Multiaccess quantum communication and product higher rank numerical range

0.41 MB·English

by  Maciej Demianowicz

QuantumInformationandComputation, Vol.0,No.0(2003) 000–000
(cid:13)c RintonPress
MULTIACCESS QUANTUM COMMUNICATION
AND PRODUCT HIGHER RANK NUMERICAL RANGE
2 MACIEJDEMIANOWICZ
1 Department of Atomic Physics and Luminescence, Faculty of Applied Physics and Mathematics,
0 Gdan´sk University of Technology, ul. Narutowicza 11/12, PL80-233 Gdan´sk, Poland &
2 National Quantum Information CenterinGdan´sk, ul. Wl . Andersa 27, PL81-824 Sopot, Poland &
[email protected]
p
e
S PAWEL HORODECKI
Department of Theoretical Physics and Quantum Informatics, Faculty of Applied Physicsand Mathematics,
4
Gdan´sk University of Technology, ul. Narutowicza 11/12, PL80-233 Gdan´sk, Poland &

National Quantum Information CenterinGdan´sk, ul. Wl . Andersa 27, PL81-824 Sopot, Poland &
]
h [email protected]
p
- KAROLZ˙YCZKOWSKI
t
n Smoluchowski Institute of Physics, Jagiellonian University,ul. Reymonta 4, 30-059 Krak´ow, Poland &
a Centerof Theoretical Physics, Polish Academy of Science, al. Lotnik´ow 32/46, 02-668 Warszawa, Poland
u & [email protected]
q
[

2
In the present paper we initiate the study of the product higher rank numerical range.
v
Thelatter,beingavariantofthe higherranknumericalrange[M.–D.Choi et al., Rep.
9
Math. Phys. 58, 77 (2006); Lin. Alg. Appl. 418, 828 (2006)], is a natural tool for
1
studyingconstructionofquantumerrorcorrectioncodesformultipleaccesschannels. We
1
reviewpropertiesofthissetandrelateittoothernumericalranges,whichwererecently
0
introducedintheliterature. Further,theconceptisappliedtotheconstructionofcodes
.
9 forbi–unitarytwo–access channelswithahermitiannoisemodel. Analyticaltechniques
0 for both outerbounding the product higher rank numerical range and determining its
2 exact shape are developed for this case. Finally, the reverse problem of constructing a
1 noisemodelforagivenproductrangeisconsidered.
:
v
Keywords: multipartyquantumchannel,producthigherranknumericalrange,quantum
i
X errorcorrection,Knill–Laflammeconditions
Communicated by: tobefilledbytheEditorial
r
a
1 Introduction
Quantum information transmission [1, 2, 3, 4, 5] inevitably involves occurrence of errors
which faced not properly can disable faithful communication. Developing fruitful and useful
strategies for combating these errors is thus one of the main challenges of the theory of
quantum channels. Many effort has been put in this direction and several techniques have
been developed to overcome destructive influence of coupling to the environment (see, e.g.,
[6, 7]). Among them, quantum error correction codes (QECC) along with the celebrated
Knill–Laflamme(KL)conditions[8]aremostwidelyrecognized(see[9]andreferencestherein).
Methods of constructing QECC for quantum communication have been previously reported
1
2 Title ...
in the literature [10, 11, 12]. There has also been a significant progress in experimental
realizations of the propositions [13, 14, 15].
Recently, a fruitful approach to solving Knill–Laflamme conditions has been put forward
[16, 17, 18]. It relies on the concept of the higher rank numerical range of an operator
and provides a systematic framework for a construction of quantum error correction codes
(QECC). As shown in Ref. [16, 17] it may capture more possibilities than the stabilizer
formalism [12].
However, all proposals analyzed so far concerned bipartite communication (see however
the related paper [21]). No general approach has been developed to treat the case of the
larger number of users of a quantum network. The main purpose of this paper is to provide
a mathematical tool of the product higher rank numerial range for designing QECC for mul-
tiaccess quantum communication. Product higher rank numerical range by the definition is
a higher rank numerical range restricted to product projections.
The paper is organizedas follows. First, we review basicideas, including errorcorrection,
of quantum communication over (multiparty) quantum channels. Further, we briefly recall
some basic features of the higher rank numerical range with applications to error correction.
Wethenmovetothemainbodyofthepresentpaperbyintroducingthenotionoftheproduct
higher rank numerical range and discussing its basic properties. In the next section we
consider construction of QECC for bi–unitary channels with a hermitian noise model. We
then demonstrateexemplaryapplicationsof ourfindings to some concreteproblems. We also
discuss the reverse problem of constructing a noise model for which a product code exists.
The manuscript is concluded with a discussion.
2 Quantum channels and quantum error correction
Here we briefly recall some basic ideas of communication over quantum channels and set the
scenario for further considerations.
2.1 Quantum channels
Quantum channel is a completely positive trace–preserving map. Every channel admits
L
the so–called Kraus (or operator–sum) representation as follows L(̺) = iAi̺A†i with
iA†iAi = 1 [22, 23]. A random unitary channel is the one which has thePrepresentation
PL(̺) = ipiUi̺Ui†, where Ui are unitary and ipi = 1, pi ≥ 0. When such a channel has
two KraPus operators, i.e., L(̺)=pU1̺U1†+(1P−p)U2̺U2†, it is called a bi–unitary channel
(BUC). This kind of channels is the main interest of the present paper.
Channels can be classified upon the number of senders and receivers using them. We
have the following types of channels according to such a classification [4, 24, 25, 26, 27]:
(i) bipartite — one sender and a single receiver, (ii) multiple access — several senders and
one receiver, (iii) broadcast — one sender and several receivers, (iv) km–user — k senders
transmit information to m receivers (k,m>1).
In our reasonings we mainly concentrate on two–access channels, that is multiple access
channels with two senders.
Due to the possibility of a global rotation U1†L(̺)U1 on the output of a channel, both in
bipartite and multiple access case one can consider a simplified BUC in general reasonings
(̺)=p̺+(1 p)U̺U . (1)
†
L −
Author(s) ... 3
For two–access channels it holds ̺=̺ ̺ , where ̺ is the input of the i–th sender.
1 2 i
⊗
2.2 Quantum error correction
QECC is a subspace of a larger Hilbert space . Equivalently, a code is defined to be the
C H
projection R onto . One says that is correctable if all states from this subspace
C C ⊆ H C
̺ = R ̺R can be recovered after an action of a channel using some decoding operation ,
C C D
that is (̺)=̺. Such recovery operation exists if and only if
D◦L
R A†iAjR =αijR (2)
C C C
forsomehermitianmatrix[L] =α . These conditionsaredue toKnillandLaflamme(KL)
ij ij
[8].
In the case of a larger number of senders we talk about local codes , that is QECC for
i
C
everysender. ItisanimmediateobservationthatKLconditionsneedonlyalittleadjustment
to serve for the case of MACs. Namely, we have (with the obvious notation):
Observation 1 Local codes are correctable for a MAC with Kraus operators A with k
i i
C { }
inputs if and only if
RC1 ⊗RC2 ⊗···⊗RCkA†iAjRC1 ⊗RC2 ⊗···⊗RCk =αijRC1 ⊗RC2 ⊗···⊗RCk (3)
for some hermitian matrix [ ] =α .
ij ij
L
This is true since the set of product codes is a subset of the set of all codes. In further parts,
we sometimes use the denotation R R (R R ) or S S for a code for a two–access
⊗ ′ M ⊗ N′ ⊗ ′
channel and talk about M N codes, where M,N denote dimensions of local codes.
⊗
In case of many usages of a channel, A are tensor products of Kraus operators in KL
i
conditions. In this paper, however, we concentrate on a single usage of a channel. For one
use ofa BUC,Eq. (1), KL conditions(2) reduceto the single condition(we write R shortly
C
as R).
RUR=λR, (4)
which for MACs takes the form
R R′UR R′ =λR R′. (5)
⊗ ⊗ ⊗
Itis usefulto introducethe notionofthe entropyofaQECC[29]. Thisentropyquantifies
the number of ancillary qubits which are needed for the recovery procedure. For a BUC the
entropy of a code is the von Neumann entropy of the matrix
p λ p(1 p)
= ± − . (6)
L λ p(1 p) 1 p
(cid:18) ± − p− (cid:19)
p
By inspection one finds that S( )=H(1/2(1+ 1 4(1 p)(1 λ2))). It is equal to zero
L − − −| |
iff λ = 1 or λ = +1 and these values correspond to DFS with trivial (identity) recovery
− p
operation a.
aItisnotcleartouswhetheritwouldmakeanysensetodefineentropyofalocalcode.
4 Title ...
2.3 Higher rank numerical range approach to bipartite QEC
Here we recall the notion of the higher rank numerical range band some of its implications
in the area of QEC [16, 17], which are important for our current purposes. The approach
we briefly describe below is the one we wish to further modify to be applicable in case of
two–access channels.
It is the form of KL conditions, which prompted the authors of Ref. [16, 17, 19, 20] to
introduce the notion of higher rank numerical range (or the rank–k numerical range) of an
operator. For an operator A, it is defined to be the following set
Λ (A)= λ C:P AP =λP (7)
k k k k
{ ∈ }
with P , where is the set of rank k projections. Elements of the set are sometimes
k k k
∈ P P
called the compression values.
Full characterization of the set Λ (A) for hermitian A has been obtained. Namely, as-
k
suming a a ... a is a spectrum of N N hermitian A, it holds that Λ (A) =
1 2 N k
≤ ≤ ≤ ×
a ,a , which is (a) a true interval whenever a < a , (b) a singleton set if
k N k+1 k N k+1
h − i −
a =a , (c) an empty set in the remaining case.
k N k+1
−
The set is also quite well understood for unitary operators [19]. We recall some results
below. Let U be nondegenerate n n unitary matrix with a spectrum spec(U) = z n
× { i}i=1
corresponding to eigenvectors v n . Let ∆ (U) be the set of λ such that for some k
{| ii}i=1 k
disjoint subsets δ ,δ ,...,δ of 1,2,...,n it holds λ conv( z ,i δ ) for all i (conv
1 2 k i j
{ } ∈ { ∈ }
stands for a convex hull). It was showncthat ∆ (U) Λ (U). The proof of this fact is
k k
⊆
constructive in a sense that it gives explicitly the projection. Since
λ= α z (8)
ij i
jX∈δi
k
with α 0 and α =1 we can choose the code P to be P = ψ ψ where
ij ≥ j∈δi ij C C i=1| iih i|
P P
ψi = √αij vj (9)
| i | i
jX∈δi
to obtain P UP = λP . One can take the subsets δ to represent triangles or, in a more
C C C i
restricted variant, sections. Both cases will be considered by us in Section 6.
Relevance of the notion to the issue of construction of QECC is obvious [16, 17].
3 Product higher rank numerical range and its basic properties
Motivated by the form of KL conditions, Eq. (3) for multiple access channels we introduce
the notion of the product higher rank numerical range. It is defined as follows
Definition 1 The k k product higher rank numerical range of an operator A is
1 2
⊗ ⊗···
defined to be
Λ (A)= λ C:R R AR R =λR R (10)
k1⊗k2⊗··· { ∈ ⊗ ′⊗··· ⊗ ′⊗··· ⊗ ′⊗···}
bIt is the generalization of the notion of the numerical range of an operator X, which is defined to be the
followingset: Λ(X)={λ∈C:hψ|X|ψi=λ,hψ|ψi=1}
cInfactforn≥3k muchstrongerresultwasproved,namely∆k=Λk buttoavoidtechnicalitiesweusehere
aweakerversion,whichissufficientforourpurposes.
Author(s) ... 5
for some R , R ,...
∈Pk1 ′ ∈Pk2
In the above, we assume that it holds k = 1 at least for a single index i . When all k ’s
i i
6
are equal to unity then one deals with the local (or product) numerical range [31], that is a
set
Λ (A)= λ C: ψ φAψ φ for ψ , φ . (11)
loc a b
{ ∈ h |h | | i| i | i∈H | i∈H }
Ifwechooseallprojectionstobethesamethenwedealwiththesymmetricproducthigher
rank numerical range Λsymm. (A). On the other hand, if only some of projections are the
k1 k2
samewecallthesetlocally⊗sy⊗m··m· etricproducthigherranknumericalrangeΛloc.symm.(A). In
k1 k2
the latter case,in a concretecase ofinterest,one needs to specify whichproject⊗ion⊗s·a··reto be
chosen the same. Obviously, the notion is meaningful only for operators acting on Cd Cd.
⊗
One has:
Fact 2 It holds
Λ (A)= Λsymm. UA U = Λsymm.(V AV ). (12)
m⊗n m⊗n 1⊗ 1⊗ † m⊗n ⊗1 †⊗1
U V
[ (cid:0) (cid:1) [
Proof: The defining equation (10) can be rewritten as [R (U RU)]A[R (U RU)] =
† †
⊗ ⊗
λ(R U RU)forsomeunitaryU. Itisequivalentto(R R)( U)A( U )(R R)=λR R
† †
from⊗which the result follows. The second case is show⊗n in a1⊗similar1m⊗anner. (cid:4)⊗ ⊗
Naturally, one can rewrite the fact for the multipartite case.
We propose to call the k k product higher rank numerical range multipartite as
1 2
⊗ ⊗···
opposed to the bipartite one, which corresponds to the k k case.
1 2
⊗
We also propose to use the dual set, the k k product codes set for A, defined as
1 2
⊗ ⊗···
follows
V (A)= R , R , :R R AR R =λR R . (13)
k1⊗k2⊗··· { ∈Pk1 ′ ∈Pk2 ··· ⊗ ′⊗··· ⊗ ′⊗··· ⊗ ′⊗···}
In a standard, i.e.non–product case, there is more than one projection (it may even be
an infinite number, when the spectrum of an operator is degenerated) corresponding to the
same compression value. This degeneracy may be removed in a product case, but we do not
know to what extent this happens in a generic case.
Although we will be interested mainly in BUC for which KL give the single equation (5)
tosolve,itisnaturaltointroducethenotionofthejointproducthigherranknumericalrange
(just as it is defined in the stadard case [30]). Namely, one defines
Definition 2 The k k joint product higher rank numerical range of operators A ,
1 2 i
⊗ ⊗···
i=1,2,...,I is defined to be the following set
Λ (A ,A ,...,A )=
k1 k2 1 2 I
⊗ ⊗···
(λ ,λ ,...,λ ) CI :R R A R R =λ R R (14)
1 2 I ′ i ′ i ′
{ ∈ ⊗ ⊗··· ⊗ ⊗··· ⊗ ⊗···}
for some R , R ,...
∈Pk1 ′ ∈Pk2
6 Title ...
Forfurtherconvenience,wealsointroducethecommonhigherranknumericalrangewhich
is defined as follows
Definition 3 The k k common product higher rank numerical range of operators
1 2
⊗ ⊗···
A , i=1,2,...,I is defined to be the following set
i
Λcomm. (A ,A ,...,A )= λ C:R R A R R =λR R (15)
k1⊗k2⊗··· 1 2 I { ∈ ⊗ ′⊗··· i ⊗ ′⊗··· ⊗ ′⊗···}
for some R , R ,...
∈Pk1 ′ ∈Pk2
Inthepresentpaper,wewillbeinterestedsolelyinthecaseswhentheproductstructureof
the projectors corresponds to the tensor product structure of Hilbert space (which is usually
uniquely determined by the problem under consideration).
What differs the standard and the product higher rank numerical ranges most strikingly
is the fact that while the former is determined solely by the eigenvalues of an operator, the
latter would also be affected by the form of eigenvectors (in applications to construction of
QECC it is enough to consider normal operators). This makes the product range difficult to
determine even with the knowledge of the standard one.
Product higher rank numerical range bears the following natural features d:
Property 3 Λ Λ
m n mn
⊗ ⊆
Property 4 Λ (A) can be empty even when Λ is non empty.
m n mn
⊗
Property 5 Λ Λ
m n loc
⊗ ⊆
Property 6 Λ (A) and V (A) are both compact sets
m n m n
⊗ ⊗
Property 7 Λ (A) Λ (A), i=1,2.
(m1+m2)⊗n ⊆ mi⊗n
Property 8 Λ (A B)=Λ (A)⊠Λ (B).
m n m n
⊗ ⊗
Property 9 Λ (A)=Λ (U VAU V ) for arbitrary unitary U and V.
m n m n † †
⊗ ⊗ ⊗ ⊗
Property 10 Λcomm.(A,B) Λjoint(A,B)
Property 11 Λsmy⊗mnm. (A) ⊆Λlocm.⊗synmm.(A) Λ (A)
m⊗n⊗p ⊆ m⊗n⊗p ⊆ m⊗n⊗p
Before we proceed, we need to recall the notion of the C–numerical range W (A) of an
C
operator A, which goes as follows [32]:
W (A)= trC U AU,U . (16)
C † †
{ ∈U}
When Us are taken to be product one deals with the local C–numerical range Wloc.(A) [33].
C
Incases consideredin the presentpaper the setis a closedinterval. With this notionin hand
we can give the following bound on the product higher rank numerical range:
Observation 12 It holds that
1
Λk1⊗k2⊗···(A)⊆WRlo⊗c.R′⊗···(cid:18)k1k2···A(cid:19). (17)
Proof: It is enough to take trace of both sides of the defining equation (10) of the product
rangetoconcludethateveryλbelongingtoΛ (A)mustalsobelongtoWloc. (A/k k ).
(cid:4) k1⊗k2⊗··· R⊗R′⊗··· 1 2···
dInwhatfollowsweuse⊠todenotetheMinkowskiproductoftwosetsonthecomplexplane,whichisdefined
asfollows: Z1⊠Z2={z:z=z1z2;z1∈Z1;z2∈Z2}.
Author(s) ... 7
Naturally, we assume that Wloc. is local according to the same cut as the higher rank
range is product.
The observation will turn out to be very useful in bounding the product higher rank
numerical range. Its application, however,will require a numerical optimization.
We single out also some chosen properties of the common range, which will serve as a
basis for one of the examples.
Property 13 Λcomm.(A, A)= 1 or .
k 1− {2} ∅
Property 14 Let Q= l i i Q , with orthonormal basis i and rank q projections
i=1| ih |⊗ i {| i} i
Q ,actond d . Thenfollowingholds: (i)ifl=d thenΛ (Q)=Λcomm.(Q ,Q ,...,Q ),
i 1⊗ 2 P 1 d1⊗k k 1 2 d1
(ii) if l<d then either Λ (Q)= or Λ (Q)= 0 with the latter holding if and only
if 0 Λcomm1.(Q ,Q ,...,Qd1⊗).k ∅ d1⊗k { }
∈ k 1 2 l
Property 15 Λcomm.(A ,A , ,A ) Λ ( K α A ) with α = 1. In particular, for
k 1 2 ··· K ⊆ k i=1 i i i i
K =2, Λcomm.(A ,A ) Λ (αA +(1 α)A ).
k 1 2 ⊆ k 1 − 2P P
ProofofProperty13: Adding(i)RAR=λRand(ii)R( A)R=λRweobtainR=2λR,
1−
thus λ = 1/2 if equation (i) has a solution with this value, if it does not — Λ (A) is empty
k
and so is Λcomm.(A, A). (cid:4)
k 1−
Proof of Property 14: (i) We look for λ obeying
d1
R R′ i i Qi R R′ =λR R′ (18)
⊗ | ih |⊗ ! ⊗ ⊗
i=1
X
for projections R and R rank, respecitevly, d and k. Since d1 i i = , it must be
′ 1 i=1| ih | 1d1
R= , thus Eq. (18) turns into
1d1 P
RQ R =λ R, (19)
1d1 ⊗ ′ i ′ 1d1 ⊗ ′
from which immediately follows
R′QiR′ =λR′ i (20)
∀
concluding the proof of this part. (ii) replacing upper limit in the sum with l < d and
1
considering diagonal terms we obtain
RQ R =λR, i=1,2,...,l (21)
′ i ′ ′
and d l equations 0 = λR. Thus λ must be equal to zero and this value must be in the
1 ′
−
common range of Q , i=1,2,...,l. The claim follows. (cid:4)
i
Proof of Property 15: Let K = 2 for simplicity. We have (i) RA R = λR and (ii)
1
RA R = λR. We multiply the first equation by α, the second by 1 α and add such
2
−
equations to obtain (iii) R(αA +(1 α)A )=λR. Thus each λ which fulfills (i-ii) for some
1 2
−
R, fulfills also (iii). The result then follows. (cid:4)
Our special interest in finding product higher rank numerical ranges of projections stems
from the special type of noise we mainly focus on in the paper (see Eq. (22)).
8 Title ...
4 Two–access quantum communication: QECC for a BUC
We now specify the type of noise we will further consider. We assume that U is hermitian.
Hermiticity of a unitary matrix implies that it must be of the form
U =P Q (22)
−
with some projections P, Q.
4.1 Zero entropy codes(λ= 1)
±
In Ref. [21] the following theorem was provede
Theorem 16 A M N DFS for a channel Eq.(1) with the noise model Eq. (22) exists if
⊗
andonlyifatleastoneofthesubspaces P and Q is(d M,d N)–decomposable.
P ≡ H Q≡ H − −
It was also shown that it can be generalized to the generic case of U allowing for DFS, i.e.,
U =P0+ keiβkQk with r(P0)≥MN.
In Ref. [21] we have applied the theorem to several concrete examples. We recall two of
P
them (in both cases =span φ )
i
Q {| i}
for the following choice of φ no 2 2 code exists
i
• ⊗
1
φ = (11 + 22 ),
1
| i √2 | i | i
1
φ = (10 + 21 ),
2
| i √2 | i | i
for the following choice the 2 2 code does exist
• ⊗
1
φ = (02 + 10 ),
1
| i √2 | i | i
1
φ = (01 + 20 ).
2
| i √2 | i | i
4.2 Higher entropy codes(λ= 1)
6 ±
We now move to the case of higher entropy codes, that is for λ= 1 in Eq. (5).
±
Let us start with some general remarks concerning the non product case, that is Eq. (4).
Assuming Eq. (22) and denoting
γ =(1 λ)/2 (23)
−
we obtain
RQR=γR, (24)
or, equivalently,
RPR=(1 γ)R (25)
−
eSee Ref. [21] for explanation of the notion of the decomposability and the justifications of conclusions
concerningexamples.
Author(s) ... 9
It is almost evident that if Eq. (24) is to be fulfilled the rankfof Q must be at least as large
as the rank of R but it cannot be too large since Eq. (25) must also hold. The theory of the
higher rank numerical range [16, 17] makes this intuition strict and we have that 0 γ 1
≤ ≤
only when r(R) r(Q) n r(R), where n is the dimension of the whole space. We will
≤ ≤ −
later prove a result of this kind for ranks ofthe reduced matrices of P andQ. It is clear that
in such situations if γ Λ(Q) then it is also the case for 1 γ for some other projection.
∈ −
In this way we have a dual pair of projections, say R and R˜. Notice that both codes have
exactly the same entropy (see Eq. (6)).
The main result of this section is the necessary condition for γ to belong to the product
higher rank numerical range of a projection.
Theorem 17 Let Q be a rank l projection. Assume R R Q R R =γR R
l M⊗ N′ l M⊗ N′ M⊗ N′
holds. Let further x x ... be eigenvalues of tr Q , and y y ... of tr Q . Then
1 2 B l 1 2 A l
≥ ≥ ≥ ≥
M r(trBQl)
x Nγ + x MN (1 γ)(1+3γ)+l MN, (26)
i i
| − | ≤ − −
i=1 i=M+1
X X p
N r(trAQl)
y Mγ + x MN (1 γ)(1+3γ)+l MN.
i i
| − | ≤ − −
i=1 i=N+1
X X p
The proof of the theorem is based on severallemmae (some concerning also the non–product
case), which we shall prove below.
Lemma 1 Fix a number n. Let R = k e e be a projection operator (the states
i=1| iih i|
e are orthonormal) onto a subspace of an n dimensional space . Denote with =
i R
| i P H B
e , e ,..., e and its orthonormal complement, so that span = . If Eq.
1 2 k ⊥ R ⊥
{| i | i | i} B {B ⊕B } H
(24) holds with 0<γ <1 then Q must have the following form in ,
R ⊥
B ⊕B
γ γ(1 γ)
−
 ... p ... 0 
γ γ(1 γ)
 − 
Q= pγ(1−γ) ... 1−γ ... p 0 , (27)
 
 γ(1 γ) 1 γ 
 − − 
 0 0 S 
 p 
 
wherenonzeroblockswithoffdiagonaltermsequaltozeroareofsizek k andS isaprojector
×
of size (n 2k) (n 2k).
− × −
Proof:If Eq. (24) holds then Q, which is hermitian, must necessarily have the form
γ A
Q= 1k (28)
A B
†
(cid:18) (cid:19)
fInthiscontext therankofamatrixisunderstoodinastandardsense.
10 Title ...
with in the basis of the eigenvalues of R. Here A is a rectangular k k matrix, while B
k ′
1 ×
denotes a positive semidefinite square matrix of size k . Since it is a projection it must be
′
true that Q2 =Q. For convenience we explicitly write the left hand side of this equation
γ2 +AA γA+AB
Q2 = 1k † . (29)
γA +BA A A+B2
† † †
(cid:18) (cid:19)
Comparison of Eqs. (28) and (29) gives us the set of conditions
γ2 k+AA† =γ k, (30)
1 1
A A+B2 =B, (31)
†
γA+AB =A. (32)
From the first one we obtain
AA =x2 , x2 =γ γ2 >0. (33)
† k
1 −
DenotingA˜=A/xwecanrewritethis asA˜A˜ = . ThusA˜A˜ mustberankk asr( )=k.
† k † k
Since for an arbitrary A˜ it holds that r(A˜A˜ ) =1r(A˜) and r(A˜) min k,k we in1fer that
† ′
≤ { }
k k as otherwise we would get a contradiction. Moreover A˜ is an isometry so we have
′
≥
k
A=xA˜=x e v (34)
i i
| ih |
i=1
X
with some orthonormal v Ck′. There are k such states in total but here only k of them
i ′
| i∈
are important. We thus have
k
A A=x2 v v . (35)
† i i
| ih |
i=1
X
From Eq. (31) we conclude that [A A,B] = 0 since an operator commutes with its own
†
function. The above equation implies that B has the following form
k
B = λ v v B (36)
i i i ′
| ih |!⊕
i=1
X
whereB livesonthesubspacespannedbysomeorthonormalset v k′ . Thesetofkar-
′ {| ii}i=k+1
bitrarynonnegativerealnumbersλ needstobedetermined. Inthebasis e , e ,..., e , v ,...,
i 1 2 k 1
{| i | i | i | i
vk , vk+1 ,..., vk′ the matrix Q reads
| i | i | i}
γ x
 ... ... 0 
γ x
 
Q= x λ1 . (37)
 
 ... ... 0 
 
 x λ 
 k 
 0 0 B 
 ′ 
 