Optimal and covariant single-copy LOCC transformation between two two-qubit states

0.36 MB·English

by  K. Bradler

Optimal and covariant single-copy LOCC transformation between two two-qubit states
K. Bra´dler
Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, Apdo. Postal 20-364, M´exico 01000, M´exico
∗
(Dated: February 1, 2008)
Giventwotwo-qubitpurestatescharacterizedbytheirSchmidtnumbersweinvestigateanoptimal
strategytoconvertthestatesbetweenthemselveswithrespecttotheirlocalunitaryinvariance. We
discuss the efficiency of this transformation and its connection to LOCC convertibility properties
between two single-copy quantum states. As an illustration of the investigated transformations we
present a communication protocol where in spite of all expectations a shared maximally entangled
pair between two participants is theworst quantum resource.
7
0
PACSnumbers: 03.67.Mn
0
Keywords: LOCCtransformation,covariantquantum channels,semidefiniteprogramming
2

n
a
J

3 I. INTRODUCTION
2

Oneofthegreatestachievementsofquantuminformationtheory(QIT)istherealizationthatquantumentanglement
3
servesasaresourceforperformingvariouscommunicationtaskswhereEkert’sscheme[1]forquantumkeydistribution
v
(QKD) or quantum teleportation [2] are the most flagrant examples. Shortly after, the question of equivalence of
7
0 different multipartite states came into question. Partially motivated by the security issues in QKD (i.e. how to
1 locally distill a shared non maximally entangled or even noised quantum state to avoid any correlations with a
1 potential eavesdropper) the problem of LOCC [4] (local operation and classical communication) convertibility [5]
1 became fundamental. We can approach the question from two extremal sides. Namely, asking whether two states
6
are LOCC transformable in an asymptotic limit or having just a single copy of an initial state at our disposal. Both
0
approaches brought the considerable progress in QIT. To name just few, in the first regime, several measures of
/
h entanglement were defined in terms of an asymptotic rate in which it is possible to convert from/to a maximally
p entangled state [3, 4]. In the second case, the connection between the Schmidt number majorization [6] and LOCC
-
t state transformation was discovered [17] or new classes of tripartite entangled states were presented [7].
n
We willtreatwithaninterestingQITparadigmwhichis socalledimpossibility transformation(or‘no-goprocess’).
a
There exist several kinds of impossible transformations stratified by the fact how the impossibility is fundamental.
u
q Quantum cloning [9] or finding the orthogonalcomplement to a given quantum state (universal NOT) [10] belong to
: the group of the highest stratum. This kind of impossibility comes from basic principles of quantum mechanics [8]
v
and can be performed just approximately [9, 10]. There are also known other examples of fundamentally impossible
i
X processes[11]. Inthelowerlevelthereexisttransformationswhicharenotforbiddenbythelawsofquantummechanics
r buttheyareimpossibleundersomeartificiallyaugmentedrequirements. Typically,weconsideronlyLOCCoperations
a as,forexample,theabovementionedsingle-copytransformationtask[17]. Inthiscase,withoutthe LOCCconstraint
there is no problem to transform one pure state to another without any limitations.
Inthispaperweusethemethods ofsemidefinite programming[20]tofindanoptimalandcompletelypositive(CP)
map for LOCC single-copy pure state transformation regarding its covariant properties. Covariance means that the
sought CP maps are universal in the sense that they do not change their forms under the action of SU(2) group
(or their products) on the input states. The covariance requirement was also added to other quantum mechanical
processes, compare e.g. [22]. In addition to the covariance, we require optimality meaning that the output state
produced by the CP LOCC map is maximally close to the required target state. The closeness is measured by the
value of the fidelity between the actual output state and the desired target state. As we will see, our problem of
covariant and optimal LOCC state transformation combines both kinds of the impossibilities mentioned above.
Thestructureofthepaperisthefollowing. InsectionIIwerecallsomebasicsfactsabouttheisomorphismbetween
quantum maps and the related group properties. The main part of this paper can be found in section III where the
optimalLOCCsingle-copystatetransformationisinvestigatedwiththe helpofsemidefinite programmingtechniques.
Section IV can be regardedas an application of the studied problem where we present a communication protocol for
the LOCC transmission of a local unitary operation from one branch of a shared two-qubit state to the second one.
We show that a maximally entangled pair does not always need to be the best quantum communication resource.
The corresponding Kraus maps for the protocol are listed in Appendix.
Electronicaddress: [email protected]
∗
2
II. METHODS
Itiswellknownthatthereexistsanisomorphismbetweencompletelypositivemaps (̺)andsemidefiniteoperators
M
R , first introduced by Jamio lkowski[12]
M
M(̺in)=Trin 11⊗̺Tin RM ⇐⇒RM =(M⊗11) P+ , (1)
(cid:2)(cid:0) (cid:1) (cid:3) (cid:0) (cid:1)
whereP+ = d−1 ii jj isamaximallyentangledbipartitestateofthedimensiond2. Theisomorphismallowsusto
i,j=0| ih |
fulfillotherwiPseadifficulttaskoftheparametrizationofallCPmapsbyputtingapositivityconditionontheoperator
R . Then, the parametrization problem is computationally much more feasible. This is not the only advantage the
reMpresentation offers. As was shown in [15], the representation is useful for the description of quantum channels (so
calledcovariantchannels)whichwewishtooptimizeregardingsomesymmetryproperties. Moreprecisely,havingtwo
representationsV1,V2 ofa unitary group,the mapM(̺) is saidto be covariantifM(̺)=V2†M(V1̺V1†)V2. Inserting
the covariance condition into Eq. (1) and using the fact that the positive operator R is unique, we get
M
RM =(V2†⊗V1T)RM(V2⊗V1∗)⇐⇒[RM,V2⊗V1∗]=0. (2)
ThespaceoccupiedbyV V canbedecomposedintoadirectsumofirreduciblesubspacesandfromSchur’slemma
2⊗ 1∗
follows that R is a sum of the isomorphisms between all equivalent irreducible representations. If we now consider
the fidelity equMation in the Jamio lkowskirepresentation
F =Tr ̺ ̺T R (3)
out⊗ in M
(cid:2)(cid:0) (cid:1) (cid:3)
thetaskisreducedonfindingthemaximumofF subjecttonon-negativityofR andotherconstraintsposedonR .
In our case, it is the trace preserving condition Trin[R ]=11 following from (M1). This can be easily reformulatedMas
a semidefinite program [20] and thus efficiently solvedMusing computers. Moreover, it is easy to put other conditions
on R such as partial positive transpose condition (PPT) and they can be easily implemented as well [21]. Recall
that Mfor two-qubit systems the PPT condition is equivalent to the LOCC requirement. Note that the usefulness of
the presented method was already shown, for example, in connection with optimal and covariant cloning [16].
In our calculation we employed the YALMIP environment [24] equipped with the SeDuMi solver [25]. One of the
advantagesofsemidefinite programmingisthe indicationofwhichparametersarezero. Then, analyticalsolutionsfor
the fidelity and even general forms of the Kraus decomposition [26] of the CP map may be found. In our problem,
usingthe propertiesofthe Jamio lkowskipositivematrixR (whicharestatedasanalmostcomputer-readytheorem
in [13, 14]) we derived the corresponding Kraus operatorsMas general as possible.
III. OPTIMAL AND COVARIANT SINGLE-COPY LOCC STATE TRANSFORMATION
Let us have an input and target state written in their Schmidt forms χ = a 00 +√1 a2 11 , ϕ =
| i | i − | i | i
c 00 +√1 c2 11 ; a,c (0,1/√2). It was shown [17] that a deterministic LOCC conversion χ ϕ is pos-
| i − | i ∈ | i → | i
sible iff a c. If we want to go in the direction where LOCC is not powerful enough we have basically two strategies
≥
at our disposal. First, in some cases we may choose a probabilistic strategy [18] also called conclusive conversion. As
an alternative,there exists a possible LOCC deterministic transformationto a state which is in some sense closestto
the required one [19]. Namely, it is a state in which the fidelity with the target state is maximal. We will follow a
relatedwayandfindhowanoptimalcovariantLOCCCPmapbestapproximatesthe idealtransformation χ ϕ .
| i↔| i
In the next sections, we consider the following parameter space a,c (0,1) both for χ and ϕ .
∈ | i | i
Adopting the covariance considerations from the previous section into our case we demand
F = ϕ (χ χ) ϕ = ϕ (χ χ ) ϕ =F , (4)
′ ′ ′ ′ ′
h |M | ih | | i h |M | ih | | i
where V χ = χ ,V ϕ = ϕ and the covariance condition (2) follows (note that quite accidentally the condition
1 ′ 2 ′
| i | i | i | i
is the same as in case of covariantcloning).
A. LOCC semicovariant transformation
Firstly, wewill be interestedin how χ canbe transformedif V1 =V2 =11 U whereU is a unitary representation
| i ⊗
of SU(2). In other words, we consider the situation where the covariance is imposed on one branch of χ (we call it
a semicovariant case). From Eq. (2) follows | i
[R ,11 U 11 U∗]=0 [R˜ ,11 11 U U]=0, (5)
M ⊗ ⊗ ⊗ ⇐⇒ M ⊗ ⊗ ⊗
3
1
0.9
0.8
0.7
0.6
F 0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8 0.9 11 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
c a
FIG. 1: The fidelity for the optimal and locally semicovariant LOCC transformation between χ = a 00 +√1 a2 11 and
| i | i − | i
ϕ =c 00 +√1 c2 11 .
| i | i − | i
where R =S†R˜ S and S =11 SWAP σY where SWAP = 00 00 + 01 10 + 10 01 + 11 11 and σY is the
Pauli Y Moperator.MWith the unita⊗rily trans⊗formed rhs in Eq. (5) t|heihdec|om|posihitio|n is| foiuhnd|in|a pihart|icularly simple
way
4
R˜ = s P a P , (6)
M ij Sij ⊕ ij Aij
iM,j=1
whereP ,P areisomorphismsbetweenequivalentsymmetricalandantisymmetricalirreduciblesubspaces,respec-
Sij Aij
tively. There are 32 free complex parameters but we know that R˜ is a nonnegative operator. It follows that a ,s
ii ii
are real and a =a ,s =s . The number of free parameters isMthus reduced to 32 real numbers. Maximizing the
ij ∗ji ij ∗ji
fidelity (3) for ̺ = χ χ,̺ = ϕ ϕ with this number of parameters is far from a possible analytical solution
in out
| ih | | ih |
but feasible in terms of semidefinite programming. For i = j it is advantageous to introduce the decomposition
6
a P +a P = [a ](P +P )+ [a ](iP iP )andsimilarlyforthesymmetricalpart. Withtheabove
ij Aij ∗ij Aji ℜ ij Aij Aji ℑ ij Aij− Aji
defined variables the fidelity to be maximized has the form (leaving out the zero parameters)
1
F = a2c2(s +a )+(1 c2)(1 a2)(s +a ))+c2(1 a2)s +(1 c2)a2s +ac (1 a2)(1 c2)a+ ,
2(cid:16) 11 11 − − 44 44 − 22 − 33 p − − 7(cid:17)
(7)
where a+ = [a ]. The resulting fidelity is depicted in Fig. 1. First, we notethat for a c the resultcorrespondsto
7 ℜ 41 ≤
the analytical result found in [19] which, for our bipartite case, has the form
2
F = ac+ (1 a2)(1 c2) . (8)
(cid:16) p − − (cid:17)
Thereasonisthattheoptimalfidelityfoundin[19]isdependentonlyontheSchmidtnumbersoftheinputandtarget
state and thus it is automatically locally covariant. If we do not consider the parameters of R which are shown
to be zero (yielded from the semidefinite program) a general form in the Kraus representationMcan be in principle
found (R can be diagonalizedwith the help of a softwarefor the symbolic manipulations). But it appears that this
decomposMition is too complex and for our purpose it is not necessary to present it. The only comment is deserved
by the identity map which covers the whole region of parameters where Eq. (8) is valid. This is in contrast with
the original work [19] where the map is not the identity due to the knowledge of parameters a,c. In reality, this
trivial map appears to be the covariant and optimal map for a bit larger region as depicted in Fig. 2. It follows that
under the realmofthe identity mapno optimal covariantCP map exists. The remaining partofthe parameterspace
4
0 0
0.2 0.2
0.4 0.4
c c
0.6 0.6
0.8 0.8
1 1
1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0
a a
FIG. 2: A2D view on Fig. 1(on theright) together with theindication where thetrivial identity is theoptimal map (thered
area on the left). The bluepart corresponds to various non-unitmaps.
of a,c shows that in spite of the allowance of the perfect deterministic conversion by the majorization criterion the
semicovariant transformation does not reach the maximal fidelity. We intentionally left out the word LOCC because
thesecondinterestingaspectisthatforthewholeparameterspacetheLOCCconditionisunnecessary. Inotherwords,
thereareonlyLOCCsemicovarianttransformationsortheidentitymapwhichisalso(trivially)LOCCsemicovariant.
We confirm the existence of another fundamental no-go process saying that it is not possible to construct a CP map
perfectly copying a partially or totally unknown quantum state to a generally different quantum state even if the
majorization criterion allows us to do it (attention to the related problem was called in [23]). The impossibility is
easy to show by considering the following tiny lemma valid not only for the investigated dimension d=2:
Let M be a unitary and covariant map, i.e. ϕ = M χ holds for two arbitrary qudits χ , ϕ . Then, from the
| i | i | i | i
covariance follows MU χ = U ϕ = UM χ [M,U] = 0. We suppose that this holds for all U SU(d) and
| i | i | i ⇐⇒ ∈
then by one of Schur’s lemma M = c11. Considering the requirement of unitarity of M it follows c = 1 and thus
ϕ = χ .(cid:3)
| i | i
We confirmed this lemma in Fig. 1 where the fidelity is equal to one only if a= c and we may reflect the calculated
optimal values of the fidelity as a refinement and quantification how much is the above process impossible.
Note that the majorization criterion [17] was developed with respect to the degree of entanglement (the Schmidt
number)butreliesonthecompleteknowledgeoftheconvertedstatewhatisatvariancewiththecovariantrequirement
where no particular state is preferred. The situation is a bit similar to quantum cloning where if we know the
preparation procedure of a state to be cloned then there is no problem to make an arbitrary number of its perfect
copies.
Another worthyaspectis that the intervalofa andc goesfromzeroto one thus coveringthe targetstates with the
same Schmidt number more than once. Nevertheless, the fidelity is different in such cases (compare e.g. the target
states 00 and 11 ). Infact,tocompletelydescribethe(semi)covariantpropertiesofthetypepresentedinthisarticle
| i | i
we should not distinguish input and target states by their Schmidt numbers but rather to fully parametrize them in
SU(2) SU(2) representation for every a,c (0,1/√2). But by relying on the lemma above we expect that this
⊗ ∈
situation does not bring anything surprising into our discussion. Also, due to the (semi)covariance we have actually
described potentially interesting transformations between χ =a 01 +√1 a2 10 and ϕ =c 01 +√1 c2 10 .
| i | i − | i | i | i − | i
B. Full LOCC covariant transformations
As the second case we investigate a full local covariancewhere, first, both qubits froman input two-qubitstate χ
| i
are rotated simultaneously and, second, both qubits are rotated independently. The covariance with respect to these
two types of rotation is required.
The covariance condition in the first case is V =V =U U and thus
1 2
⊗
[R ,U U U U ]=0 [R˜ ,U U U U]=0. (9)
∗ ∗
M ⊗ ⊗ ⊗ ⇐⇒ M ⊗ ⊗ ⊗
5
1
0.8
0.6
F
0.4
0.2 1
0 0.9
0.8
1
0.9 0.7
0.8 0.6
0.7 0.5
0.6
0.5 0.4
0.4 0.3
0.3 0.2
0.2
0.1 0.1 a
c 00
FIG. 3: The fidelity for the optimal and full locally covariant LOCC transformation between χ = a 00 +√1 a2 11 and
| i | i − | i
ϕ =c 00 +√1 c2 11 .
| i | i − | i
Employing the fact that
2
SU(2)⊗j=41/2 = cJD(J) (10)
JM=0
with c (2,3,1) we find the basis vectors of all irreducible subspaces (summarized in Tab. I) and construct isomor-
J
∈
phisms P between equivalent species
2 cJ
R˜ = d P . (11)
M JM=0kM,l=1 Jkl Dk(Jl)
Choosingthe parametersd we requireR to be a semidefinite matrix. We calculate the fidelity for the same kind
Jkl
of input/target states from the previous subMsection yielding
2 1 1
F = ac+ (1 a2)(1 c2) d + d + c2(1 a2)+(1 c2)a2 d . (12)
022 211 211
(cid:16) p − − (cid:17) (cid:18)3 6 (cid:19) (cid:0) − − (cid:1)
Running an appropriate semidefinite program for maximizing F we are able to get analytical results both for the
fidelity and the CP map in the Kraus form. It appears that many of the coefficients d are zero and thus Eq. (12)
Jkl
simplifiesaswellastheconstraintsgivenbythetracepreservingcondition. AsfarastheLOCCconditionthesituation
here is that the CP maps with and without the posed condition are different but both give the same optimal fidelity.
It can be shown that the LOCC condition in this case is just a dummy constraint determining the value of a free
parameter in the resulting map (see the parameter d in Eq. (14)). Then
011
2 1 2 3
F =max ac+ (1 a2)(1 c2) , ac+ (1 a2)(1 c2) + c2(1 a2)+a2(1 c2) (13)
(cid:20)(cid:16) p − − (cid:17) 10(cid:16) p − − (cid:17) 5(cid:0) − − (cid:1)(cid:21)
andthe correspondinggraphis in Fig. 3. It is noteworthythat there arejust twotypes of CP covariantmaps for two
investigatedintervals of a,c correspondingto the different fidelity functions in (13). The identity map is the first one
6
TABLE I: Orthogonal basis vectors of all irreducible subspaces of SU(2)⊗j=41/2.
Total momentum J Irreduciblesubspace D(J) Basis vectors
kl
0 D(0) 1 01 10 01 10
11 2| − i| − i
0 D(0) 1 `0011 1 01+10 01+10 + 1100 ´
22 √3 | i−2| i| i | i
1 D(1) 1 01 10 00
11 √2| − i| i
1 01 10 01+10
2| − i| i
1 01 10 11
√2| − i| i
1 D(1) 1 00 01 10
22 √2| i| − i
1 01+10 01 10
2| i| − i
1 11 01 10
√2| i| − i
1 D(1) 1(00 01+10 01+10 00 )
33 −2 | i| i−| i| i
1 (0011 1100 )
−√2 | i−| i
1(01+10 11 11 01+10 )
−2 | i| i−| i| i
2 D(2) 0000
11 | i
1(00 01+10 + 01+10 00 )
2 | i| i | i| i
1 (0011 + 1100 + 01+10 01+10 )
√6 | i | i | i| i
1(01+10 11 + 11 01+10 )
2 | i| i | i| i
1111
| i
and the conclusion from the previous case holds. The second map is described by the set of the Kraus operators
0 1 1 0 0 0 0 0 0 0 0 0
−
0 0 0 0 0 0 0 0 0 1 1 0
A1 = 1−d3011 0 0 0 0,A2 = 1−d3011 0 0 0 0,A3 = 1−1d2011 0 −1 1 0,
q q q
−
0 0 0 0 0 1 1 0 0 0 0 0
   −   
0 0 0 0 1 0 0 0 0 0 0 0
0 1 1 0 0 1 1 0 1 0 0 0
A4 = √d2011 0 −1 −1 0,A5 = √1100 −−1 −−1 0,A6 =q230−−1 0 0 0,
0 0 0 0 0 0 0 1  0 1 1 0
     
0 1 1 0 0 0 0 1 0 0 0 0
− −
0 0 0 1 0 0 0 0 0 0 0 0
A = 3  ,A = 3 ,A = 3 , (14)
7 20 0 0 0 1 8 5 0 0 0 0 9 5 0 0 0 0
q q q
0 0 0 0 0 0 0 0 1 0 0 0
     
where d011 is a free parameter from the decomposition (11). The trace-preserving condition 9i=1A†iAi = 11 is
satisfied [28]. P
Letus proceedto the secondcasewhere weconsiderindependent unitary rotationsonbothqubits ofthe pair,that
is V =V =U U . Derived analogously as before, it follows
1 2 1 2
⊗
[R˜ ,U U U U ]=0 (15)
1 1 2 2
M ⊗ ⊗ ⊗
with the decomposition in a particularly simple form
R˜ =p P P +p P P +p P P +p P P , (16)
1 A A 2 A S 3 S A 4 S S
M ⊗ ⊗ ⊗ ⊗
whereP ,P aretheprojectorsintoasymmetricalandsymmetricalsubspaces,respectively[27]. Theresultingfidelity
A S
equation (again independent on the LOCC condition) can be derived analytically
2 1 2 4
F =max ac+ (1 a2)(1 c2) , ac+ (1 a2)(1 c2) + c2(1 a2)+a2(1 c2) (17)
(cid:20)(cid:16) p − − (cid:17) 9(cid:16) p − − (cid:17) 9(cid:0) − − (cid:1)(cid:21)
withthe picture lookingsimilarlyasinFig.3. Theachievedfidelityis evenlowerdue tothe strongerrequirementson
the covariancepropertiesin Eq.(15) in comparisonwith Eq.(9). As in the previouscase, there are two maps for two
7
different fidelity functions, one of them being the identity map. The Kraus decomposition of the nontrivial map is
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
10 1 0 0  √2 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0
A1 = 30 −0 1 0,A2 =−03 0 0 0,A3 = √2 0 0 0,A4 =0 0 0 0,A5 =0 2 0 0
 −    − 3     3 
0 0 0 1  0 0 √2 0  0 √2 0 0 2 0 0 0 0 0 0 0
   3   3  3   
(18)
and A6 =A†2,A7 =A†3,A8 =A†5,A9 =A†4.
IV. COVARIANT LOCC COMMUNICATION PROTOCOL
Letustrytoapplythepreviousconsiderationstothesolutionofthefollowingcommunicationproblem. Supposethat
the two-qubit state χ =a 00 +√1 a2 11 was locally and unitarily modified on Alice’s side and then distributed
| i | i − | i
between Alice and Bob. Next imagine that the distributor of this state is confused and oblivious and he wanted
originally to modify Bob’s part of the state. Moreover, he forgot which unitary modification was done. Since Alice
andBobareseparatedtheonlypossibilitytorectifythedistributor’smistakeisLOCCcommunicationbetweenthem.
In other words, they would like to perform the following transformation
LOCC
χ′ =(U 11) χ (11 U) χ = ϕ′ (19)
| i ⊗ | i → ⊗ | i | i
such that the LOCC transformationwill be equally and maximally successful irrespective of U. Generally, this is the
problem of sending an unknown local unitary operation between branches of a shared bipartite state. Notice that if
χ is a maximally entangled state then the task changes to finding a transpositionof the unitary operationU due to
| i
the well known relation
(U 11) 00+11 =(11 UT) 00+11 . (20)
⊗ | i ⊗ | i
The covariantcondition in the Jamio lkowskirepresentationreads
[R ,11 U U∗ 11]=0 (21)
M ⊗ ⊗ ⊗
using decomposition (6) and the unitary modification R˜ =SR S† with S =(11 SWAP σY)(11 11 SWAP).
Again, the figure of merit is the fidelity which now has tMhe formM ⊗ ⊗ ⊗ ⊗
1
F = a4(s +a )+(1 a2)2(s +a ) +a2(1 a2)(s +s +a+ s+), (22)
2 11 11 − 44 44 − 22 33 7 − 7
(cid:0) (cid:1)
wherea+ = [a ],s+ = [s ]. OnemayfindageneralformofthismapintermsoftheKrausoperatorsinAppendix.
7 ℜ 41 7 ℜ 41
If we first run the corresponding semidefinite program without the LOCC condition we get the fidelity equal to one
for all a. This has a reasonable explanation because if we allow the nonlocal operations there exists a universal and
always successful unitary operation – SWAP. The inspection of the particular R confirms this inference. After
imposing the LOCC condition the resulting fidelity is depicted in Fig 4. This resMult is noteworthy because we see
that the LOCC CP map is the most successful for the factorized states (a = 0,1 F =2/3) while it holds F = 1/2
∼
for the maximally entangled states. The reasonlies in Eqs. (20) and (19). If χ is a maximally entangled state then
| i
a local unitary action passes the whole local orbit whereas for non-maximally entangled states the unitary action on
one branch is not sufficient for the attainment of all possible partially entangled states characterized by the same
Schmidt number a. We may conclude with an intriguing claim that in case of our protocol it is better for Alice and
Bob to share a factorized state instead of a maximally entangled state. Let us stress that the optimal map is not
trivially identical for any value of the parameter a in the input state χ .
| i
V. CONCLUSION
InthisworkwestudiedtheLOCCtransformationsbetweentwo-qubitbipartitestatescharacterizedbytheirSchmidt
numbers. In addition to the obvious CP requirement, we looked for the covariant maps which maximize the fidelity
between an input and a target state. Moreover, we supposed that we had just a single copy of the input state at
our disposal. The studied covariancecan be divided into two groups: so called semicovariancewhere we required the
independence of the input state regarding the action of SU(2) representationon one of the input qubits. The second
investigatedpossibilityweretwocasesoffullcovarianceconditionwherethe independenceandoptimalityofthestate
8
0.66
0.64
0.62
0.6
F 0.58
0.56
0.54
0.52
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a
FIG. 4: The fidelity of the protocol for ‘handing over’ a local unitary operation between branches of a partially entangled
two-qubit pair. Theentanglement of theshared pair is characterized bytheSchmidt number a.
transformation had been examined with respect to two (equivalent and nonequivalent) SU(2) representations acting
on both branches of the input bipartite state.
We employedthe methods of semidefinite programming which, in spite of being a numericalmethod, enables us to
find totally or partially general analytical solutions for the fidelity and for the corresponding LOCC CP maps. We
have found that, first, due to the covariance conditions there are no possible perfect state transformations even if
the majorizationcriterionallowsthem andwith the calculatedoptimalfidelity wequantifiedthe ‘maximalallowance’
of the considered transformations. Second, we have shown that there only exist LOCC covariant transformations.
Hence, sincethis conditionisunnecessarythis kindoftransformationcanbe ratedasanotherbasicprocessforbidden
bythelawsofquantummechanics. Wehavealsoconnectedourworkwiththeearlierworksonsocalledfaithfulsingle-
copy state transformations [19]. Notably, for the corresponding subset of the investigated parameter area the same
analyticalresults for the fidelity werederivedbut under the localunitary covariantcircumstances. Consequently, the
formsoftheparticularCPmapsaredifferentfrompreviouslyderivedputtingthisproblemintoadifferentperspective.
Finally,weillustratedthesemethodsonanapplicationofthecommunicationprotocolforLOCC‘handingover’ofa
localunitaryoperationfromonebranchofasharedtwo-qubitbipartitestatetoanotherwithoutitsactualknowledge.
Intriguingly,is has been shownthat the best results (in terms of the fidelity between aninput and a targetstate) are
achieved if both parties share one of the considered factorized states 00 or 11 and not the maximally entangled
| i | i
state.
Even if for general multipartite states the PPT condition used here is not equivalent to the LOCC condition, the
described methods might be useful for this kind of study as well, for example, to help clarifying the role of the PPT
operations and the transformation properties of these states.
Acknowledgments
The author is very indebted for discussions and support from R. J´auregui and for comments from R. Demkowicz-
Dobrzan´ski.
APPENDIX A
Considering
p1,2 = −s11+s44±√s211−2s2+s11s44+s244+4(s+7)2 (A1)
7
p3,4 = −a11+a44±√a211−2a2+a11a44+a244+4(a+7)2 (A2)
7
9
and
1/2
1
d = s + s + s2 2s s +s2 +4(s+)2 (A3)
1 √2(cid:18) 11 44 q 11− 11 44 44 7 (cid:19)
1 1/2
d = s + s s2 2s s +s2 +4(s+)2 (A4)
2 √2(cid:18) 11 44− q 11− 11 44 44 7 (cid:19)
1/2
1
d = a + a + a2 2a a +a2 +4(a+)2 (A5)
3 √2(cid:18) 11 44 q 11− 11 44 44 7 (cid:19)
1 1/2
d = a + a a2 2a a +a2 +4(a+)2 (A6)
4 √2(cid:18) 11 44− q 11− 11 44 44 7 (cid:19)
d = √s (A7)
5 22
d = √s (A8)
6 33
we may write the Kraus operators for the problem in Sec. IV as
p 0 0 0 0 0 0 0
1
0 0 p 0  1 0 0 0
A1 = √d12√11+p21 0 1 −01 0,A2 = √d12√11+p21 −0 0 0 0,
 −   
0 0 0 1  0 1 0 0
   
1 0 0 0 0 0 0 0
0 0 1 0   1 0 0 0
A4 = √d22sig√n(1p+2−p21p1)0 p1 −0 0 ,A5 = √d22sig√n(1p+2−p21p1)−0 0 0 0,
   
0 0 0 p   0 1 0 0
1
 −   − 
p 0 0 0 1 0 0 0
3
−
0 0 p 0  0 0 1 0
A7 = √d32√11+p23 0 1 03 0,A8 = √d42sig√n(1p+3−p23p4) 0 p3 −0 0,
   
0 0 0 1  0 0 0 p 
3
   
0 1 0 0 0 0 0 0 0 0 0 1
−
0 0 0 1 0 1 0 0 0 0 0 0
A = d5 ,A =d ,A =d ,
9 √20 0 0 0 10 50 0 0 0 11 50 0 0 0
     
0 0 0 0 0 0 0 0 0 0 0 0
     
0 0 0 0 0 0 0 0 0 0 0 0
 1 0 0 0 0 0 0 0 0 0 0 0
A12 = √d62−0 0 0 0,A13 =d60 0 0 0,A14 =d60 0 1 0, (A9)
     
 0 1 0 0 1 0 0 0 0 0 0 0
     
and A3 =A†2,A6 =A†5. The maps satisfy 1i=41A†iAi =11 if the trace preserving condition on the Jamio lkowskimap
is posed. Similarly to Eq. (14), the KrausPoperators are not in their apparent LOCC form but can be transformed
into it.
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