Unified quantum no-go theorems and transforming of quantum states in a restricted set

0.47 MB·

by  Ming-Xing Luo

Unified quantum no-go theorems and transforming of quantum states in a restricted
set
Ming-Xing Luo1,2, Hui-Ran Li2, Hong Lai3, Xiaojun Wang4
1 Information Security and National Computing Grid Laboratory,
Southwest Jiaotong University, Chengdu 610031, China
2 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
3 School of Computer and Information Science, Southwest University, Chongqing 400715, China
4 School of Electronic Engineering, Dublin City University, Dublin 9, Ireland
The linear superposition principle in quantum mechanics is essential for several no-go theorems
suchastheno-cloningtheorem,theno-deletingtheoremandtheno-superposingtheorem. Itremains
7
an open problem of finding general forbidden principles to unify these results. In this paper, we
1
investigate general quantumtransformations forbidden or permitted by thesuperposition principle
0
2 for various goals. First, we prove a no-encoding theorem that forbids linearly superposing of an
unknown pure state and a fixed state in Hilbert space of a finite dimension. Two general forms
r includetheno-cloningtheorem,theno-deletingtheorem,andtheno-superposingtheoremasspecial
a
M cases. Second, we provide a unified scheme for presenting perfect and imperfect quantum tasks
(cloning and deleting) in a one-shot manner. This scheme may lead to fruitful results that are
completely characterized with the linear independence of the input pure states. The generalized
9
upper bounds for the success probability are proved. Third, we generalize a recent superposing of
1
unknownstates with fixedoverlaps when multiple copies of theinput states are available.

]
h PACSnumbers: 03.67.-a,03.65.Ta,03.67.Hk
p
-
t
n I. INTRODUCTION unknown states. A recent theorem [17, 18] shows that
a it is impossible to create a superposition state of un-
u known states using a universal quantum transformation
q In quantum mechanics, the no-cloning theorem as a
[17]. This no-gotheorem is different from the no-cloning
[ well-known fact asserts that an arbitrary unknown pure
theorem [1], the no-deleting theorem [10] or the special
state[1]ornoncommutingmixedstate[2]cannotbeper-
2 no-superposing theorem that forbids superposing an un-
fectlycreatedusingauniversalquantumtransformation.
v known state and its orthogonal complement [19]. It re-
This non-trivialexception is derivedfrom the superposi-
6
mainsachallengetoexploreintrinsicconnectionsamong
6 tion principle of quantum states and may be implied by
these no-go theorems.
1 the no-communication theorem which forbids the clas-
4 sical information transmission using only quantum en- In comparison to recovering perfect resultant, Buˇzek
0 tanglement. Although the no-cloning theorem prevents and Hillery provided a universal quantum transforma-
1. several important extensions of classical results such as tionfor imperfectcloning [20]. The partialtraceoneach
0 classicalerrorcorrections[3,4],itplaysakeyroleinvar- copy of their resultant is a mixed state with the opti-
7 ious quantum applications, especially in quantum secure mal fidelity [21–24]. Their scheme had triggered lots
1 communication. Theno-cloningtheoremplaysakeyrole of investigations on quantum cloning tasks that include
:
v incertifyingtheunconditionalsecurityofquantumcryp- N M cloning of qubits [21], d-dimensional universal
→
i tography [5–9]. A universal cloning machine will violate cloning [25–28], asymmetric cloning [29–33], mixed-state
X
the no-teleportation theorem that forbids converting a cloning [2, 34–36], experimental cloning [37–44], and rel-
r quantum state into classical bits. Unitary cloning also ative discussions and applications [45, 46]. The cloning
a
implies the possibility of deleting quantum information fidelities of these imperfect schemes may be improvedto
which is impossible. In fact, with the quantum super- unit if input states are chosen from a finite set of the
position principle, unknown quantum states of a finite whole space. In particular, the linear independence of
dimension cannot be unitarily transformed into a fixed input pure states has been proved to be a sufficient and
state,i.e.,thequantumno-deletingtheorem[10]. Ingen- necessary condition for perfect cloning with a unitary
eral, the no-cloning theorem has no relationship to the and post-selection process [47, 48]. The optimal success
no-deleting theorem because the post-selection permit- probability is the same as that of state discriminations
ted in the cloning task is forbidden in the deleting task. [49–52]. The linear independence has also been used for
the task of superposing multiple copies of input states
Thekeyofthe no-cloningtheoremandtheno-deleting
[53]. Similar resultsholdforperfect deleting. These per-
theoremis the superpositionprinciple of quantumstates
fect schemes may be unified in a one-shot manner using
[11]. It is also essential for nontrivial quantum applica-
multiple copies of input states [54].
tions including Shor’s factoring algorithm [12], Grover’s
searchalgorithm[13],quantum cryptography[14], quan- Recently, a constructive scheme [17] shows that a uni-
tum metrology [15], and Boson sampling [16]. However, versal superposing transformation exists for unknown
this principle is unsuitable for the task of superposing states chosen from two sets where all the states of each
2
sethave afixed overlapwith a knownstate. The scheme II. UNIFIED NO-GO THEOREMS
shows a new insight of the linear independence, i.e., a
superposing scheme exists for a collection (generally un-
In this section, we explore general forms of the no-
countable)offinitesetsconsistingoflinearlyindependent
cloning theorem, the no-deleting theorem and the no-
states. This is a great improvement for the quantum
superposing theorem.
tasks definedon a finite setof quantumstates. It should
be interesting to investigate whether or not similar gen-
eralizations hold for other tasks of cloning or deleting
A. No-encoding theorem
[47, 48, 53, 54].
In this paper, our motivation is to generalize the no-
Inwhatfollows we only considerpure states. In quan-
cloning theorem [1], no-deleting theorem [10] and no-
tum mechanics, a pure state provides a probability dis-
superposingtheorem[17]. Althoughthey aredifferentin
tribution for the value of eachobservable. In the mathe-
general and have no explicit relationships among them,
matical formulation, from Riesz representation theorem
we hope to present a unified form which may provide
[55] each pure state corresponds to a ray in a Hilbert
new insights of these no-go theorems. Moreover, new
space while each observable quantity is associated with
constructions will be proposed to generate different de-
a self-adjoint (or Hermitian) and positive semi-definite
siredstates usingthe linearindependence ofinput states
operator. In this paper,we only considerthe normalized
thatbelong to restrictedsets. The maincontributionsof
pure states. Thus each pure state is uniquely denoted
this paper are listed as follows:
by a density matrix ρ on Hilbert space H with the di-
ψ
mension dim(H) = n 2, where ψ is its vector repre-
(1) Wepresentnewno-gotheorems. Thefirstoneisthe sentative up to a globa≥l phase fact|orieiθ, i.e, ρ =ρ .
no-encoding theorem that forbids linearly super- ψ eiθψ
Denote the vector ψ φ if ρ =ρ , i.e., ψ = φ up
posing of an unknown pure state and a fixed state | i∝| i ψ φ | i | i
to a global phase.
(knownorunknown)inthesamespaceofafinitedi-
Let (H ,H )be the setofcompletely positive(CP)
mensionfromauniversalquantumtransformation. maps wCPhich1tran2sform pure states on Hilbert space H
Its generalizations include the no-cloning theorem, 1
into pure states on Hilbert space H . All the uni-
the no-deleting theorem and the no-superposing 2
tary maps, which are denoted as SU(H ), are CP maps
theorem as special cases. 1
that preserve the inner product of the Hilbert space.
In particular, from Stinespring’s Theorem [56], each
(2) We present several schemes to unify the perfect (H ,H ) may be realized by a unitary map
1 2
andimperfecttasksofcloninganddeletingforpure F ∈ SCUP(H H H ) and a projection map
1 A P
statesinafiniteset. Alltheschemesarecompletely U ∈(H H ⊗H ,⊗H ), where H is an ancillary sPpac∈e
1 A P 2 A
characterized by the linear independence of input CanPdH ⊗isano⊗therancillaryspace(namedasprobespace
P
states. Generalizedupper bounds are provedusing
forsimplicity)usedfordefiningprojectors. Foreachmap
the matrix norm and state metric, respectively. : H H with Hilbert spaces H and H , denote
1 2 1 2
F : ρ →ρ for ρ on H and ρ on H as its detailed
ψ φ ψ 1 φ 2
(3) We propose a scheme to superpose pure states in Fmappin7→g of (ρ )=ρ .
ψ φ
a finite set. We also present some generalized su- F
perposing schemes when multiple copies of input Theorem 1 Let α,β be nonzero complex constants sat-
states are available. isfying α2+ β 2 =1,andρ beafixedpurestate(known
φ
or unkn|ow|n)|on| Hilbert space H with dim(H) 2. There
Therestofthepaperisorganizedasfollows. InSec. II, does not exist CP map (H2,H) for all≥pure states
we present new no-go theorems. The first one is the no- ρ on H such that F ∈CP
ψ
encoding theorem. Namely, an arbitrary unknown state
and a fixed state in the same space of a finite dimension (ρ ρ )=ρ , (1)
ψ φ ϕ
F
cannotbelinearlysuperposedusingauniversalquantum
transformation with a nontrivial probability. This the- where the vector representative of ρ is given by ϕ
ϕ
| i ∝
orem will be further extended for multiple copies of the √r(αψ +β φ ) and r is a normalization constant de-
| i | i
purestates. InSec. III,severalquantumtransformations pendent of α,β, ψ and φ .
| i | i
will be presented to unify imperfect and perfect tasks of
cloninganddeleting withcontrollableerrors. The gener- The result in Theorem 1 holds for a fixed pure state
alizedboundsofthesuccessprobabilitywillbeprovedus- ρ which may be known or unknown (randomly chosen
φ
ing the matrix norm and state metric, respectively. Sec. fromH). Thistheoremshowsthatonecannotuseafixed
IV contributes new superposing schemes that create the purestatetolinearlysuperpose(orlinearlyencode)with
linear superposition of unknownpure states chosenfrom an unknown state, which may be randomly chosen from
afiniteset,orarestrictedsubspacewhenmultiplecopies H according to the Haar measure. It is also a general
of the input states are available. The last section con- no-superposing theorem that forbids the creation of the
cludes this paper. linearsuperpositionofanunknownstateandafixedstate
3
in the same space. Theorem 1 is not a corollary of the ρ . Φ˜ and Φ˜ aretherespectivevectorrepresentative
1 0 1
| i | i
non-superposingTheorem[17]whileitisconverselytrue. of the failure states ρ and ρ that should be annihi-
Φ˜0 Φ˜1
Proof of Theorem 1. Note that each quantum op- lated by the projector P P , i.e, tr(ρ P P ) =
eration can be represented by a unitary operation and tr(ρ P P )=0. | 1ih 1| Φ˜0| 1ih 1|
post-selection of projective measurements [48]. Thus FoΦ˜r1|an1yihpu1|re state ρ with the vector representative
h
a CP map is represented by a unitary transformation h a0 +b1 with a2+ b2 =1, fromEq.(3), we can
and the post-selection of projective measurements on a |obita∝in | i | i | | | |
joint system consisting of the input states and ancillary
states. Assume that there exists a CP map satis- U h Σ P p′′ ϕ 0 P + p′′ Φ˜′′ , (6)
fying Eq.(1). From Stinespring’s Theorem [56F], there | i| i| 0i∝ 0| 1i| i| 1i 1| i
exist two pure states ρΣ (including the pure state ρφ where the vector reppresentative ϕ1 ispdefinedby ϕ1
| i | i∝
as its subsystem) and ρΣ′ on ancillary Hilbert space √r′′(αh + β 0 ) and r′′ is a normalization constant.
HA, one probe state ρP on probe space HP, a unitary Φ˜′′ is|thie vec|toir representative of the failure state ρ
map : H H H H H H and a pro- | i Φ˜′′
A P A P thatshouldbeannihilatedbytheprojector P P ,i.e.,
jectiveU map⊗ : H⊗ HA→ HP⊗ H⊗ HA such that tr(ρ P P )=0, and p′′ is the success p|ro1biahb1il|ity of
(ρψρΣρPP)=ρϕ⊗ρΣ′ ho⊗lds for→all the⊗states ρψ on H. recoΦv˜′e′r|in1gihpu1r|e state ρ . 0
PIn◦thUe following, we only need to prove the nonexistence ϕ1
Moreover, using the linearity of quantum operations,
of the unitary map .
Now, assume therUe exists a unitary map (H it follows from Eqs.(4) and (5) that
H H ,H H H ) for all the states ρU ∈onCHP suc⊗h
A P A P ψ U h Σ P
⊗ ⊗ ⊗ 0
that | i| i| i
ϕ 0 P +a√p Φ˜ +b p′ Φ˜ , (7)
∝ | 2i| i| 1i 1| 0i 1| 1i
(ρ ρ ρ )=ρ , (2)
U ψ Σ P0 Ψ p
where the vector ϕ2 is given by ϕ2 = a√p0 0 +
where the vector representative of ρ is given by Ψ | i | i | i
√p0√r(α|ψi+β|φi)|Σ′iA|P1i+√p1Ψ|Φ˜i and r is a| nio∝r- b p′0|χ′i. FromEqs.(6)and(7),wegeta( pp0′0eiθ−α)+
malization constant. In Eq.(2), Σ′ is the vector repre- βpbeiθ = αβ with θ [0,2π]. This leads toqthe fixed val-
| i ∈
sentativeofancillarystateρΣ′ thatmaybedependentof ues of a and b from |a|2+|b|2 =1, where p0,p′0, |α| and
α,β, ψ and φ . P0 and P1 are the respective vector β are some fixed constants. This is a contradiction to
repres|enitative|oif o|rthiogona|l puire states ρP0 and ρP1 on t|h|e assumption of ρh. (cid:3)
probe space H . p is the success probability of recover-
P 0
ing a superposition state with the vector representative
√r(αψ +β φ ) that is dependent on α,β, ψ , φ , and B. Unified no-go theorems
| i | i | i | i
satisfies p >0 and p +p =1. Φ˜ is the vector repre-
0 0 1
| i
sentative of a general failure state ρ , that will be anni-
Φ˜ Theorem 1 presents the nonexistence of linear super-
hilatedbythe projector P P ,i.e,tr(ρ P P )=0.
| 1ih 1| Φ˜| 1ih 1| posing of an unknown state and a fixed state in Hilbert
Here, we only consider the special case of ρΣ′ = ρ0. space of a finite dimension. Similar results hold for dif-
The proof of the general state ρΣ′ is shown in Appendix ferent input states and output states.
A. In this case, Eq.(2) may be rewritten into
U ψ Σ P √p χ 0 P +√p Φ˜ , (3) Theorem 2 Let α,β be complex constants satisfying
| i| i| 0i∝ 0| i| i| 1i 1| i α2 + β 2 = 1 and β = 1, k be an integer, and ρ be
Φ
| | | | | | 6
whereU isthematrixrepresentativeoftheunitarymap- a fixed pure state (known or unknown) on Hilbert space
ping and the vector representative of ρχ is defined by Hn with dim(H) 2. Then, there does not exist CP map
U ≥
χ √r(αψ +β φ ). (Hk,Hn) for all the states ρ⊗k on Hk such that
| iF∝or simpl|iciity, w|e oinly consider ρ =ρ . A similar re- F ∈CP ψ
φ 0
sultmaybeprovedusingagenerallyfixedstateρφ,which (ρ⊗k)=ρ (8)
will be discussed in the next subsection. Since Hilbert F ψ Ψ
space of qubit states is a subspace of general Hilbert
if n,k, β and satisfy one of the following conditions:
space H with dim(H) > 2, it is sufficient to prove the F
nonexistence of Eq.(3) for the qubit space. For the basis (i) n 1 and 0< β <1;
states ρ ,ρ on H (qubit space) as input states, Eq.(3) ≥ | |
0 1
leads to (ii) n>k and β =0;
U 0 Σ P0 √p0 0 0 P1 +√p1 Φ˜0 , (4) (iii) n < k, β = 0, the output state is restricted to
| i| i| i∝ | i| i| i | i
U|1i|Σi|P0i∝ p′0|χ′i|0i|P1i+ p′1|Φ˜1i, (5) mρΨaρp0⊗;k+n−m, and F is restricted to the unitary
where the vector represpentative χ′ is depfined by χ′
α1 +β 0 . Inthese equations,p| aindp′ arethe r|espiec∝- where the vector representative of ρ is given by Ψ
ti|veisucc|esisprobabilityoftransfo0rmingp0urestateρ and √r(αψ ⊗n+β Φ ) and r is a normΨalization const|anit.∝
0
| i | i
4
Theorem 3 Let α,β be nonzero complex constants sat-
isfying α2+ β 2 =1 and β =1, and ρ be a fixedstate
φ
(known|or| un|kn|own) on H|ilb|e6rt space H with dim(H)
2. Then there dose not exist CP map (Hk,Hn≥) NoN-Toch-locelnooirnneigmng NToh-deeolreetming
F ∈ CP Theorem
for all the states ρ⊗k on Hk such that Theorem 2 Theorem 3
ψ
Theorem1
(ρ⊗k)=ρ⊗n (9)
F ψ ϕ No-superposing
Theorem
if n,k, β and satisfy one of the following conditions:
F
(i) n 1 and 0< β <1;
≥ | |
(ii) n>k and β =0;
FIG.1: (Coloronline)Schematicrelationshipsofdifferentno-
go theorems. The set bounded by the black line denotes the
(iii) n < k, β = 0, the output state is restricted to
results ensured by Theorem 2. The set bounded by the red
ρ⊗ϕnρ⊗0k−n, and F is restricted to unitary map; line denotes theresults ensured by Theorem 3.
where the vector representative of ρ is given by ϕ
ϕ
| i ∝
√r(αψ +β φ ) and r is a normalization constant. with the matrix representativeU for all the states ρ on
| i | i ψ
H such that
Theorem 2 may be viewed as no-superposing of the
product state ρ⊗ψk and a fixed state ρΦ while Theorem 3 U|ϕi⊗k|Σi|P0i
may be viewed as the combinationof the no-cloning and √p Ψ 0 ⊗N+k−n P +√p Φ˜ , (10)
0 1 1
the no-encoding of an unknown pure state and a fixed ∝ | i| i | i | i
where the vector representative of ρ is given by Ψ
state. Moreover, Theorem 2 may be viewed as the no- Ψ
cloningofρ⊗n withafixederrorterm Φ whileTheorem √r(αϕ ⊗n + β Φ ) and r is a normalization con|stain∝t.
ψ | i Σ is|thie vecto|r riepresentative of an ancillary state ρ
3 may be viewed as the no-cloning of ρ with a fixed er- Σ
ψ |oniHilbertspaceH =HN withN >n,and P , P are
ror term φ . Theorems 2 and 3 are different except the A 0 1
| i | i
| i the respective vector representative of orthogonal states
specialcaseofβ =0,whichleadstothe no-cloningtheo-
ρ and ρ on probe space H with dim(H ) 2.
rem [1] (n> k) or the no-deleting theorem [10] (n <k). P0 P1 P P ≫
The relationships among Theorems 2 and 3 and previ- |Φ˜i is the vector representative of the failure state ρΦ˜
ous no-go theorems [1, 10, 17] are schematically shown thatshouldbeannihilatedbytheprojector P1 P1 ,i.e.,
| ih |
in Fig.1. From this Figure, the set defined by respec- tr(ρΦ˜|P1ihP1|)=0.
tive Theorem 2 and 3 includes the no-cloning theorem For any state with the vector representative eiθ ϕ , it
| i
[1], the no-deleting theorem [10], Theorem 1, or the no- follows from Eq.(10) that
superposition theorem [17] as a special case. Formally, U(eiθ ϕ ⊗k Σ P )
0
we obtain the implications as follows: | i | i| i
p′ Ψ′ 0 ⊗N+k−n P + p′ Φ˜′ , (11)
∝ 0| i| i | 1i 1| i
Theorem 2 Theorem 1
⇒ where the vecptor representativeof ρΨ′ ips givenby Ψ′
The no-superposing Theorem, √r′(αeinθ ϕ ⊗n +β Φ ) and r′ is a normalization| cion∝-
⇒
| i | i
Theorem 2 The no-cloning Theorem, stant. Φ˜′ is the vector representative of a general fail-
⇒ | i
Theorem 2 The no-deleting Theorem, urestateρΦ˜′ thatshouldbe annihilatedbythe projector
⇒
P P .
Theorem 3 ⇒ Theorem 1 | 1Niohte1|that ϕ eiθ ϕ ,i.e,thepurestateρ isphys-
eiθϕ
The no-superposing Theorem, | i∝ | i
⇒ ically undiscriminating from the pure state ρϕ. From
Theorem 3 The no-cloning Theorem, Eq.(10) we obtain
⇒
Theorem 3 ⇒ The no-deleting Theorem. U(eiθ ϕ ⊗k Σ P0 ) U ϕ ⊗k Σ P0 . (12)
| i | i| i ∝ | i | i| i
Proof of Theorem 2. Since Theorem1 is a specialcase From Eqs.(11) and (12) we obtain β =0 which is a con-
of this Theorem, the proof of Theorem 1 for a general tradiction to the assumption of β. (cid:3)
fixed state ρ is included in the following proof. The proof of Theorem 3 is shown in Appendix B.
φ
When β = 0 and is restricted to the unitary map,
F
Theorem 2 reduces to the no-deleting theorem. When
β =0, Theorem 2 reduces to the no-cloning theorem for III. QUANTUM TRANSFORMATION OF PURE
n>k. In the following, we only need to provethe result STATES IN A FINITE SET
for 0< β <1.
| |
The proof of Theorem 1 suggests a method to prove Although the no-cloning theorem forbids perfectly
thistheorem. Now,wepresentanotherproofwithsimple cloning unknown states, Buˇzek and Hillery [20] pro-
evaluations. Assume that there exists a unitary map vided a cloning transformationfor unknownqubit states
U
5
such that the partial traces on the original qubit and Theorem 4 (Schur Product Theorem) [57]. The
on the cloned qubit have the same density matrix as Hadamard product of two positive-semidefinite matri-
ρ=fρψ+(1 f)ρψ⊥ with the optimal fidelity f =5/6. ces (or positive-definite) is also positive-semidefinite (or
−
Here, ρψ⊥ is the orthogonal state of the input state ρψ. positive-definite).
The cloning transformation of Buˇzek-Hillery triggered
lots of investigations on the imperfect quantum cloning, Lemma 1 Let A be a positive definite matrix in matrix
see reviews [45, 46]. spaceCn×n andB beaHermitematrixinthesamespace.
Similar results hold for cloning pure states in a fi- Then A B is positive definite if
−
nite set. In fact, let S = ρ , ,ρ , where ρ is
on Hilbert space H with d{imψ(1H·)·· mψm. }There exisψtjs a kBk2 <kA−1k−21 (13)
CP map (S H ,S2) [47,≥48] for all the states
A
in S suchFth∈atCFP(ρψ⊗iρΣ) = ρψiρψi if the vector repre- rwihce.re k·k2 is the induced norm from the Euclidean met-
sentatives of ρ ,ρ , ,ρ are linearly independent,
where ρ is anψa1ncψil2lar·y··staψtenon Hilbert space H with
Σ A Proof of Lemma 1. For a positive definite matrix
dim(H ) m. Note that a CP map may be represented
by a uAnit≥ary evolution and post-selection of projective A [57], all the eigenvalues λ1,λ2,··· ,λn are positive
constants. From the definition of the induced norm,
measurement. So, the cloning transformation may be
viewedasanimperfectcloningbeforeperforminFgthepro- wmeinhaλve,λkA, k2 ,=λ mawxh{eλr1e,λA2−,·1·d·e,nλont}esatnhdekinAv−e1rske−21ma=-
jective measurement. This scheme has been extended to { 1 2 ··· n}
trix of A. For a Hermite matrix B, all the eigenval-
probabilistically recover ρ⊗2, ρ⊗3, , ρ⊗M for a large
integerM fromtheinput sψtiatesψρi ·,·i·=1ψ,i2, ,m[53]. ues λ′1,λ′2,··· ,λ′n are nonnegative and satisfy kBk2 =
Moreover,combinedwiththequanψtiumdeletin·g··machine, mthaexre{λe′1x,iλst′2s,·a··n,oλn′nz}e.roIfvAec−torBvis neCgnatsivuechsetmhia-tdevfi(nAite,
i.e., ρ ρ ρ , Feng et al. [54] proved that there ex- ∈ −
istsaψCiPψmi a7→p ψsiforthesamegoalusingtheinputstates B)v† ≤0. It means that
tρoψ⊗ikth,eiq=ua1n,t2u,m·F··st,amte. dTishcersiemoinpatitmioanl[s4c9h–e5m2]e.sIanrceormelpaatreid- kA−1k−21 ≤vAv† ≤vBv† ≤kBk2,
sonto the imperfect cloning scheme[20], the errorterms which contradicts the inequality of (13). (cid:3)
induced by these CP maps on a finite set are not re-
strictedtothe orthogonalstatesofthe input states. The
general transformation of the pure states in a finite set B. General quantum transformation of finite set
depends on the linear independence of the input states.
Thesuccessprobabilityisdeterminedbyapropermetric
In this section, the imperfect and perfect transfor-
inequality [48, 53, 54]. mation of the pure states in a finite set will be uni-
Inthissection,ourgoalistopresentthegeneraltrans- fied. In detail, let S be a finite set defined by S =
formations of a finite set of the pure states beyond pre- ρ⊗k,ρ⊗k, ,ρ⊗k Hk, where dim(H) 2 and k
vious probabilistic cloning or deleting [47, 48, 53, 54]. {is aψ1posψit2iv·e·i·nteψgmer}. ⊂ ≥
Theorem 5 Let α and β be complex constants satis-
ij ij
A. Hadamard product of matrices fying α 2+ β 2 = 1 and M+k α 2 = 0, and ρ
| ij| | ij| j=1 | ij| 6 Φij
be a known pure state on Hilbert space Hj, then there ex-
P
For completeness, some necessary definition and ists a CP map j (Hk,Hj) for all the states on S
F ∈ CP
propositionsofHadamardproduct(Shurproduct)ofma- such that
trices are firstly presented as follows [57].
(ρ⊗k)=ρ (14)
Fj ψi ϕij
Definition 1 Let A = [a ] and B = [b ] be
ij m×n ij m×n iff the vector representatives of ρ ,ρ , ,ρ are
matrices in matrix space Cm×n. The Hadamard product ψ1 ψ2 ··· ψm
linearly independent, where the vector representative of
of A and B is defined as A B =[a b ] Cm×n.
◦ ij ij m×n ∈ ρϕij is given by |ϕiji ∝ √rij(αij|ψii⊗j +βij|Φiji) and
r is a normalization constant, i = 1,2, ,m; j =
ij
Proposition 1 The Hadamard product of matrices is ···
1,2, ,M +k, and M is a positive integer.
commutative, associative and distributive. ···
In Theorem 5, the CP maps exist for S when the vec-
Proposition 2 The Hadamard product of two symmet- tor representatives of S are linearly independent. If S is
ric (Hermite) matrices is symmetric (Hermite). known, a CP map [14] exists for recovering the states in
S from a new set ρ ,ρ , ,ρ . Hence, our con-
{ ψ1 ψ2 ··· ψm}
These Propositions may easily follow from their defi- dition will be replacedby the linear independence of the
nitions. Moreover,the Hadamardproduct maintains the vectors ψ , ψ , , ψ foralargek. Thisisbecause
1 2 m
positive semidefinite property as the following theorem. that the| veict|orire·p··res|entaitives of S should be linearly
6
independent when k is large using the Gershgorin circle normalized vector φ CK with K 2 such that
ij
theorem [58] for its Gram matrix A◦k. α ψ +β φ |ψ⊥i ∈whichis the orth≥ogonalto ψ ,
ij| ii ij| iji∝| i i | ii
Before we present the proof, some useful results may the Corollary 2 leads to the following result with similar
be firstlyfollowedfromTheorem5. When β =0 for all proof.
ij
i=1,2, ,m andj =1,2, ,M+k, the generalmap
in Eq.(14··)·reduces to the un·ifi··ed cloning [54]. If βij = 0 Corollary 3 Let ρφij be a known pure state on H, then
holds for all i = 1,2, ,m and j = 1,2, ,k1 with there exists a CP map j (Hk,Hk1+j) for all the
1 k < M +k, Theo·r·e·m 5 reduces to a s·im··ultaneous states in S such that F ∈ CP
1
≤
perfect and imperfect cloning as follows.
(ρ⊗k)=ρ⊗k1ρ⊗j (17)
Corollary 1 Let α ,β be complex constants satisfying Fj ψi ψi ψi⊥
ij ij
|αij|2+|βij|2 =1 and Mj=+1k|αij|2 6=0, k1 be an integer iffthevectorrepresentativesofρψ1,ρψ2,··· ,ρψm arelin-
satisfying 1 k < M +k, and ρ be a known pure early independent, where i=1,2, ,m; j =1,2, ,L,
state on Hilb≤ert1spacePHj, then theΦriej exists a CP map L=M +k k1 and M is a posit·i·v·e integer. ···
j (Hk,Hk1+j) for all the states in S such that −
F ∈CP Proof of Theorem 5. The proof consists of two
(ρ⊗k)=ρ⊗k1ρ (15) parts. First, we prove that if there exists a CP map
Fj ψi ψi ϕij satisfying Eq.(14), then the vector representatives of
j
F
iff the vector representatives of ρψ1,ρψ2,··· ,ρψm are lin- ρψ1,ρψ2,··· ,ρψm are linearly independent. These CP
early independent, wherethevector representative ofρϕij mapswillbepresentedinaunifiedformwithanontrivial
is given by ϕij √rij(αij ψi ⊗j+βij Φij ) and rij is a probability in the following. In fact, from Stinespring’s
| i∝ | i | i
normalization constant, i = 1,2, ,m; j = 1,2, ,L, Theorem [56], assume that there exists a unitary map
··· ···
L=M +k k and M is a positive integer. with matrix representative U, an ancillary state with
1
− tUhe vector representative Σ on H , N +1 orthogonal
A
FromEq.(15),onemaysimultaneouslyproducek1 per- stateswiththevectorrepre|seintatives P0 , P1 , , PN
fect copies and j imperfect copies of ρψi when k1 > k. on HP such that | i | i ··· | i
Moreover, one may delete k k copies and clone j im-
1
−
perfect copies of ρ when k < k. The imperfect terms M+k
|Φiji(i=1,2,··· ,ψmi ;j =1,21,··· ,L)areindependentof U|ψii⊗k|Σi|P0i ∝ √pij|ϕji|0i⊗M+k−j|Pji
the input states. When k1 < k, one may keep k1 copies Xj=1
roefmNρψaoiitnebeedtihnkag−tunkfoc1rhcaoanprgibeeidstrfaarnroydmpcTeorhmfeoporlmreexmacC5o.PnsmtaanptsFαˆoinj,tβˆhiej +s=MXN+k+1√qis|Φ˜isi|Psi (18)
satisfying αˆ 2 + βˆ 2 = 1, and normalized vectors
ψ , φ | iCj|K wi|thijK| 2, there exist complex con- where HA = HM be an ancillary Hilbert space with
s|taiint|s αiji,∈β anda norm≥alizedvector Φ CnK such M > k, and HP be a probe space with dim(HP)
ij ij ij ≥
g|tαheianjte|2rαa+liljy||ψβriiejif|⊗o2nr=m+e1dβ.iaFjs|rΦofomijlilosw∝ims.i(lαˆairj|pψrioio+f,|tβˆhijis|iφf∈iajcit)⊗mnawyibthe Nsntoar+tmes1a,lii>ze=dMs1t,+a2t,ek·.·ρ·Φ|,iΦmjij;tihjai=st tis1h,ei2n,vd·ee·cp·teo,nMrdree+nptrkeo.sfentphtijaetiiisvnepthuoetf
success probability of producing n imperfect copies of
Cos|fαoanroitjirHsa|o2fly,ll+ilttnahh|grβeeyin1js|t2≤2taht=Lekere1se1t<iαenaxniMjiSds,tβ+sPsuijkcaMj,hb=Cea+1tnPhkcdao|αtmmρiφpjai|lpj2exb6=Fecjoa0n∈,kskntC1aoPnwbte(nsHapsknua,trHiienskftsy1et+gianejtger) ρsafitnψejasidt.iePss|Φρ˜dMj=Φ˜ei=sifi+s1iMnkoiespndi+tjHhb+ekykv⊗P+eHcjNs1t=,=AoMMrw+r0ie+ktpMh+rk1e+tshq+keie−sn2jpt=,ar0·to1·ibMv·,aei,+bNo=kifl−i.ta1jy,Eg2qea,inPsc·eh·jtr·haC,aPlMPtfjasima+lutairkspe-,,
F F | i h | ◦| ih |◦U
j =1,2, ,M +k.
Fj(ρ⊗ψik)=ρ⊗ψik1ρ⊗ϕijj (16) c A,cssu,m·e··t,hcatthseurcehextihstata vψecto⊗rk|ψ=ji⊗k amndcconψsta⊗ntks.
iffthevectorrepresentativesofρψ1,ρψ2,··· ,ρψm arelin- F1rjom2jEq··.(·18)mwje have | ji i=1 ij| ii
early independent, wherethevector representative ofρ P
ϕij
is defined by ϕij √rij(αij ψi +βij φij ) and rij is a M+k
normalization| conis∝tant, i = 1|,2i, ,m|, ji= 1,2, ,L U Ψj √pjs ϕs 0 ⊗M+k−s Ps
··· ··· | i ∝ | i| i | i
and L=M +k−k1. Xs=1
N
The no-superposing theorem [19] forbids superposing + √qjt Φ˜jt Pt (19)
an unknown state and its orthogonal state in Hilbert | i| i
t=M+k+1
space of a finite dimension. However, Corollary 2 sug- X
gests a perfect superposing for unknown states in a fi- where the vector representative of ρ is given by
ϕs
nite set. If we choose complex constants αij,βij and a ϕs √ris(αis ψi ⊗s+βis Φis ) and the vector Ψj =
| i∝ | i | i | i
7
ψ ⊗k Σ P . Moreover, from the linearity of quantum A◦k M+kG B G† can be diagonalized by a unitary
| ji | i| 0i − s=1 s s s
operations, we obtain that matrix V as follows:
P
M+k
M+k m
U Ψj √cij√pis ϕs 0 ⊗M+k−s Ps V(A◦k− GsBsG†s)V† =diag(γ1,γ2,··· ,γm) (24)
| i ∝ | i| i | i s=1
s=1 i=1 X
X X
N m where γ ,γ , ,γ denote all the eigenvalues of A◦k
1 2 m
+ √cij√qit Φ˜it Pt (20) M+kG B G··†·andγ >0,j =1,2, ,m. DefineQ −=
t=MX+k+1Xi=1 | i| i V†sd=i1ag(xs1ℓ,sx2ℓs, ,xjmℓ)V wherexj·ℓ·i·snonnegativeaℓnd
P N ···
satisfies x =γ ,j =1,2, ,m. Hence,the
Note that Eqs.(19) and (20) must be equal to each ℓ=M+k+1 jℓ j ···
matricesQ ,Q , ,Q arepositivedefinite
other for all the states ρψi (ump to a phase factor), i = from ThPeorMem+k+41andMth+ekn+2ph·y·s·icallNy realizable.
1,2,··· ,m. It follows that i=1√cijpis|ϕsi∝√pjs|ϕsi Before completing the proof, we need to prove the in-
for all the integers s satisfying 1 s M +k. These
P ≤ ≤ equality (23). Using the triangle inequality, it easily fol-
equalities imply that c = 1 and c = 0 for i = j be-
tcehaaeurlsyveeitnchtdeoerpveercnetpdorerens|tΦe.nijtiatiisjvjiensdoepfeρnψd1e,nρtiψj2o,f·i·n·p,uρtψsmta6taerse. lSino-, lSsoaowt,isstfhytehirnaegtekxEiPqst.(Msn2=o3+1n)kzGwershoBemnsGatt†shrkie2cye≤ssGaPtsis(Msfs=y+1=kthk1eG,2sf,ok·l22l·o·×w,iMknBg+skikn2)-.
Now,weneedtoprovetheexistenceoftheunitarymap equality
satisfying Eq.(18). For the input states ρ⊗k and ρ⊗k,
ψi ψj M+k
taking the inner product of the representative vectors of G 2 B < (A◦k)−1 −1 (25)
the output states [60], from Eq.(18) we have k sk2×k sk2 k k2
s=1
X
M+k which is easily guaranteed. This completes the proof. (cid:3)
akij = √pis(√risαisasijα∗js√rjs Theinequality(23)oritsweakform(25)haspresented
animplicitboundofthe efficiencyintermsofthe matrix
s=1
X
+αisλij,sβj∗s+βisλ∗ji,sα∗js+βisβj∗s)√pjs norm. An entry-pair inequality is obtained from Eq.(22)
using the state metric [19] as follows
N
+ √qiℓqjℓ, (21) M+k
ℓ=MX+k+1 Dij,k ≥ pij,s[αij,s(rij,sDi2j,s−2rij,s+2)
s=1
wherea := ψ ψ andλ := ψ ⊗s Φ . Briefly,we X
ij i j ij,s i js 2max λ , λ ) (26)
h | i h | | i ij,s ji,s
obtain a matrix equation using Definition 1 as follows: − {| | | |}
where p = (p + p )/2, α = α α , β =
ij,s is js ij,s is js ij,s
A◦k =M+kGs(ΛsA◦sΛ†s+Cs)G†s+ N Qℓ, (22) t|β=isβkjs|o,rrisj.,s T=he√rpirsorojsf,isansdhoDwinj,tin=A2|p(p1e−nd|iax|ijC|t). Twhitihs
Xs=1 ℓ=MX+k+1 inequality has generalized the bound of the probabilis-
tic cloning [14] or the state discrimination [49]. It also
where A◦t denotes the t-fold Hadamard power of ma-
reducestotheboundoftheprobabilisticcloninginauni-
trix A = [a ] , C = [α λ β∗ + β λ∗ α∗ +
ij m×m s is ij,s js is ji,s js fiedform[19,48,53,54]forspecialconstantsofλ , α
βisβj∗s]m×m, Qℓ = [√qiℓqjℓ]m×m, and diagonal matri- and β . ij,s ij
ij
ces Gs = G†s = diag(√p1s,√p2s,··· ,√pms) and Λs = Remark. From the cloning scheme [48], the final
diag(α1s√r1s,α2s√r2s, ,αms√rms). states belong to a finite set of M+kS , where S =
From Lemma 1 [4·8·]·, it is sufficient to prove ∪j=1 j j
ρ⊗j,ρ⊗j, ,ρ⊗j . Similar results hold for S [54].
Eq.(22) with physically available matrices QM+k+1, M{ oψr1eovψe2r,·f·r·omψTmh}eorem 5, the final states belonkg to a
Q , ,Q fortheexistenceoftheunitarymapin
EqM.(+1k8+).2 T··h·e mNatrix A◦k is positive definite from The- new set ∪Mj=+1kS˜j, where S˜j denotes an infinite vector set
α ψ ⊗j +β Φ , α 2+ β 2 =1,i=1,2, ,m
orem 4 because the vectors ψ , ψ , , ψ are lin- ij i ij ij ij ij
early independent. The ma|tr1iices| 2iMs=·+·1·kΛ|sAm◦isΛ†s and wS{˜ithm|afiyxiebdevveicetwo|resd|Φaisij|ai ∈“|cHomj.p|rInes|sgeedomuentirtys,pehaecrhe·”·s·uwbsiteh}t
C are Hermite. From Lemma 1, when the efficiency j
s P two axes of ψ ⊗j and Φ , see a schematic example
matrices G (s = 1,2, ,M +k) satisfy the following i ij
s | i | i
··· shown in Fig.2.
inequality
M+k
G B G† < (A◦k)−1 −1, (23) IV. SUPERPOSITION OF PURE STATES IN
k s s sk2 k k2 RESTRICTED SETS
s=1
X
the matrix A◦k M+kG B G† is positive definite, Recentschemesshowthatonemaysuperposeunknown
where B = Λ A−◦sΛ†s=+1 C s. sSo,s the Hermite matrix stateswithafixedprojectorontoaknownstate[17]. Our
s s s s
P
8
unitary map with the matrix representative U, an ancil-
1
larystatewithvectorrepresentative Σ onHilbertspace
|Φ1〉 H = HM with M > m, 3 orthogo|nail states with the
A
vector representatives P , P , P on H such that
0.5 | 0i | 1i | 2i P
|ψ〉
2 U ψ ψ Σ P
i j 0
| i| i| i| i
β 0 |ψ〉 ∝ √pij|ϕiji|0i|P1i+√qij|Φiji|P2i (28)
1
Here, p is the successprobability ofrecoveringa super-
ij
posed state ρ with the vector representative ϕ
−0.5 √rij(αij ψi +ϕiβjij ψj ), andsatisfies pij =pij wh|ichijcio∝r-
| i | i
|Φ〉 respondstotheswappingoftheinputstates. Φij isthe
2 | i
vectorrepresentativeofanormalizedfailurestateρ on
−−11.5 −1 −0.5 0 0.5 1 1.5 H HAwiththeprobabilityqij thatsatisfiesqij+piΦji=j 1,
α ⊗
i,j =1,2, ,m.
···
For the input states ρ ρ and ρ ρ , taking the
ψi1 ψi2 ψj1 ψj2
inner product ofthe vectorrepresentativesof the output
SF1IGin. 2C:2.(CHoelorer,otnhleinvee)ctSocrhsem|ψa1tiic=re|0pireasnednt|aΦti1vies=of√12S˜(1|0ain+d states from Eq.(28), we can obtain
|1i), |ψ2i = q23|0i+q13|1i and |Φ2i = √12(|0i−|1i). S1 = ai1j1ai2j2 = √pi1i2ri1i2hi1i2,j1j2√pj1j2rj1j2
{|cβiρriψc|2l1e,=sρ.ψ12,}ii=sa1,se2t}oifstawoseptocionntss.isS˜ti1ng=o{fαail|lψtihi+e pβoi|iΦntisi,o|αni|t2w+o whhere aks==α∗hψαk|+ψas√i,qi1+hi2i1qβij21∗,jj2β1jb2i1ai2=,j1j2h+Φijα|∗Φβj1j2ai, (a2n9+d)
βi∗1iα2,j1aj2 . Itmi1ayj1bei1bjr1ieflydi2escjr2ibie2jd2asami1atjr2ixi1ejq2ua-
i2 j1 i2j1
goalinthissectionistoproposenewsuperposingschemes tion as follows
for the input states in different restricted sets. A=ΛHΛ†+Q, (30)
where the matrices A,H,Q are respectively defined by
A. Pure states in a finite set
A=[a a ] ,
i1j1 i2j2 m2×m2
Inthissubsection,weinvestigatethesuperposedstates Λ=diag(√p11,√p12, ,√pmm)m2×m2,
···
bweitlohngdiinmg(Hto) a fimni.teNsoetteSth=at{tρhψe1,lρinψe2a,·r·i·n,dρeψpmen}d⊂encHe H = √ri1i2rj1j2hi1i2,j1j2 m2×m2,
of the input s≥tates is a special case of the fixed overlap Q=[(cid:2)√qi1i2qj1j2bi1i2,j1j2]m2(cid:3)×m2,
[17, 59]. Unfortunately, the scheme proposed by Osz- andthe rowsorcolumnsofthe matricesA,H,Qarerep-
maniec et al [17] is only sufficient. New schemes present resented by two-bit series i i and j j , respectively.
1 2 1 2
thenecessityofthesuperpositionschemewhentheinput It is sufficient to prove Eq.(30) with physically realiz-
states belong to a finite set. able matrix Q for the existence of the unitary map in
Eq.(28). A is positive definite and ΛHΛ† is Hermite.
Theorem 6 Let αij,βij be nonzero complex constants From Lemma 1, A ΛHΛ† is positive definite when Λ
satisfying αij 2+ βij 2 =1, then there exists a CP map satisfies −
(H|2,H|) for| all|the states in S such that
F ∈CP ΛHΛ† < A−1 −1. (31)
k k2 k k2
(ρ ρ )=ρ (27)
F ψi ψj ϕij Therefore, there exists a unitary matrix V such that
iff the vector representatives of ρψ1,ρψ2,··· ,ρψm are lin- V(A ΛHΛ†)V† =diag(λ1,λ2, ,λm2), (32)
early independent, wherethevector representative ofρ − ···
ϕij
is defined by ϕij √rij(αij ψi +βij ψj ) and rij is a where λ1,λ2, ,λm2 denote all the eigenvalues of A
normalization| conis∝tant, i,j =|1,i2, ,|m.i ΛHΛ† and a·r·e· positive. Denote Q = V†diag(λ−,
1
···
λ , ,λ )V. So, Q is positive from Theorem 4 and
2 m2
Proof of Theorem 6. Note that the vector represen- then··p·hysically realizable. This completes the proof. (cid:3)
tatives ψ ψ , ψ ψ , , ψ ψ are linearly de- The bound of the efficiency may be proved from
1 1 1 2 m m
| i| i | i| i ··· | i| i
pendent iff the vectors ψ , ψ , , ψ are linearly Eq.(29) as follows
1 2 m
| i | i ··· | i
dependent. Similar to the proof of Theorem 5, the ne-
2
cessity is easily followed from the linearity of quantum D D p [r (16 D
operations. Now we need to prove the existence of the i1j1 i2j2 ≤ 3 i1i2,j1j2 i1i2,j1j2 − i1j1
D D D ) 6]
CP map. Assume there exists a CP map satisfying − i2j1 − i1j2 − i2j2 −
F
Eq.(27). From Stinespring’s Theorem[56], there exists a +4D +4D , (33)
i1j1 i2j2
9
whereri1i2,j1j2 =√ri1i2rj1j2, pi1i2,j1j2 =(pi1i2+pj1j2)/2 0.9
andD =2(1 a ). Thisisanewboundoftheaverage
ij ij
−| |
efficiencyp incomparisontotheboundinEq.(26) 0.8
i1i2,j1j2
because Theorem 6 cannot be proved from Theorem 5. 0.7
Combined with the probabilistic cloning or deleting of
0.6
Theorem 5, we obtain a general theorem as follows:
β20.5
Theorem 7 Let αij,βij be nonzero complex constants αα130.4
satisfying α 2 + β 2 = 1, then there exists a CP
ij ij
map | |(Hk | H|k,Hs) for all the states in S = 0.3
F ∈ CP ⊗
ρ⊗k,ρ⊗k, ,ρ⊗k Hk such that 0.2
{ ψ1 ψ2 ··· ψm}⊂
0.1
(ρ⊗kρ⊗k)=ρ (34)
F ψi ψj ϕij,s 00 0.2 0.4 0.6 0.8 1
α
iff the vector representatives of ρ⊗k,ρ⊗k, ,ρ⊗k are 1
ψ1 ψ2 ··· ψm
linearly independent, where the vector representative of
ρϕij,s is defined by |ϕij,si∝√rij,s(αij|ψii⊗s+βij|ψji⊗s) FIG. 3: (Color online) Schematic figure of c1 vias the x-axis
and rij,s is a normalization constant, i,j = 1,2, ,m; component α1. Here, the vector X = x1 x2 x3 , we get
··· | i | i| i| i
s=1,2, ,M +2k and M is an integer. that α1α3β2 = c1,β1β3α2 = c2, where αi = xiψ and
··· βi = xiφ , i = 1,2,3. The blue line denotes t|hhe|grai|ph of
The proof is shown in Appendix E. {(α1,|ch1)||α1i|∈[0,1]} with |x1i=|0i,|x2i= √12(|0i+|1i) and
x3 = 1 . The red line denotes the graph of (α1,c1)α1
[|0,1i]}w|itih|x1i=|0i,|x2i= √12(|0i+|1i) and|{x3i= 21(||0i+∈
B. Pure states in a restricted set √31 ). The green line denotes the graph of (α1,c1)α1
[0,1|]}iwith|x1i=|1i,|x2i= √12(|0i+|1i) and|{x3i= 21(||0i+∈
In this subsection, we generalize a previous overlap √31 ).
| i
condition [17] and obtain new superposing schemes of
unknown states when multiple copies of the input states
are available. where the vector representative of ρϕ′ is defined
by ϕ′ √r′(αeiθ3 ψ + βeiθ4 φ ) with eiθ3 =
Theorem 8 Letα,β benonzerocomplexconstantssatis- | i ∝ | i | i
fying α2+ β 2 =1, ρX be a known pure state in Hilbert |hhXX||φφii||φφii||ψψii| and eiθ4 = |hhXX||ψψii||ψψii||φφii| and r′ is a nor-
space|H|3 wi|th|dim(H) 2, and ρ be an unknown qubit malization constant.
µ
≥
state with vector representative µ =α0 +β 1 .
| i | i | i
(i) ThereexistsaCPmap (C2 H2 H2,H)for Before proving this theorem, we want to present com-
all the states ρ ,ρ onFH∈sCaPtisfyi⊗ng th⊗e following parisons of the condition in Eq.(35) to the fixed overlap
ψ φ
condition of Theorem2 [17]. It can be easily shownthat
inequalities
the new condition in Eq.(35) holds for the fixed over-
tr(ρ ρ ρ ρ )=c >0, lap condition [17]. However, the converse is not true.
X ψ φ ψ 1
⊗ ⊗ ⊗ In particular, taking X = x x x as an example,
tr(ρX ρφ ρψ ρφ)=c2 >0, (35) | i | 1i| 2i| 3i
⊗ ⊗ ⊗ we obtain that c = α α β and c = β β α , where
1 1 3 2 2 1 3 2
such that αi = xi ψ and βi = xi φ , i = 1,2,3. An evaluated
|h | i| |h | i|
exampleisshowninFig.3. Thereexistsuperposedstates
(ρ ρ⊗2ρ⊗2)=ρ , (36) that satisfy Eq.(35) with the fixed state ρ while they
F µ ψ φ ϕ X
do not satisfy the condition of Theorem 2 [17] with the
bwyherϕe the v√ecrt(oαreirθe1pψrese+ntaβteiivθe2 φof)ρwϕitihs edieθfi1ne=d vfiaxleudesstoafteαρxf1o.rIstomisebcec.ause that there are two different
| i ∝ | i | i 1 1
hX|φi|ψi|φi and eiθ2 = hX|ψi|φi|ψi and r is a nor- Proof of Theorem 8. The proof procedure is similar to
|hX|φi|ψi|φi| |hX|ψi|φi|ψi|
malization constant. that of the superposing scheme [17]. Define an auxiliary
normalized vector v =√c(√c1 0 +√c2 1 ), where c is
(ii) There exists a CP map ′ (C2 H2 H2,H) a normalization con|sitant. Let |=i | i , where
for all the states ρ ,ρ Fin H∈CsaPtisfyi⊗ng ⊗ F F5F4F3F2F1
ψ φ
:ρ V ρV†,
tr(ρ ρ ρ ρ )=c′ >0, F1 7→ 1 1
X ⊗ ψ⊗ ψ ⊗ φ 1 :ρ V ρV†,
tr(ρX ⊗ρφ⊗ρφ⊗ρψ)=c′2 >0, (37) F2 :ρ 7→ V2ρV2†,
such that F3 7→ 3 3
:ρ V ρV†,
F4 7→ 4 4
F′(ρµρ⊗ψ2ρ⊗φ2)=ρϕ′, (38) F5 :ρ 7→ tr1345(ρ), (39)
10
r P where g(1,2, ,2k 1) denotes the permutation of in-
m v ··· −
dex pairs(1,2, ,2k 1)accordingto the permutation
rryf S2,3 oρpΦe1,r2a,·t··o,2rkg−1∈:=P2·ρk·⊗ψ−·k1.⊗Lρe−⊗φtkρ−Ψ11.,2,···,2k−1 :=ρ⊗ψk−1⊗ρ⊗φk and
ry PX Theorem 9 Let α,β be nonzero complex constants sat-
r S4,5 isfying α2 + β 2 = 1, ρX be a known pure state in
f Hilbert |sp|ace H| 2|k−1 with dim(H) 2, and ρ be an
µ
unknown qubit state on C2 with v≥ector representative
µ α0 +β 1 . For τ ,τ P . Then there ex-
FIG. 4: (Color online) Schematic circuit of CP mapping F. i|stis∝a CP| imap | i 1(C22 ∈Hk2k−H1 k,H) for all pure
Here, PX and Pv denote the measurement induced by the Fτ1,τ2 ∈CP ⊗ ⊗
operator X X and v v, respectively. denotes the con- states ρ⊗k,ρ⊗k in Hk satisfying
ψ φ
| ih | | ih | •
trolling system.
tr(ρ ρ )=c >0,
X ⊗ Ψτ1(1,2,···,2k−1) 1
tr(ρ ρ )=c >0, (43)
where X ⊗ Φτ2(1,2,···,2k−1) 2
such that
V = 0 0 I I I I + 1 1 I I S ,
1 4,5
| ih |⊗ ⊗ ⊗ ⊗ | ih |⊗ ⊗ ⊗
V2 = |0ih0|⊗I⊗I ⊗I⊗I +|1ih1|⊗S2,3⊗I ⊗I, Fτ1,τ2(ρµρ⊗ψkρ⊗φk)=ρϕ, (44)
V = I I P ,
3 X
⊗ ⊗ where the vector representative of ρ is defined by ϕ
V4 = Pv⊗I⊗I ⊗I⊗I. (40) √r(αeiθ1 ψ + βeiθ2 φ ) with eiθ1 =ϕ hX|Φτ2(1,2,···,2|k−i1)∝i
Here, Pv = |vihv|, PX = |XihX|, I denotes the identity and eiθ2 |=ihX|Ψτ1(1,|2,·i··,2k−1)i and r i|shXa|Φnτo2(r1m,2,a··l·i,z2ka−ti1o)in|
operator,Si,j denotestheswappingoperatoroftwopure |hX|Ψτ1(1,2,···,2k−1)i|
states in the i-th and j-the subsystems, and tr () is constant.
1345
·
thepartialtraceoverallthefactorsexceptthesecond. Its
In Theorem 9, Ψ is the vector rep-
schematic circuit is shown in Fig.4, where one swapping | τi(1,2,···,2k−1)i
resentative of the pure state ρ , and
operationS3,4 is firstly usedto obtain ρµρψρφρψρφ from Ψτi(1,2,···,2k−1)
ρv⊗ρ⊗ψ2⊗ρ⊗φ2. F iscompletelypositiveandtracenonin- s|Φtaτtie(1,ρ2,···,2k−1)i is t.hTehveecptroororfeopfrtehseisntthateiovreemofitshseimpiularer
creasing. The finalstate may be evaluated forward. The Φτi(1,2,···,2k−1)
to that of Theorem 9, see Appendix G.
uniqueness is similar to that in Ref.[17] except some dif-
In general, we can superpose n independent states as
ferent constructions, see Appendix E. The success prob-
ability is given by follows. Let ρΨj1,2,···,nk−1 := ⊗nt=1ρ⊗ψtkjt, where kjt is an
integer satisfying k = k for j = t and k = k 1 for
jt jt
ps = tr[F(ρµ⊗ρ⊗ψ2⊗ρ⊗φ2)] j =t. 6 −
= tr[ (ρ ρ ρ ρ ρ )]
µ ψ φ ψ φ
F Corollary 4 Let ρ be a known pure state on Hilbert
c c X
= c11+2c2Nϕ2. (41) sstpaatceeiHnnCkn−1wiwthithvedctimor(Hre)pr≥ese2n,taatnidveρµµ b=e annu−n1kαnowjn.
The map cannot be rescaled to increase the For any permutations τ ,τ , ,τ P| i , thje=r0e exji|stis
success proFbability because of operator inequality a CP map 1 2 ·(C··n nH∈nk,nHk−k1)Pfor all pure
(V4V3V2V1)†(V4V3V2V1) I⊗5. If coefficients α and β states ρ⊗k,ρF⊗τk1,,τ2,···,,τρn⊗∈k iCnPHk sa⊗tisfying
arefixed,onemayconstr≤uctanewCPmapwithahigher ψ1 ψ2 ··· ψn
probability, see Appendix F. (cid:3) tr(ρ ρ )=c >0,i=1,2, ,n (45)
It is possible to generalize the superposing map for X ⊗ Ψiτi(1,2,···,nk−1) i ···
F
k copies of the superposed states. The fixed state is de-
such that
fined in Hilbert space H2k−1. For convenience, we make
puseermoufttahteiopnerompuertaattiioonnsgoronuapfiPn2kit−e1sceotnwsiistthin2gkof a1lletlhee- Fτ1,τ2,···,τn(ρµ⊗(⊗ni=1ρ⊗ψik))=ρϕ, (46)
ments. For each permutation g ∈ P2k−1, there−exists a where the vector representative of ρϕ is defined by |ϕi∝
tgieonnergalsuswchaptphiantgSogperatio(nHS2kg−i1n,dHu2cke−d1b).yItnhepparetrimcuultaar-, √r jn=−01αjeiθj|ψji with eiθj = |hhXX||ϕϕττjj((11,,22,,······,,nnkk−−11))ii| and
let P2k−1 be a perm∈utCaPtion operated on the index set r isPa normalization constant.
1,2, ,2k 1 . For each state ρ H2k−1
{ ··· − } φ1,2,···,2k−1 ∈ In this corollary, ϕ is the vector repre-
(where subscripts 1,2, ,2k 1 denote the indexes of | τj(1,2,···,nk−1)i
··· − sentativeofthepurestateρ ,j =1,2, ,n.
subsystems), then we define ϕτj(1,2,···,nk−1) ···
The proof is similar to that of Theorem 8, see Appendix
S :ρ ρ , (42) H.
g φ1,2,···,2k−1 7→ φg(1,2,···,2k−1)