Separability of Bosonic Systems

0.13 MB·English

by  Nengkun Yu

Separability of Bosonic Systems
Nengkun Yu1,2∗
1Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada
2Department of Mathematics & Statistics, University of Guelph, Guelph, Ontario, Canada
In this paper, we study the separability of quantum states in bosonic system. Our main tool
here is the “separability witnesses”, and a connection between “separability witnesses” and a new
kindofpositivityofmatrices—“PowerPositiveMatrices”isdrawn. Suchconnectionisemployedto
demonstrate that multi-qubit quantum states with Dicke states being its eigenvectors is separable
if and only if two related Hankel matrices are positive semidefinite. By employing this criterion,
weareable toshowthat suchstate isseparable if andonly if it’spartial transpose isnon-negative,
5 which confirms the conjecture in [Wolfe, Yelin, Phys. Rev. Lett. (2014)]. Then, we present a class
1 of bosonic states in d⊗d system such that for general d, determine its separability is NP-hard
0 although verifiableconditions for separability is easily derived in case d=3,4.
2
PACSnumbers: 03.65.Ud,03.67.Hk
n
a
J
Introduction—Entanglement, first recognized as a of great interest is to study the entanglement of bosonic
3
spooky feature of quantum machinery by Einstein, system [12–16]. For N-qubit bosonic system, a natural
1
Podolsky,andRosen[1],liesattheheartofquantumme- basisisN-qubitDickestates(unormalized)whicharede-

] chanics. It has been discovered that entanglement plays fined as,
h
anessentialroleinvariousfundamentalapplicationsand
p
protocols in quantum information science, such as quan- |D i:=P |0i⊗n⊗|1i⊗N−n ,
- N,n sym
t tum teleportation, superdense coding and cryptography
n
(cid:0) (cid:1)
a [2–4]. Moreover,highorderofmultipartiteentanglement with P being the projection onto the Bosonic (fully
sym
u hasbeenshowntoberequisitetoreachthemaximalsen- symmetric) subspace, i.e., P = 1 U , the
q sitivity in metrological tasks [5]. sum extending over all permustyamtion oNpe!ratπo∈rsSNU πof the
[ P π
Inmultipartitesystems,aquantumstate iscalledsep- N-qubit systems. It is worth to note that the entangle-

1 arableifitcanbewrittenasastatisticalmixtureofprod- mentofpureDickestatehasbeenwidelystudiedrecently
v
uct states, otherwise, it is entangled. Research on sepa- [17–24].
7
5 rabilitycriteria,thatis,oncomputationalmethodstode- InthisLetter,wefocusontheproblemoftheseparabil-
9 termine whether a given state is separable or entangled, itycriterionforquantumstatesinbosonicsystembycon-
2 turns out to be a a cumbersome problem and essential sidering separable witnesses. We first draw a connection
0 subject in quantum information theory. Starting from between the separable witnesses of general multi-qubit
.
1 the famous PPT (Positive Partial Transpose) criterion bosonicstatesanda newtype ofpositivityofmatrices—
0 [6], a considerable number of different separability crite- what we called “Power Positivity”. This connection is
5
rions have been discovered (see the references in [7, 8]). employed to study the separability of N-qubit quantum
1
One fundamental tool of detecting entanglement is en- stateswhichbeingthemixtureofDickestates. Inpartic-
:
v tanglement witnesses [10, 11], which is equivalent to the ular,theseparablewitnessesofsuchstatescorrespondsto
i
X method of positive, but not completely positive maps. diagonal “Power Positive” matrices, that are just poly-
Entanglement witnesses are observables that completely nomials whose value of is always non-negative for non-
r
a characterize separable states and allow to detect entan- negative variable. By employing the characterization of
glement physically. Their origin stems from Hyperplane non-negative polynomials, an easily evaluated complete
separation theorem of geometry: the convex sets can be criterionforthe separabilityofmixture ofDickestatesis
described by hyperplanes. In particular, a witness is an demonstrated. Moreover,we show that any suchsepara-
observable, which is non-negative for separable states, blestatecanbewrittenasthemixtureof(N+1)(N+2)
and it can have a negative expectation value for entan- productstates. We then study the separabilityofa class
gled states. of states whose eigenvectors are generalized d⊗d Dicke
Despite great efforts and considerable progress have states. Itis provedthatthe separabilityproblemof such
been made, the physical understanding and mathemat- statesisNP-completeforgenerald,althoughverysimple
ical description of its essential characteristics remain criterion is demonstrated for d=3,4.
however highly nontrivial tasks, especially when many- Main Results—IntheN-quditsystemH ⊗H ⊗···⊗
1 2
particle systems are analyzed. Moreover, it was shown H withdbeingthedimensionofeachHilbertspaceH ,
N i
byGurvits[9]thatthis problemisNP-hard. However,it the bosonic space is a subspace that spanned by pure
is still possible to havecomplete criterionfor the separa- quantum states which are invariant under the swap of
bility of some interesting certain situations. A problem anytwosubsystemsamongallN subsystems,i.e.,forthe
2
swap operator exchanging the two qudits system F , Notice thatthe setofseparablewitnessesformsa con-
i,j
vex compact set. In order to check the separability of
S :≡{|ψi:|ψi=Fi,j|ψi, for all i, j and Swap F}. bosonicstates,one wayis to parameterizethe setofsep-
arable witnesses, at least the set of extreme points of
A mixed state ρ is called bosonic if its support is a sub-
separable witnesses.
spaceofbosonicspacewherethesupportofρ,Supp(ρ),is
For simplicity, we mainly focus on the separable wit-
the subspace spanned by the eigenvectors corresponding
nesses of N-qubit bosonic system.
to its non-zero eigenvalues. In other words, ρ = F ρ =
i,j Notice that any Hermitian W = P HP corresponds
S S
ρF holds for any 1≤i,j ≤N.
i,j toaHermitianmatrixM :=(m ) asfollows
i,j (N+1)×(N+1)
One very simple observation is that, if a bosonic state
ρisseparable,i.e.,thereexistproductstates(unnormal-
N
ized) ⊗Nk=1|αjkihαjk| such that W := mi,j|D]N,iihD]N,j|
i,j=0
N X
ρ=XO|αjkihαjk|, where we employe the dual basis of Dicke states as
j k=1
then we can choose product states |αji⊗N, that is |D]N,ni:= Nn −1Psym |0i⊗n⊗|1i⊗N−n ,
(cid:18) (cid:19)
N (cid:0) (cid:1)
ρ=XO|αjihαj|. that is, hDN,m|D]N,ni=δm,n.
j k=1 NowwecanderivetheconditionforW beingseparable
witness: tr(Wα⊗N) ≥ 0 holds for all one-qubit |αi is
To see this, one only need to observe that
equivalent to
N N
O|αjki∈S ⇒∃|αji,O|αjki=|αji⊗N. tr(W|0ih0|⊗N)≥0⇔mN,N ≥0,
k=1 k=1 tr{W[(|1i+z|0i)(h1|+z∗h0|)]⊗N}≥0,
Now, we introduce the separable witnesses for bosonic
forallz ∈C. Onecanobservethatthesecondcondition
system as a useful tool: For N-qudit system, a Hermi-
indicates the first one as |z|→∞.
tian operator W is called a separable witness of bosonic
system if W = PSWPS with PS being the projection of Observethat(|1i+z|0i)⊗N = Nj=0zj|DN,ji,weknow
the bosonic space S and it satisfies that that the second condition given above is just
P
tr(Wα⊗N)≥0,for all α=|αihα|. ~z†M~z ≥0 for all ~z =(1,z,z2,··· ,zN)T ∈CN+1.
Thisiswhatwecalled“PowerPositiveMatrix”,whichis
The importance of separability witness is due to the fol-
far different from “Semi-definite”, “Complete Positve”.
lowing proposition.
Unfortunately, we are not able to give a complete de-
Proposition 1. A bosonic state ρ is separable if and scription of the set of “Power Positive Matrices”, even
only if tr(Wρ) ≥ 0 holds for all separability witness W one can easily conclude that it is a superset of “Semi-
of bosonic system. definite Matrices”.
Althoughitisdifficulttochecktheseparabilityofgen-
Remark: This proposition can be generalized to the eral N-qubit states, we demonstrate an easily verified
separability of quantum states lying in fixed subspace. analytical condition for the separability of the following
Proof:—Theonlyifpartsimplyfollowsfromtheabove generaldiagonalsymmetricstateswhichisnecessaryand
observation about the structure of separable states of sufficient.
bosonic system. To show the validity of the if part, In N-qubit bosonic system, one can naturally define
we assume the existence of entangled bosonic state ρ the following class of quantum states, so called the gen-
such that tr(Wρ) ≥ 0 holds for all separability witness eral diagonal symmetric states, GDS [16],
W of bosonic system. Notice that the set of separable
states of bosonic system is convex and compact. En- N
ρ= χ |D ihD |,
tangled ρ does not lie in this set, by hyperplane sep- n N,n N,n
aration theorem, one can conclude that there exists a nX=0
H such that tr(Hρ) < 0 and tr(Hα⊗N) ≥ 0 holds where χ represent the eigenvalues in the eigen-
n
for all α. Therefore, tr(Wα⊗N) = tr(PSHPSα⊗N) = decomposition of ρ.
tr(HPSα⊗NPS) = tr(Hα⊗N) ≥ 0 for W = PSHPS, NoticethatanyGDSstateρenjoysthesymmetrythat
then W is a separability witness. On the other hand, for all diagonal qubit unitary U =diag{1,eiθ},
θ
tr(Wρ) = tr(P HP ρ) = tr(HP ρP ) = tr(Hρ) < 0,
S S S S
which contradicts to the assumption. (cid:4) ρ=U⊗Nρ U†⊗N.
θ θ
3
Thus, for any separability witness W, we have Proposition 4. Extreme point of the diagonal separable
witnesses for GDS has one of the following forms
tr(Wρ)=tr(WU⊗Nρ U†⊗N)=tr(U†⊗NWU⊗Nρ),
θ θ θ θ
^ ^
S = a a |D ihD |,
i j N,i+j N,i+j
W is a “diagonal” separable witness and tr(W ρ) =
tr(0Wρ) with 0 0≤Xi,j≤N2
^ ^
T = b b |D ihD |,
i j N,i+j+1 N,i+j+1
W = 1 2πU†⊗NWU⊗Ndθ = N m |D]ihD]|. 0≤i,Xj≤N2−1
0 2π θ θ k,k N,k N,k
Z0 k=0 with a ,b ∈R.
X k k
Here, a separable witness W = Ni,j=0mi,j|D]N,iihD]N,j| Now we are ready to show our main result,
is called diagonal if m = 0 for i 6= j. According to
i,j
Proposition 1, we know that: P Theorem 1. The GDS stateρ= Nn=0χn|DN,nihDN,n|
is separable if and only if the following two Hankel Ma-
Proposition 2. A general diagonal symmetric state ρ trices [25] M ,M are positive semPi-definite, i.e.,
0 1
is separable if and only if tr(W ρ) ≥ 0 for all diagonal
0
separable witness W . χ χ ··· χ
0 0 1 m0
χ χ ··· χ
Recalltheconceptof“PowerPositiveMatrix”,W0 isa M0 := ··1· ··2· ··· m··0·+1 ≥0, (1)
sneopna-rnaebglaetiwvietnfoerssailflazn∈dCon.lTyhifis isNk=eq0umivka,kle|zn|t2ktois always χm0 χm0+1 ··· χ2m0 
P  χ1 χ2 ··· χm1 
χ χ ··· χ
g(r)≥0 for all r ≥0, M1 := ··2· ··3· ··· m··1·+1 ≥0, (2)
for real coefficient polynomial g(x) := Nk=0mk,kxk,  χm1 χm1+1 ··· χ2m1−1 
whosevalueg(x)isalwaysnon-negativefornon-negative
P where m :=[N] and m :=[N+1].
x. The characterization of such polynomials is accom- 0 2 1 2
plished by the following proposition.
Proof:—According to Proposition 2 and Proposition
4, ρ is separable if and only if tr(W ρ) ≥ 0 holds for
Proposition 3. A real coefficient polynomial g(x) satis- 0
anyextremepointW ofthediagonalseparablewitnesses
fies that g(r) ≥ 0 for all r ≥ 0 if and only if there exist 0
for GDS, that is, for all ~a = (a ,··· ,a )T ∈ Rm0+1,
real coefficient polynomial P (x),Q (x) such that 0 m0
i i ~b = (b ,··· ,b )T ∈ Rm1the following quadratic forms
1 m1
g(x)= xP2(x)+ Q2(x). are non-negative,
i i
i i
X X tr(Sρ)= χ a a =~aTM ~a≥0,
i+j i j 0
Proof:—The if part is simple. To show the validity 0≤iX,j≤m0
of the only if part, we use the fundamental theorem of tr(Tρ)= χ b b =~bTM ~b≥0.
i+j−1 i j 1
algebra,
1≤iX,j≤m1
g(x)=a0 (x−zk)lk. It is equivalent to the non-negativity of real Hankel Ma-
trices M ,M . (cid:4)
Y 0 1
For non real root zk, we know that for all real r, Now we are going to present the rigorous proof of the
conjecture from [16].
(r−z )(r−z¯)=(r−Re(z ))2+Im2(z )≥0.
k k k k
Theorem 2. The GDS stateρ= N χ |D ihD |
n=0 n N,n N,n
For non-positive zk, we know that for all r ≥0, isseparableifandonlyifitisPPT.Moreprecisely, ifand
only if it is PPT under the partialPtranspose of m =[N]
0 2
r−zk =r+(−zk)≥0. subsystems.
For positive zk, its power lk must be even. Proof:—First, it is sufficient to consider the partial
Thus, expanding g(x) = a0 (x−zk)lk confirms the transposeofthefirstm0subsystemsbynoticingthesym-
only if part. (cid:4) metricinthebosonicsystem. Assumeρispositiveunder
Q
Invoking the relation between the diagonal separable the partial transpose of m = [N] subsystems, accord-
0 2
witnessW andthe polynomialg(x), onecandeduce the ing to Theorem 1, we only need to show M ,M ≥ 0 of
0 0 1
following, Eq.(1,2).
4
OnecanwriteρΓ inbasis|D i|D iwith0≤j ≤ where Rd stands for the d-dimensional vector space
m0,j m1,k +
m ,0≤k ≤m by verifying the following equations, whose entries are all non-negative.
0 1
Itiswidelyknownthatthedecisionproblemoncheck-
n
ing the completely positivity ofgivenmatrixis NP-Hard
|D i = |D i|D i, for n≤m ,
N,n m0,j m1,n−j 0
for general d while for d=3,4, checking that the matrix
j=0
X is positive semidefinite and has all entries ≥ 0 is both
m0
|D i = |D i|D i, for n>m , necessary and sufficient. Formally,
N,n m0,j m1,n−j 0
j=Xn−m1 Theorem 3. It is NP-Hard to decide whether a given
d ⊗ d GDS state is separable. On the other hand,
where m =N −m .
Since ρ1Γ ≥ 0, the0n the restriction of ρΓ on subspace ρ = di,j=1χi,j|ψi,jihψi,j| is separable if and only if
χ=(χ ) is semi-definite positive.
spanned by {|D i|D i,0 ≤ j ≤ m } is still non- ij d×d
m0,j m1,j 0 P
negative, direct calculation leads us to the fact that this In other words, we have the following: PPT criterion
is just M0 ≥0. is not sufficient for detect the entanglement for bosonic
On the other hand, the restriction of ρΓ on subspace states, even for GDS states unless at least P = NP,
spanned by {|Dm0,j−1i|Dm1,ji,1 ≤j ≤m1} is still non- which is highly impossible.
negative, direct calculation leads us to the fact that this Conclusion—In this paper, we study the separability
is just M1 ≥0. problem of bosonic system. An analytical condition for
Invoking Theorem 1, we can conclude that ρ is sepa- the separability of n-qubit states whose eigenstates are
rable. (cid:4) Dickestatesisdemonstrated. Forbipartitequditsystem,
One can have the following interesting corollary, we presenta class of standardbosonic states,for general
Corollary 1. GDS state ρ = N χ |D ihD | is d, its separability is NP-hard, while for d = 3,4, the
n=0 n N,n N,n condition of separability is provided.
positive under the partial transpose of m = [N] subsys-
P 0 2 It is still not clear that whether there exist easily ver-
tems, then it is positive under the partial transpose of
ified analytical conditions for the separability of general
arbitrary subsystems.
n−qubit bosonic states?
Inthe following,weintroduceaclassofbipartite GDS We thank Prof. John Watrous, Prof. Debbie Leung
states, and study the separability of such states: In d⊗ andProf. BeiZengfortheircomments. NYissupported
d system, one can define the following general diagonal by NSERC, NSERC DAS, CRC, and CIFAR.
symmetric states,
d
ρ= χ |ψ ihψ |,
i,j i,j i,j
iX,j=1 ∗ Electronic address: [email protected]
[1] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47,
|iii if i=j, 777 (1935).
with |ψ i := being some basis
i,j (|iji+|jii otherwise. [2] C. Bennett and S.J. Wiesner, Phys. Rev. Lett 69, 2881
(1992).
of the bosonic subspace of d⊗d system, i.e., the sym-
[3] C. H. Bennett, G. Brassard, C. Crpeau, R. Jozsa, A.
metric subspace. Peres, and W. K. Wootters, Phys. Rev. Lett 70, 1895
Noticethatρ=(U⊗U)ρ(U⊗U)† holdsforalldiagonal
(1993).
quditunitaryU. Then, ρisseparableifandonlyifthere [4] C.H.BennettandG.Brassard,ProceedingsofIEEEIn-
exist qudit states |αki= dj=1xk,j|ji such that ternational Conference on Computers, Systems and Sig-
nal Processing, 175, 8 (1984).
P [5] C. Gross, T. Zibold, E. Nicklas, J. Esteve, and M. K.
ρ = (U ⊗U)α⊗2(U ⊗U)†dU
k Oberthaler, Nature(London) 464, 1165 (2010).
Xk Z [6] A. Peres, Phys.Rev.Lett. 77, 1413 (1996).
= |x |2|x |2|ψ ihψ |. [7] L.M. Ioannou, Quant.Inf. Comp. 7, 335 (2007).
k,i k,j i,j i,j
[8] R.Horodecki,P.Horodecki,M.Horodecki,K.Horodecki,
k
X Rev. Mod. Phys. 81 865-942,(2009).
⇔χ: = (χij)d×d = x~kx~kT. [9] L. Gurvits, Journal of Computer and System Sciences,
k 69, 3 (2004).
X
[10] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.
where dU ranging over all diagonal unitaries, and x~k = Lett. A 223, 1 (1996); M. Horodecki, P. Horodecki, and
(|xk,1|2,··· ,|xk,d|2)T ∈Rd. R. Horodecki, Phys. Lett.A 283, 1 (2001).
Recall that the cone of complete completely positive [11] B. M. Terhal, Phys.Lett. A 271, 319 (2000).
matrices [26, 27] is define as [12] G. T´oth and O. Gu¨hne, Applied Physics B 98, 617
(2009); G. T´oth and O. Gu¨hne, Phys. Rev. Lett 102,
C ={ y~ y~ T :y~ ∈Rd}, 170503 (2009); O. Gu¨hne and G. T´oth, Physics Reports
k k k + 474, 1 (2009).
i
X
5
[13] K. Eckert, J. Schliemann, D. Bruss and M. Lewenstein, [22] T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L.
Annalsof Physics 299, 88-127 (2002). Lamata, and E. Solano, Phys. Rev. Lett 103, 070503
[14] J.Tura,R.Augusiak,P.Hyllus,M.Kus,J.Samsonowicz (2009).
andM.Lewenstein, Phys.Rev.A85, 060302(R) (2012). [23] T. Bastin, C. Thiel, J. vonZanthier, L. Lamata, E.
[15] M. Cramer, A. Bernard, N. Fabbri, L. Fallani, C. Fort, Solano, andG.S.Agarwal, Phys.Rev.Lett.102, 053601
S.Rosi,F.Caruso,M.InguscioandM.B.Plenio,Nature (2009).
Communications 4, 2161 (2013). [24] W.Wieczorek,N.Kiesel, C.Schmid,andH.Weinfurter,
[16] E. Wolfe, S.F. Yelin, Phys. Rev. Lett. 112, 140402 Phys. Rev.A 79, 022311 (2009).
(2014). [25] J. R. Partington, An Introduction to Hankel Opera-
[17] N.Yu,Phys. Rev.A 87, 052310 (2013). tors,London Mathematical Society Student Texts 13,
[18] W. Du¨r, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, Cambridge University Press.
062314 (2000). [26] H. Diananda, Mathematical Proceedings of the Cam-
[19] N.Yu,E. Chitambar, C. Guo, and R. Duan,Phys. Rev. bridge Philosophical Society, 58, 17 (1961).
A 81, 014301 (2010). [27] L. J. Gray and D. G. Wilson. Linear Algebra Appl, 31,
[20] N. Yu, C. Guo, and R. Duan, Phys. Rev. Lett 162, 119-127, (1980).
160401 (2014). [28] K.G. Murty and S.N. Kabadi, Mathematical Program-
[21] R. Hu¨bener, M. Kleinmann, T-C Wei, C. Gonz´alez- ming, 39,117-128 (1987).
Guill´en,andO.Gu¨hne,Phys.Rev.A80,032324(2009).