i
k
does not exceed k.
This map can be now straightforwardly extended to map entangled pure states of the
enlargedspaceintotheoriginalonebyusingtheSchmidtdecompositionsaccordingtothesplit
(H ⊗Kk)⊗(H ⊗Kk) (Remark 7) and applying the map to each Schmidt term separately.
A B
Definition 6: By J we define the extension of J˜ on all pure entangled states |Ψi in
k k
(H ⊗Kk)⊗(H ⊗Kk):
A B
l l
J :|Ψi= λ |A i⊗|B i7→|J (Ψ)i:= λ |J˜ (A ⊗B )i. (15)
k i i i k i k i i
i=1 i=1
X X
We call the action of the map J “lifting down”.
k
Remark 8: Notice thatfor all|ψi∈H ⊗H withSchmidt rankk or less,itholds that:
A B
|J (I (ψ))i=|ψi. Thus,whenrestrictedtothesestates,themapJ istheinversemapofI .
k k k k
9
3.3 Connection between the expectation values of operators
After the definitionofthe twomaps (the “lifting up” mapI andthe “lifting down”mapJ )
k k
oneobservesthatthereisacloserelationbetweentheexpectationvalueofanoperatorinthe
originalspace and the expectationvalue of the correspondingoperatorin the enlargedspace.
Lemma 5: Given two arbitrary pure product states |A B i,|A B i ∈
1 1 2 2
(H ⊗Kk)⊗(H ⊗Kk) the following equation:
A B
hA |⊗hB |S |A i⊗|B i=hJ (A ⊗B )|S|J (A ⊗B )i (16)
1 1 k 2 2 k 1 1 k 2 2
holds.
Proof: We use Remark 3 to express S as a function of S, and express each pure state |A i
k i
(respectively |B i) in its Schmidt decomposition according to Remark 7. The expectation
i
value of S is thus given as:
k
k k k
hA B |S |A B i = λ µ ha b |S( νξ |e f i) hc d |ssihtt|g h i
1 1 k 2 2  ∗i ∗j i j l m l m  i j l m
i,j=1 l,m=1 s,t=1
X X X
 
k k
= F λ µ ha b | S G νξ |e f i
 i∗j ∗i ∗j i j   lm l m l m 
i,j=1 l,m=1
X X
= hJ (A ⊗B )|S|(J (A⊗B )i,  (17)
k 1 1 k 2 2
where F := k hss|c d i and G := k htt|g h i (cid:3).
ij s=1 i j ij t=1 i j
P P
So far, we have defined the action of the maps on pure states and operators and we have
shown how the maps permit to “jump” from the originalspace to the enlarged one (and vice
versa). We have also shown the relation between the expectation value of an operator in the
originalspaceandthecorrespondingoperatorintheenlargedspace. Weproceednowtoshow
that a Schmidt number witness (k-SW) acting in H ⊗H corresponds to an entanglement
A B
witness (2-SW) acting on (H ⊗Kk)⊗(H ⊗Kk). The main results of our paper are stated
A B
in the following two theorems.
Theorem 2:
i) Given two arbitrary operators S,ρ ∈ B(H ⊗H ) and an arbitrary decomposition
A B
ρ= l p |φ ihφ |,itholdsthat Tr (Sρ)= Tr (S Γ ),whereΓ := l p |I (φ )ihI (φ )|.
i=1 i i i k k k i=1 i k i k i
ii)GiventwoarbitraryoperatorsS ,Θ∈B((H ⊗Kk)⊗(H ⊗Kk))andanarbitrarydecom-
P k A B P
positionΘ= p |Φ ihΦ|,itholdsthat Tr (S Θ)= Tr (Sθ),whereθ := |J (Φ )ihJ (Φ )|.
i i i k i k i k i
P P
Proof: (i) See Remark (6). (ii) The proof is a concatenation of the previous remarks (cid:3).
Theorem 3:
i)IfSisak-SWactingonH ⊗H ,thenS isa2-SWactingon(H ⊗Kk 1)⊗(H ⊗Kk 1).
A B k 1 A − B −
ii) If S is a 2-SW, then S is an n-SW w−ith k ≤n≤2k.
k 1
−
10