Entanglement Witnesses for Indistinguishable Particles

1.5 MB·English

by  A. Reusch

Entanglement Witnesses for Indistinguishable Particles
A. Reusch, J. Sperling,∗ and W. Vogel
Arbeitsgruppe Theoretische Quantenoptik, Institut fu¨r Physik, Universita¨t Rostock, D-18051 Rostock, Germany
(Dated: April 22, 2015)
We study the problem of witnessing entanglement among indistinguishable particles. For this
purpose,wederiveasetofequationswhichresultsinnecessaryandsufficientconditionsforprobing
multipartite entanglement between arbitrary systems of Bosons or Fermions. The solution of these
equationsyieldstheconstructionofoptimalentanglementwitnessesforpartialandfullentanglement
indiscreteandcontinuousvariablesystems. Ourapproachunifiestheverificationofentanglementfor
distinguishableandindistinguishableparticles. Weprovidegeneralsolutionsforcertainobservables
tostudyquantumentanglementinsystemswithdifferentquantumstatisticsinnoisyenvironments.
PACSnumbers: 03.67.Mn,05.30.-d
5
1
0
I. INTRODUCTION the Schmidt decomposition [9]. Whenever a single ten-
2
sorproductstateissufficienttoexpandapurestate,itis
r
p Nonlocal correlations among many particles or quan- separable. In analogy, entanglement between IP is char-
A tizedfieldsareonekeyelementofthequantumnatureof acterized by the Slater decomposition [15–17]. A pure
Fermion or Boson state is separable, if it is a single an-
physics [1–4]. Applications in metrology use this quan-
1 tisymmetric or symmetric product state, given in terms
tum feature to beat classical limitations [5–8]. Quan-
2
of Slater determinants or permanents, respectively; see
tumentanglementhasbeenalsostudiedasaresourcefor

also [18, 19] for a slightly different approach. A classical
] quantum information technologies [9]. For a fundamen-
h mixtureofproductstatesextendsthecorrespondingdefi-
tal characterization, a lot of attention has been devoted
p nitionstomixedquantumstates. Ifsucharepresentation
to verify entanglement between distinguishable particles
- is impossible, the state under study is entangled.
t (DP) or multiple degrees of freedom [10, 11].
n
Indistinguishable particles (IP), on the other hand, Irregardless of the product for constructing compound
a
u are indispensable for understanding the properties of Hilbert spaces, ⊗,∧,∨, one can detect quantum correla-
q many-particle quantum systems. For IP systems having tion via so-called entanglement witnesses [15, 20, 21]. A
[ a certain spin statistics [12], however, even the notion witnessisanobservablewhichisnon-negativeforallsep-

of entanglement itself has no generally accepted defini- arable states, and may be negative for entangled states.
2
v tion [13]. For example, the two-Fermion or two-Boson Such criteria have been successfully applied to exper-
5 state, imentally probe quantum correlations [22–27]. In the
9 same dimensions, the characterization and application
5 |↑(cid:105)⊗|↓(cid:105)−|↓(cid:105)⊗|↑(cid:105)∼=|↑(cid:105)∧|↓(cid:105), of entanglement in systems of IP gained an increasing
2 |↑(cid:105)⊗|↓(cid:105)+|↓(cid:105)⊗|↑(cid:105)∼=|↑(cid:105)∨|↓(cid:105), (1) importance during the past years; see, e.g., [28–40]. In
0 particular, the entanglement of multiple qubits, realized
.
1 is a Bell-like entangled state using the tensor product ⊗, in spin systems, has been investigated [41–44], and the
0 question of how to extract entanglement for applications
and, at the same time, it is a product state in the notion
5 from systems subjected to the Pauli principle has been
oftheantisymmetricproduct∧orsymmetricproduct∨,
1
addressed [45, 46]. Recently, a method for the construc-
: respectively. Thisambiguityoriginatesfromthefactthat
v tionofoptimizedmultipartiteentanglementwitnessesfor
the(anti)symmetrizationrequirementof(Fermion)Boson
i DP has been proposed [47] and applied to perform a full
X systems has formally the same structure as a nonlocal
entanglement analysis of experimentally generated mul-
superposition. The left-hand side of Eq. (1) is closely re-
r
a lated to the well established theory of entanglement be- timode states [48].
tweendistinguishablesubsystems[14]. Hence,wewillfo- Inthepresentcontribution, wederiveequationswhich
cus on the right-hand side [15–17]. That is, for Fermions allow the construction of optimized, necessary, and suf-
andBosons,thestates(1)areseparableproductstatesin ficient entanglement probes for Boson and Fermion sys-
theexterioralgebraandsymmetricalgebra, respectively. tems. The formalism is applicable to arbitrary numbers
Ingeneral,wewillfocusonthefollowingquestion: ”How of particles, partially and fully entangled states, and dis-
does one certify quantum entanglement which does not crete and continuous variable quantum systems. Our
rely on (anti)symmetrization?” method unifies the detection of entanglement for DP
For bipartite pure states, the relation between entan- and IP. Furthermore, we explicitly construct witnesses
glement for DP and the tensor product is represented by to demonstrate the strength of this technique to certify
entanglement for different spin statistics in noisy envi-
ronments.
This work is structured as follows. In Sec. II, we dis-
∗Electronicaddress: [email protected] cussthenotionofmultipartiteseparablestatesofIP.The
2
construction of corresponding entanglement witnesses is |bk(cid:105) is, in general, not a product state in H⊗Nk. Finally,
derived in Sec. III. Bipartite examples are studied for IP a K-separable (anti)symmetric vector is defined as
andrelatedtothesystemsofDPinSec.IV.SectionVis
devotedtoestablishspinstatisticsindependentwitnesses |ψ±(cid:105)=Πˆ±|ψ (cid:105)=Πˆ±|b ,...,b (cid:105). (8)
K K 1 K
in multimode systems. We conclude in Sec. VI.
Since the permutation is applied in (8), the initial or-
dering of the Hilbert spaces in (7) does not play a role,
II. K-SEPARABLE FERMION AND BOSON i.e., the partition (N1,...,NK) relates to the Hilbert
STATES spacesequenceH⊗N1⊗···⊗H⊗NK forDPor–replacing
⊗ by ∧,∨ – for IP. Let us point out that (N ,...,N )
1 K
and (N(cid:48),...,N(cid:48) ) define, in general, different partitions
The formulation of multipartite entanglement for DP 1 K
if these tuples are not identical up to a permutation of
is based on the tensor product structure of compound
Hilbert spaces H⊗N. A N-partite quantum state σˆ is indices. For example, the three partition (2,3,1) of a six
mode system describes the same partitioning as (1,2,3),
fully separable, if it can be written as a convex combina-
but it differs from (2,2,2).
tion of product states of the subsystems [14],
AnexplicitexampleofpartialseparabilityofDPisthe
σˆ =(cid:90) dP(a ,...,a )|a1,...,aN(cid:105)(cid:104)a1,...,aN|. (2) tripartite state:
1 N (cid:104)a ,...,a |a ,...,a (cid:105)
1 N 1 N |Ψ(cid:105)=|0(cid:105)⊗(|1(cid:105)⊗|2(cid:105)+|3(cid:105)⊗|4(cid:105))∈H⊗3, (9)
Here, |a ,...,a (cid:105) = |a (cid:105) ⊗ ··· ⊗ |a (cid:105) are, in general,
1 N 1 N
unnormalizedN-partiteproductvectors,andP isaclas- using a single-mode orthonormal basis {|0(cid:105),...,|4(cid:105)}.
sical probability distribution. Similarly, one can construct partially separable states of
A fundamental postulate of quantum mechanics for DP in (8), e.g.,
Bosons or Fermions is that the quantum states are sym-
metric or antisymmetric upon exchange of the subsys- |Ψ+(cid:105)∼=|0(cid:105)∨(|1(cid:105)∨|2(cid:105)+|3(cid:105)∨|4(cid:105))∈H∨3
(10)
tems, respectively. This restricts the physical states to and |Ψ−(cid:105)∼=|0(cid:105)∧(|1(cid:105)∧|2(cid:105)+|3(cid:105)∧|4(cid:105))∈H∧3.
the (anti)symmetric subspace of the N-fold tensor prod-
uctHilbertspace,H∧N,H∨N ⊂H⊗N. Aprojectionfrom In Appendix B we prove – independently from the
the tensor product space to these subspaces is given by method to be developed later on – that these projected
the permutation operators Πˆ±, states,|Ψ±(cid:105)=Πˆ±|Ψ(cid:105),areindeedpartiallyseparable,K =
2, and not fully separable, K (cid:54)=3.
Πˆ±|a ,...,a (cid:105)= (cid:88) (±1)|σ||a ,...,a (cid:105), (3) Basedonthe(anti)symmetricproduct,onegetsagen-
1 N N! σ(1) σ(N) eral definition of N-separability for IP [33, 49]. Addi-
σ∈SN tionally, a N-Fermion or N-Boson quantum state σˆ is K
where |σ| and (±1)|σ| denote the parity and the sign of separable, 1 ≤ K ≤ N, if it can be written as a convex
the permutation σ ∈S , respectively. Other projections combination of (anti)symmetric product states,
N
might be similarly studied, which allow a generalization
to other parastatistics. Now, the (anti)symmetric prod- σˆ=(cid:90)dP(b ,...,b )Πˆ±|b1,...,bK(cid:105)(cid:104)b1,...,bK|Πˆ±.
uct states can be identified as [49] 1 K (cid:104)b ,...,b |Πˆ±|b ,...,b (cid:105)
1 K 1 K
(11)
|a (cid:105)∧···∧|a (cid:105)∼=Πˆ−|a ,...,a (cid:105), (4)
1 N 1 N
|a (cid:105)∨···∨|a (cid:105)∼=Πˆ+|a ,...,a (cid:105). (5) Exchangingthetensorproduct⊗byeitherthesymmetric
1 N 1 N
product ∨ or the antisymmetric product ∧, cf. Eqs. (4)
In general, for every vector |ψ(cid:105) ∈ H⊗N, we get the and(5),keepstheseparabilitydefinition(2)ofDPstruc-
(anti)symmetric vector in the projected subspace as turally preserved. If a state cannot be written according
to definition (11) for K = N, then entanglement be-
|ψ±(cid:105)=Πˆ±|ψ(cid:105). (6) tweenIPiscertifiedbeyondanycorrelationthatcanarise
from the (anti)symmetrization requirement of the quan-
See Appendix A for the symmetrization of operators.
tum statistics itself.
Equations (4) and (5) define fully or N-separable
Fermions and Bosons, respectively. More involved is the
notion of K-separable states; see [10, 11] for introduc-
III. CONSTRUCTION OF ENTANGLEMENT
tions. InsystemsofDPaK-separablevector|ψ (cid:105)∈H
K N WITNESSES
is defined as a product vector
|ψ (cid:105)=|b (cid:105)⊗···⊗|b (cid:105)=|b ,...,b (cid:105), (7) A. Separation of entangled states
K 1 K 1 K
for positive integers (N1,...,NK), |bk(cid:105) ∈ H⊗Nk, and Since Eq. (11) defines a closed, convex set of states,
(cid:80)K N = N. The tuple (N ,...,N ) defines a par- the Hahn-Banach separation theorem is applicable [50,
k=1 k 1 K
titioning of the N-fold Hilbert space. Further note that 51]. It states that for any closed, convex subset C of a
3
Banach space and a point, x ∈/ C, there exists a linear WedefinetheabbreviationXˆ foraHermitianoperator
bk
and continuous functional f that separates these sets: Xˆ, acting on H⊗N, by the relation
f(x) > sup f(c). In our case the Banach space is
c∈C
the set of Hermitian trace-class operators and the dual (cid:104)x|Xˆ |y(cid:105)=(cid:104)b ,...,b ,x,b ,...,b |Xˆ (17)
space is isomorphic to the set of bounded operators, i.e., bk 1 k−1 k+1 K
f(ρˆ)=tr(ρˆLˆ) for some Hermitian Lˆ. This means for the ×|b1,...,bk−1,y,bk+1,...,bK(cid:105),
problem under study that for any K-entangled state of
IP,(cid:37)ˆ=Πˆ±(cid:37)ˆΠˆ±, existsaboundedHermitianoperatorLˆ, for all |x(cid:105),|y(cid:105)∈H⊗Nk. Now, the derivative reads as
such that
tr(Lˆ(cid:37)ˆ)>sup{tr(Lˆσˆ):for all σˆ in Eq. (11)}. (12) 0= ∂∂(cid:104)bgk| = ∂(cid:104)∂bk|(cid:104)bk(cid:104)|b(kΠˆ|(±ΠˆLˆ±Πˆ)±bk)|bbkk|(cid:105)bk(cid:105) (18)
Related approaches to identify entanglement of DP can =(Πˆ±LˆΠˆ±)bk|bk(cid:105) − (cid:104)bk|(Πˆ±LˆΠˆ±)bk|bk(cid:105)(Πˆ±) |b (cid:105).
be additionally found in Refs. [20, 21, 47, 52]. Since (cid:104)b |(Πˆ±) |b (cid:105) (cid:104)b |(Πˆ±) |b (cid:105)2 bk k
the separation theorem ensures the existence of such an k bk k k bk k
operator, we have a necessary and sufficient condition
Further, inserting the definition of g in Eq. (15) yields
in terms of observables Lˆ probing multipartite entangle-
mentbetweenIPinfiniteandinfinitedimensionalspaces. (Πˆ±LˆΠˆ±) |b (cid:105)−g(Πˆ±) |b (cid:105)
However,findingtheproperLˆ foragivenstate(cid:37)ˆandde- 0= bj j bj j . (19)
termining the least upper bound of the right-hand side (cid:104)bj|(Πˆ±)bj|bj(cid:105)
of inequality (12) are cumbersome problems. In the fol-
lowing, we will propose a method to address the latter Let us summarize the conducted optimization which
aspect. results in Eq. (19). The derived set of algebraic equa-
tions,
(cid:0)Πˆ±LˆΠˆ±(cid:1) |b (cid:105)=g(cid:0)Πˆ±(cid:1) |b (cid:105) for j =1,...,K, (20)
B. Witness construction through optimization bj j bj j
definetheseparabilityeigenvalue(SEvalue)equationsfor
It is sufficient to take the least upper bound on the
IP. The common eigenvalue g is denoted as the SEvalue,
right-hand side of inequality (12) over all product vec- and the (anti)symmetric product vector Πˆ±|b ,...,b (cid:105)
tors, being the extremal points of the given convex set 1 K
is the corresponding separability eigenvector (SEvector)
of K-separable states. Moreover, the operator Πˆ± plays
for IP.
the role of the identity in the (anti)symmetric subspace.
The SEvalue equations for IP represent a system of K
Combining these facts allows us to write condition (12)
coupled eigenvalue equations. Remarkably, it turns out
in terms of the expectation value of the entanglement
that they have the same structure as the corresponding
witness operator:
equationsforDP[47]. ForDP,weusetheN-foldidentity
Wˆ =GΠˆ±−Πˆ±LˆΠˆ±, (13) ˆ1 that replaces the projector Πˆ± in (20). Even though
± wefindastrongrelationtothecaseofDP,cf.[47,54],we
(cid:40) (cid:41)
(cid:104)b ,...,b |Πˆ±LˆΠˆ±|b ,...,b (cid:105) want to point out that neither the distinguishable case
G=sup 1 K 1 K , (14)
(cid:104)b ,...,b |Πˆ±|b ,...,b (cid:105) includes the indistinguishable one, nor vice versa. This
1 K 1 K isduetothepropertiesofthenoninvertibleoperatorΠˆ±.
WefoundthattheSEvaluegcorrespondstoanoptimal
where the least upper bound is taken over all product
expectationvalueofLˆ forK-separableBosonorFermion
vectors (7); see also [15, 19]. Interestingly, the simple
states. Therefore, we get the bound in the entanglement
modification Πˆ± → ˆ1 in Eqs. (13) and (14) yields the
criterion (12) as
corresponding construction of witnesses for DP [47, 53].
TheleastupperboundinEq.(14)definesanoptimiza-
sup{tr(Lˆσˆ):for all σˆ in Eq. (11)}=sup{g}, (21)
tion of the Rayleigh quotient,
i.e., the initial convex optimization problem is solved by
(cid:104)b ,...,b |Πˆ±LˆΠˆ±|b ,...,b (cid:105)
g = 1 K 1 K →G, (15) the largest SEvalue g of the equations (20). Now, the
(cid:104)b ,...,b |Πˆ±|b ,...,b (cid:105) entanglementconditionforthestate(cid:37)ˆmaybewrittenas
1 K 1 K
under the constraint that the denominator exists, i.e., (cid:104)Lˆ(cid:105)=tr((cid:37)ˆLˆ)>sup{g}. (22)
Πˆ±|b ,...,b (cid:105)=(cid:54) 0. The optimization is carried out as a
1 K
derivative of the Rayleigh quotient: Alternatively,thisconditioncanbewrittenintermsof
witnesses constructed from Eqs. (13) and (14):
∂g
0= for j =1,...,K. (16)
∂(cid:104)b | Wˆ =GΠˆ±−Πˆ±LˆΠˆ±, with G=sup{g}, (23)
j ±
4
which reads as tr((cid:37)ˆWˆ ) < 0. Similarly, a witness can IV. BIPARTITE EXAMPLE
±
be constructed using the lower bound of the Rayleigh
quotient (15) as Inafirstapplicationofourintroducedmethod,weaim
(cid:104) (cid:105) at witnessing bipartite entanglement. As our observable
Wˆ± =Πˆ± Lˆ−inf{g}ˆ1 Πˆ±. (24) we may choose the rank one operator:
Hence, by solving the algebraic problem (20) of ob- Lˆ =|ψ(cid:105)(cid:104)ψ|, (30)
servables Lˆ, we are able to construct, in principle, any
being defined by a two-mode vector |ψ(cid:105) ∈ H⊗H. For
optimal entanglement witnesses for multiple correlated
DP, we have the well-known Schmidt decomposition [9]
Bosons or Fermions. Moreover, the SEvalue equations
to represent this vector,
for IP might be also used for a numerical optimization if
ananalyticalsolutionisnotavailable. Sincethecriterion
d d
in (22) and the witnessing approach are equivalent, we |ψ(cid:105)= (cid:88) ψ |i,j(cid:105)= (cid:88)λ |u ,v (cid:105), (31)
i,j n n n
study from now on solely the former one.
i,j=1 n=1
intermsoforthonormalsets{|u (cid:105)}d and{|v (cid:105)}d as
n n=1 n n=1
C. Further properties of the SEvalue equations wellasnon-negativecoefficientsλ ≥0. ForaFermionor
n
BosonstatewegettheSlaterdecomposition–asstudied,
EquivalenttotheformoftheSEvalueequationsforIP for example, in Refs. [19, 49, 55] – as
in(20),onemightformulateasecondform. Thesolutions
g andΠˆ±|b1,...,bK(cid:105)oftheHermitianoperatorLˆ canbe |f(cid:105)=Πˆ−|ψ(cid:105)= (cid:88)d f |i,j(cid:105)
found by solving i,j
i,j=1
Πˆ±LˆΠˆ±|b1,...,bK(cid:105)=gΠˆ±|b1,...,bK(cid:105)+|χ(cid:105), (25) (cid:98)d/2(cid:99)
(cid:88)
= κ (|w ,w (cid:105)−|w ,w (cid:105)), (32)
n 2n−1 2n 2n 2n−1
with the perturbation term |χ(cid:105), which has to fulfill for
n=1
all j =1,...,K and for all |x(cid:105)∈H⊗Nj an orthogonality
d d
relation of the form |b(cid:105)=Πˆ+|ψ(cid:105)= (cid:88) b |i,j(cid:105)= (cid:88)κ(cid:48) |w(cid:48),w(cid:48)(cid:105), (33)
i,j n n n
(cid:104)b ,...,b ,x,b ,...,b |χ(cid:105)=0 (26) i,j=1 n=1
1 j−1 j+1 K
for f = −f , orthonormal |w (cid:105), (cid:98)x(cid:99) denoting the
to be equivalent with the first form in (20). This treat- i,j j,i n
largest integer less or equal to x, and κ ≥ 0; and for
ment transforms the coupled set of equations of the first n
b =b with orthonormal |w(cid:48)(cid:105) and κ(cid:48) ≥0.
form (20) into a single, but perturbed, eigenvalue equa- i,j j,i n n
In Appendix C, the solution of the SEvalue equation
tion of the second form (25). Note that the perturbation
for IP is explicitly computed for the considered observ-
|χ(cid:105) is an element of the (anti)symmetric subspace, since
able(30). Here, letussummarizetheresults. Itisworth
Πˆ±[LˆΠˆ±−gˆ1]|b ,...,b (cid:105)=|χ(cid:105).
1 K pointing out that the nontrivial solutions, i.e., g (cid:54)= 0,
Another important property of the SEvalue equations
have forms which are directly related to the decomposi-
for IP is the behavior under certain transformations of
tions (32) and (33). Namely, we get for Fermions
theobservable. LocalunitariesUˆ andshiftsofLˆ,leading
to a transformed observable SEvalues: g =2κ2, (34)
n n
(cid:104) (cid:105)†(cid:104) (cid:105)(cid:104) (cid:105) SEvectors: |w (cid:105)∧|w (cid:105), (35)
Lˆ(cid:48) = Uˆ⊗N λ Lˆ+λ Πˆ± Uˆ⊗N , (27) 2n−1 2n
1 2
and for Bosons:
with λ , λ ∈ R\{0}, can be directly passed onto the
1 2
solutions. If the SEvalue g together with the SEvector SEvalues: g =κ(cid:48) 2 and g =κ(cid:48)2+κ(cid:48)2, (36)
n n k,l k l
Πˆ±|b1,...,bK(cid:105) is a solutions of the SEvalue equations SEvectors: |w(cid:48)(cid:105)∨|w(cid:48)(cid:105) and |w+ (cid:105)∨|w− (cid:105), (37)
for IP of Lˆ, then the operator Lˆ(cid:48) has the corresponding n n k,l k,l
solutions: respectively, with |w± (cid:105) = (cid:112)κ(cid:48)|w(cid:48)(cid:105)±i(cid:112)κ(cid:48)|w(cid:48)(cid:105). Now,
k,l k k l l
SEvalue: g(cid:48) =λ g+λ , (28) theentanglementcondition(22),canbewritteninterms
1 2
of the fidelities:
SEvector: Πˆ±|b(cid:48),...,b(cid:48) (cid:105)=Uˆ⊗NΠˆ±|b ,...,b (cid:105)
=Πˆ±Uˆ1⊗N|b ,.K..,b (cid:105). 1 K (29) (cid:104)b|(cid:37)ˆB|b(cid:105)> max (cid:8)κ(cid:48)k2+κ(cid:48)l2(cid:9) (38)
1 K 1≤k<l≤d
or (cid:104)f|(cid:37)ˆ |f(cid:105)> max (cid:8)2κ2(cid:9), (39)
Hence,bysolvingtheSEvalueequationsforIPforagiven F n
1≤n≤(cid:98)d/2(cid:99)
Lˆ we gain the solutions for a whole class of observables.
Additionally, we get for λ = 1 and λ = 0 that the for bipartite, mixed or pure entangled states of Bosons,
1 2
SEvalues are invariant under local transformations. (cid:37)ˆ , or Fermions, (cid:37)ˆ . Note that, for the case of DP, we
B F
5
alsogettheseparableboundfromthedecomposition(31) Since the structure of (anti)symmetric product states is
as G=max {λ2}, using the results in [52]: related to Bell-like states, cf. Eq. (1), we also consider
1≤n≤d n
Schmidt rank (SR) two states for DP. The calculation
SEvalues: gn =λ2n, (40) of the corresponding bounds is done in [58] and applied
SEvectors: |u (cid:105)⊗|v (cid:105). (41) in [59]. For any p in gray area of the plot SR>2, we can
n n
conclude that more than two tensor-product states have
Let us apply the method to characterize the entangle- to be superimposed to describe the state (42). Thus our
ment of the pure state |ψ(cid:105) which is mixed with white approachallowsthedetectionofdifferentformsofentan-
noise, glement based on a single observable.
ˆI
ρˆ=pˆI|ψ(cid:105)(cid:104)ψ|ˆI+(1−p) , (42)
trˆI V. MULTIPARTITE EXAMPLE
with p∈[0,1] being a noise parameter,ˆI∈{ˆ1,Πˆ+,Πˆ−}, In the following we will study a multipartite entangle-
and the second term being separable. For p = 0, we get ment test, which is even independent of the spin statis-
the separable stateˆI/trˆI, which is a uniformly weighted tics. We further assume dimH = ∞ given by the or-
mixture of all normalized product state ˆI|a1,a2(cid:105) with thonormal single-mode basis {|n(cid:105)}∞n=0. The observable
1 = (cid:104)a ,a |ˆI|a ,a (cid:105); cf. Eqs. (2) and (11). Replacing ˆI is
1 2 1 2
withotherprojectionswecouldadditionallystudyentan-
Lˆ=|1,...,N(cid:105)(cid:104)N+1,...,2N|+|N+1,...,2N(cid:105)(cid:104)1,...,N|.
glement for other parastatistics, e.g., for anyons [56, 57],
(43)
using the same treatment as presented for Bosons and
Fermions.
For a state ρˆ, the observable Lˆ measures an interference
term of the form
distinguishable,SR(cid:62)1 distinguishable,SR(cid:62)2
1.0 1.0
(cid:104)Lˆ(cid:105)=(cid:104)N+1,...,2N|ρˆ|1,...,N(cid:105)
0.8 0.8
0.6 0.6 +(cid:104)1,...,N|ρˆ|N+1,...,2N(cid:105). (44)
p p
0.4 0.4
In Appendix D, we solve the SEvalue equations for DP
0.2 0.2
and IP. The obtained maximal bound for K-separable
0.0 0.0
5 10 15 20 5 10 15 20 states is
d d
sup{g}=(1/2)K−1. (45)
indistinguishable,Bosons indistinguishable,Fermions
1.0 1.0
Notethatthisboundisevenindependentofthequantum
0.8 0.8
statistics. Therefore, the entanglement condition (12) in
0.6 0.6 this case states: Whenever the interference (cid:104)Lˆ(cid:105) for N-
p p
0.4 0.4
particle system of DP, Bosons, or Fermions exceeds the
0.2 0.2 bound (1/2)K−1, we have certified that the state cannot
0.0 0.0 be K separable.
5 10 15 20 5 10 15 20
The observable (43) may be applied to a GHZ-type
d d
state [60],
FIG. 1: Mixing in terms of p for the state (42) depending
(cid:113) ∞
on d = dimH is plotted. As long as p is in the gray shaded |q(cid:105)= ν(ˆI)ˆI(cid:88)(cid:112)1−|q|2qn|nN+1,...,(n+1)N(cid:105), (46)
area, we successfully detected entanglement. The coefficients
in Eqs. (31) and (33) for DP and Bosons, respectively, are n=0
chosentobeequalλk =κ(cid:48)k =d−1/2(k=1,...,d). Inthecase with ˆI ∈ {ˆ1,Πˆ+,Πˆ−}, ν(Πˆ±) = N!, and ν(ˆ1) = 1.
ofFermions,wechooseκ =(2(cid:98)d/2(cid:99))−1/2 (k=1,...,(cid:98)d/2(cid:99)),
k This state is of a GHZ-type structure, because for each
see Eq. (32), yielding a different behavior for even and odd
mode j holds that the individual vectors |nN +j(cid:105) are
dimensions d.
orthonormal for different n. Using the transformations
Tˆ |n(cid:105)=|nN+j(cid:105) in (D1) of Appendix D, it can be di-
j
rectly seen that the state in (46) is a GHZ-type of state,
In Fig. 1, we compare different quantum statistics
|q(cid:105) = ˆI(Tˆ ⊗···⊗Tˆ )(cid:80)∞ λ |n,...,n(cid:105). In addition,
regarding their entanglement properties for the mixed 1 N n=0 n
the pure state might be perturbed due to a randomly
state (42) in dependence on the dimensionality of the
distributed parameter q (|q|<1):
single particle’s Hilbert space, d=dimH. We apply the
testoperatorin(30). Aslongas(cid:104)Lˆ(cid:105)>sup{g}(grayarea (cid:90)
inFig.1),wehaveidentifiedentanglementforthemixing ρˆ= d2qp(q)|q(cid:105)(cid:104)q|, (47)
parameter p for DP (plot: SR>1), Bosons or Fermions. |q|<1
6
where p is a classical probability distribution. for any number of particles. The optimization of these
The identification of multipartite entangled Bosons criteria is based on a set of generalized eigenvalue equa-
and Fermions as well as DP is shown in Fig. 2 for a de- tions. These equations yield a structural unification of
phasing channel, i.e., the amplitude r = |q| is fixed and entanglement for distinguishable particles, Fermions and
phase ϕ = argq is randomized. This uniform dephasing Bosons, anditcanbegeneralizedtootherparastatistics.
in the interval ϕ∈[−δ,+δ] results in the density matrix We analyzed and compared a number of examples for
studyingthedifferencesandsimilaritiesbetweenfulland
(cid:90) +δ dϕ partialseparabilityaswellasentanglementinsystemsof
ρˆ= |rexp[iϕ](cid:105)(cid:104)rexp[iϕ]| (48)
2δ distinguishable and indistinguishable particles. For in-
−δ
stance, the determination of entanglement in bipartite
∞
= (cid:88) (1−r2)rn+n(cid:48)sinc[δ(n−n(cid:48))]ν(ˆI) and multipartite as well as discrete and continuous vari-
ablequantumsystemsdemonstratethewiderangeofap-
n,n(cid:48)=0
plications of our technique even in the presence of noise.
׈I|nN+1,...,(n+1)N(cid:105)(cid:104)n(cid:48)N+1,...,(n(cid:48)+1)N|ˆI,
This also shows that our method is not limited to bi-
partitions or small numbers of particles. Moreover, the
with sinc[x] = sin[x]/x (sinc[0] = 1). Hence we have for
construction of spin-statistics independent entanglement
thisstate,forallˆI∈{ˆ1,Πˆ+,Πˆ−},andforallK partitions
probes has been established. Such witnesses can detect
(N ,...,N ) the entanglement condition
1 K entanglement of a quantum system independent of sym-
metrization effects.
(cid:104)Lˆ(cid:105)=2(1−r2)rsinc[δ]>(1/2)K−1. (49)
We presented an approach which allows one to con-
struct entanglement criteria, in principle, from almost
As long as the expectation value in Fig. 2 is above the
allobservables. However,thequestionwhichmeasurable
dashed lines, we certified that the state cannot be a K-
quantity is able to witness the entanglement of a partic-
separableone. Notethatforafulldephasing, δ =π,this
ular state is open and requires further studies. Beyond
stateisdiagonalinproductstatesand,therefore,separa-
theherepresentedfullanalyticalapproach,numericalim-
ble. For no dephasing, δ = 0, we have a pure GHZ-type
plementations may allow one to generate more sophisti-
entangled state. This example demonstrates the general
catedentanglementprobesfordetectingentanglementin
possibilitytoconstructspinstatisticsindependententan-
more general scenarios. Therefore, we believe that our
glement tests with our approach.
approach will provide a versatile tool to characterize en-
tanglement in future experiments, with applications to
1.0
Bose-Einstein condensates or ultra-cold Fermi systems.
0.8
Acknowledgments
0.6
(cid:96)L (cid:92) K(cid:61)2 ThisworkwassupportedbytheDeutscheForschungs-
(cid:88)
0.4 gemeinschaft through SFB 652.
K(cid:61)3
0.2
K(cid:61)4 Appendix A: Symmetrization of operators
K(cid:61)5
0.0
0 Π4 Π2 3Π4 Π
For an N-fold Hilbert space H⊗N, the projection op-
∆ erators Πˆ+ and Πˆ− are defined as
FIG. 2: Expectat(cid:144)ion value in(cid:144)Eq. (49√) is (cid:144)plotted for N = Πˆ± = (cid:88) (±1)|σ|Pˆ , (A1)
5 (solid curve), an amplitude |q| = 1/ 3, and a uniformly N! σ
distributedphaseintheintervalargq∈[−δ,δ]. If(cid:104)Lˆ(cid:105)isabove σ∈SN
thedashedlineK,thenthestatecannotbeaK-separableone with Pˆ |a ,...,a (cid:105) = |a ,...,a (cid:105) for any permu-
σ 1 N σ(1) σ(N)
– independent of the quantum statistics.
tation σ ∈ S . It holds (Πˆ±)† = Πˆ±. We may study
N
Hermitian operators in a product basis operator expan-
sion, given by terms of the form
Xˆ =Yˆ ⊗···⊗Yˆ , (A2)
1 N
VI. CONCLUSIONS
with Yˆ = Yˆ† for j = 1,...,N. The symmetric form of
j j
In summary, we derived a method which allows the Xˆ is defined as
caonndstFreurcmtiioonnso.fTenhteasnegnleemceesnstapryroabnedsisnuffisycsiteenmtscoofnBdiotsioonnss Xˆ(sym) = (cid:88) N1!Yˆσ(1)⊗···⊗Yˆσ(N). (A3)
are capable of determining full and partial entanglement σ∈SN
7
We claim withtheorthogonalvectors|Φ (cid:105)=|1,2(cid:105)±|2,1(cid:105)+|3,4(cid:105)±
0
|4,3(cid:105), |Φ (cid:105) = |2,0(cid:105)±|0,2(cid:105), |Φ (cid:105) = |0,1(cid:105)±|1,0(cid:105), |Φ (cid:105) =
1 2 3
Πˆ±XˆΠˆ± =Xˆ(sym)Πˆ± =Πˆ±Xˆ(sym). (A4) |4,0(cid:105)±|0,4(cid:105),and|Φ (cid:105)=|0,3(cid:105)±|3,0(cid:105). Thereducedstate
4
– tracing with respect to the first subsystem – is
The first equality can be directly computed, since for all
|a1,...,aN(cid:105) holds: ρˆred =tr1|Ψ±(cid:105)(cid:104)Ψ±| (B4)
|Φ (cid:105)(cid:104)Φ |+|Φ (cid:105)(cid:104)Φ |+|Φ (cid:105)(cid:104)Φ |+|Φ (cid:105)(cid:104)Φ |+|Φ (cid:105)(cid:104)Φ |
Πˆ±XˆΠˆ±(cid:79)N |a (cid:105)=Πˆ± (cid:88) (±1)|σ| (cid:79)N (cid:104)Yˆ |a (cid:105)(cid:105) = 0 0 1 1 26 2 3 3 4 4 ,
j N! j σ(j)
and it has a rank of five, rank(ρˆ )=5.
j=1 σ∈SN j=1 red
For proving that the states |Ψ±(cid:105) cannot be fully sepa-
= (cid:88) (±1)|σ|+|τ| (cid:79)N (cid:104)Yˆ (cid:105)(cid:79)N |a (cid:105) rable,|Ψ+(cid:105)(cid:29)|a1(cid:105)∨|a2(cid:105)∨|a3(cid:105)and|Ψ−(cid:105)(cid:29)|a1(cid:105)∧|a2(cid:105)∧|a3(cid:105),
N!2 τ(j) τ(σ(j)) let us study the properties of the reduced density matrix
σ,τ∈SN j=1 j=1 of fully separable Boson and Fermion states. We have
= (cid:88) N1!(cid:79)N Yˆτ(j) (cid:88) (±N1)!|µ|Pˆµ(cid:79)N |aj(cid:105) |s±(cid:105)=Πˆ±(|a1(cid:105)⊗|a2(cid:105)⊗|a3(cid:105))
τ∈SN j=1 µ∈SN j=1 1 1 1
= |a (cid:105)⊗|s (cid:105)+ |a (cid:105)⊗|s (cid:105)+ |a (cid:105)⊗|s (cid:105), (B5)
=Xˆ(sym)Πˆ±(cid:79)N |a (cid:105), (A5) 6 1 1 6 2 2 6 3 3
j
with |s (cid:105) = |a ,a (cid:105)±|a ,a (cid:105), |s (cid:105) = |a ,a (cid:105)±|a ,a (cid:105),
j=1 1 2 3 3 2 2 3 1 1 3
and |s (cid:105)=|a ,a (cid:105)±|a ,a (cid:105). Consequently, the range or
3 1 2 2 1
whereweusedasubstitutionµ=τ◦σ and(±1)|τ|+|σ| = image of the partially reduced operator is
(±1)|τ◦σ|. The second equality in (A4) follows from the R=Im(tr |s±(cid:105)(cid:104)s±|)⊆span{|s (cid:105),|s (cid:105),|s (cid:105)}=R(cid:48), (B6)
fact that Xˆ and Πˆ± are Hermitian operators, 1 1 2 3
where the linear span R(cid:48) has a dimensionality of three
Πˆ±Xˆ(sym) =Πˆ±XˆΠˆ± =(cid:0)Πˆ±XˆΠˆ±(cid:1)† (if |s1(cid:105),|s2(cid:105),|s3(cid:105) are linearly independent) or less. Since
dimR ≤ 3, it follows that the rank of the partially
=(cid:0)Πˆ±Xˆ(sym)(cid:1)† =Xˆ(sym)Πˆ±. (A6) reduced operator of any fully separable state of IP is
bounded by three:
For Xˆ = ˆ1 ⊗ ··· ⊗ ˆ1, we get from (A4) that Πˆ± is
idempotent. An observable Lˆ which solely acts on the rank(ρˆ(cid:48)red)≤3, for all (B7)
(cid:104) (cid:105)
correspondingsubspaces(H∨N orH∧N)shouldfulfillthe ρˆ(cid:48) =tr Πˆ±|a ,a ,a (cid:105)(cid:104)a ,a ,a |Πˆ± .
red 1 1 2 3 1 2 3
commutation relation [Lˆ,Πˆ±] = 0. From (A4) follows
that this is fulfilled for every Lˆ =Lˆ(sym). Finally,weconcludethatthestates|Ψ±(cid:105)cannotbefully
separablebecauseofrank(ρˆ )=5>3. Thus,thestates
red
in(B1),usingEq.(9),areauthenticexamplesofpartially
Appendix B: Existence of partial separable Bosons separablestatesofFermionsandBosonsthatarenotfully
and Fermions separable.
Let us prove that the set of partially separable states
Appendix C: Bipartite observable
of IP includes more elements than the fully separable
ones. For this reason, we consider the orthonormal basis
{|0(cid:105),...,|4(cid:105)} and the three-partite vector |Ψ(cid:105) ∈ (cid:0)C5(cid:1)⊗3 In the following two subsections, the solution of the
inEq.(9). Itispartiallyseparableinthetensorproduct, SEvalue equation for IP will be explicitly computed for
i.e., |Ψ(cid:105) = |0(cid:105)⊗|Φ(cid:105). Applying the symmetrization or theconsideredobservable(30). Forsimplicity,wewillas-
anti-symmetrization operator, Πˆ±, we get sumeinthefollowingLˆ =|f(cid:105)(cid:104)f|(Lˆ =|b(cid:105)(cid:104)b|)forthetwo-
Fermion (two-Boson) system which yields Lˆ = Πˆ−LˆΠˆ−
|Ψ+(cid:105)=Πˆ+|Ψ(cid:105)∈(cid:0)C5(cid:1)∨3 and |Ψ−(cid:105)=Πˆ−|Ψ(cid:105)∈(cid:0)C5(cid:1)∧3, (B1) (Lˆ =Πˆ+LˆΠˆ+).
or, more explicitly, we have the expansion
1. Solution for Fermions
1
|Ψ±(cid:105)= [|0,1,2(cid:105)±|0,2,1(cid:105)+|0,3,4(cid:105)±|0,4,3(cid:105)
6
First, we give a unitary state representation for arbi-
+|1,2,0(cid:105)±|1,0,2(cid:105)+|3,4,0(cid:105)±|3,0,4(cid:105) trary d-dimensional (d ∈ N∪{∞}) pure states of two
+|2,0,1(cid:105)±|2,1,0(cid:105)+|4,0,3(cid:105)±|4,3,0(cid:105)] (B2) Fermions. We start with a Fermion state,
1
=6[|0(cid:105)⊗|Φ0(cid:105)+|1(cid:105)⊗|Φ1(cid:105)+|2(cid:105)⊗|Φ2(cid:105) |f(cid:105)= (cid:88)d f |i,j(cid:105), with f =−f , (C1)
i,j i,j j,i
+|3(cid:105)⊗|Φ (cid:105)+|4(cid:105)⊗|Φ (cid:105)], (B3)
3 4 i,j=1
8
and introduce the skew-symmetric coefficient matrix for all n = 1,...,(cid:98)d/2(cid:99). For the special case L =
m,n
κ κ , we get the maximal SEvalue as
Mˆ =(f ) =−MˆT. (C2) m n
f i,j i,j f
G=max{2κ2}. (C9)
n
Using the Autonne-Takagi factorization in Ref. [55], we n
find the Slater decomposition of this coefficient matrix, Note that trivial solutions, g = 0, can be obtained by
|k(cid:105)∧|l(cid:105) with (k,l)(cid:54)=(2n,2n+1).
Mˆ =UˆDˆUˆT, with Uˆ†Uˆ =ˆ1 (C3)
f
d/2 (cid:20) (cid:21)
and (d even): Dˆ =(cid:77)κ 0 +1 , 2. Solution for Bosons
j −1 0
j=1
Again, we first give the state representation according
(d−1)/2 (cid:20) (cid:21)
or (d odd): Dˆ = (cid:77) κ 0 +1 (cid:77)[0], to Ref. [55] for arbitrary d-dimensional pure state of two
j −1 0 Bosons. This means that the symmetric state
j=1
d
with Dˆ being a block diagonal matrix containing anti- |b(cid:105)= (cid:88) b |i,j(cid:105), with b =b , (C10)
i,j i,j j,i
diagonal 2×2 blocks and κ ≥0. This yields Eq. (32) in
j i,j=1
the form
can be identified with a symmetric coefficient matrix
(cid:98)d/2(cid:99)
|f(cid:105)=Uˆ ⊗Uˆ (cid:88) κn(|2n−1,2n(cid:105)−|2n,2n−1(cid:105)), (C4) Mˆb =(bi,j)i,j =MˆbT and Mˆb =UˆDˆUˆT, (C11)
n=1
with Uˆ†Uˆ = ˆ1 and Dˆ = diag[κ(cid:48),...,κ(cid:48)] ≥ 0. Thus the
1 d
with an orthonormal single-mode basis {|1(cid:105),|2(cid:105),...} and symmetric Slater representation (33) of the state is
Uˆ|k(cid:105) = |w (cid:105). Since the SEvalues are invariant under
k
unitary separable operations Uˆ⊗Uˆ, we assume, without |b(cid:105)=Uˆ ⊗Uˆ (cid:88)d κ(cid:48) |n,n(cid:105), with Uˆ|n(cid:105)=|w(cid:48)(cid:105). (C12)
loss of generality, that Uˆ =ˆ1. n n
n=1
Now, we consider more general operators having an
expansion as As in the previous example for Fermions, let us con-
sider the more general operator
(cid:98)d/2(cid:99)
Lˆ = (cid:88) Lm,n(|2m−1,2m(cid:105)−|2m,2m−1(cid:105)) Lˆ = (cid:88)d L |m,m(cid:105)(cid:104)n,n|. (C13)
m,n
m,n=1
m,n=1
×((cid:104)2n−1,2n|−(cid:104)2n,2n−1|), (C5)
The relation Lˆ = Πˆ+LˆΠˆ+ simplifies the SEvalue equa-
which includes the special case L = κ κ of pro-
m,n m n tions for Bosons in the second form to
jection operator Lˆ = |f(cid:105)(cid:104)f|. Since Lˆ = Πˆ−LˆΠˆ−, the
1
SEvalue equations for Fermions in the second form (25) Lˆ|a ,a (cid:105)=g (|a ,a (cid:105)+|a ,a (cid:105))+|χ(cid:105). (C14)
1 2 2 1 2 2 1
read as
Using γ =(cid:104)n,n|a ,a (cid:105), we can now write
1 n 1 2
Lˆ|a ,a (cid:105)=g (|a ,a (cid:105)−|a ,a (cid:105))+|χ(cid:105). (C6)
1 2 2 1 2 2 1 d (cid:34) d (cid:35)
Lˆ|a ,a (cid:105)= (cid:88) (cid:88)L γ |m,m(cid:105). (C15)
Using γ =((cid:104)2n−1,2n|−(cid:104)2n,2n−1|)|a ,a (cid:105), we get 1 2 m,n n
n 1 2
m=1 n=1
Lˆ|a ,a (cid:105) (C7) Hence one class of solutions with |a (cid:105)=|a (cid:105) is given by
1 2 1 2
 
(cid:98)(cid:88)d/2(cid:99) (cid:98)(cid:88)d/2(cid:99) Πˆ+|a1,a2(cid:105)=|n,n(cid:105)∼=|n(cid:105)∨|n(cid:105),
=  Lm,nγn(|2m−1,2m(cid:105)−|2m,2m−1(cid:105)).
g =L , (C16)
n,n
m=1 n=1
(cid:88)
|χ(cid:105)= L |m,m(cid:105).
m,n
We find that Lˆ|a ,a (cid:105) is already diagonalized in the
1 2 m(cid:54)=n
form (C4). Hence, the orthogonality of |a ,a (cid:105) to the
1 2
Unlike in the Fermion case, we have to take a brief
perturbation |χ(cid:105) is fulfilled if
look on the decomposition of product states of Bosons.
1 Namely the state Πˆ+|a ,a (cid:105) for any |a (cid:105) =(cid:54) |a (cid:105) has a
Πˆ−|a ,a (cid:105)= (|2n−1,2n(cid:105)−|2n,2n−1(cid:105))∼=|2n−1(cid:105)∧|2n(cid:105), 1 2 1 2
1 2 2 decomposition, cf. Eq. (C12), as
g =2L , (C8)
n,n Πˆ+|a ,a (cid:105)=Uˆ(cid:48)⊗Uˆ(cid:48)(λ(cid:48)|1,1(cid:105)+λ(cid:48)|2,2(cid:105)), (C17)
(cid:88) 1 2 1 2
|χ(cid:105)= Lm,n(|2m−1,2m(cid:105)−|2m,2m−1(cid:105)), for |a (cid:105)=Uˆ(cid:48)(cid:104)(cid:112)λ(cid:48)|1(cid:105)+(−)i(cid:112)λ(cid:48)|2(cid:105)(cid:105).
m(cid:54)=n 1(2) 1 2
9
Hence, we get a more involved set of solutions of withν(Πˆ±)=N!andν(ˆ1)=1. Duetothisfact,wemay
Eq. (C15) in the form (for k (cid:54)=l): define the K-separable vectors
Πˆ+|a ,a (cid:105)=λ(cid:48)|k,k(cid:105)+λ(cid:48)|l,l(cid:105), (C18) |v ,...,v (cid:105)=|1,...,N(cid:105)
1 2 k l 1 K
(D4)
|χ(cid:105)= (cid:88) (Lm,kλ(cid:48)k+Lm,lλ(cid:48)l)|m,m(cid:105), (C19) and |w1,...,wK(cid:105)=|N+1,...,2N(cid:105),
m(cid:54)=k,l
which are orthogonal for Fermions, Bosons, and DP and
where the coefficients λ(cid:48) and λ(cid:48) have to be determined. any partition (N ,...,N ).
k l 1 K
We insert (C18) and (C19) into (C14), As a last fact before we solve the SEvalue equations
forthisoperator,letusrecallanexampleofthestandard
LˆΠˆ+|a1,a2(cid:105)−|χ(cid:105)=gΠˆ+|a1,a2(cid:105), (C20) eigenvalue problem:
and find that the remaining terms to be computed are Mˆ =m|m (cid:105)(cid:104)m |+m∗|m (cid:105)(cid:104)m |, (D5)
w v v w
(L λ(cid:48) +L λ(cid:48))|k,k(cid:105) (cid:18) m∗ (cid:19) √
k,k k k,l l |µ (cid:105)= |m (cid:105)± |m (cid:105) / 2, (D6)
+(L λ(cid:48) +L λ(cid:48))|l,l(cid:105)=g(λ(cid:48)|k,k(cid:105)+λ(cid:48)|l,l(cid:105)). (C21) ± w |m| v
l,k k l,l l k l
µ =±|m|, (D7)
This is a standard eigenvalue problem in C2, which has ±
the solutions
with complex m (cid:54)= 0, orthonormal {|m (cid:105),|m (cid:105)}, and
w v
g± =Lk,k+Ll,l±∆, ∆=(cid:113)(L −L )2+4|L |2, |µ±(cid:105) being the eigenvectors of Mˆ to the eigenvalues µ±.
2 k,k l,l k,l Now, let us use the first form of the SEvalue equation
λ(cid:48) =2L and λ(cid:48) =L −L ±∆, (C22) for IP and DP of the operator (43). Since the spanned
k k,l l l,l k,k
subspace of Lˆ is span{|v ,...,v (cid:105),|w ,...,w (cid:105)}, let us
with the Hermiticity condition L =L∗ . 1 K 1 K
l,k k,l expand
Again,intheparticularcaseL =κ(cid:48) κ(cid:48) ,wegetthe
m,n m n
simplified solutions |b (cid:105)=β |v (cid:105)+β |w (cid:105). (D8)
j v,j j w,j j
g =λ2, g− =0, and g+ =κ(cid:48)2+κ(cid:48)2. (C23)
n k l WegetforthejthSEvalueequationthetwocomponents
Combiningthesolutionsoftheform|a1(cid:105)∨|a1(cid:105)and|a1(cid:105)∨ (cid:104)v |(cid:0)ˆILˆˆI(cid:1) |b (cid:105)=g(cid:104)v |(cid:0)ˆI(cid:1) |b (cid:105),
|a2(cid:105)aswellasusingthefactthatκ(cid:48)k2+κ(cid:48)l2 ≥κ(cid:48)l2, weget j bj j j bj j (D9)
the maximal SEvalue as (cid:104)w |(cid:0)ˆILˆˆI(cid:1) |b (cid:105)=g(cid:104)w |(cid:0)ˆI(cid:1) |b (cid:105).
j bj j j bj j
G=max{λ2 +λ2}. (C24)
k(cid:54)=l k l Equivalently, we get by a rescaling with ν(ˆI)
(cid:89)(cid:0)β∗ β (cid:1) β =g (cid:89)(cid:0)|β |2+|β |2(cid:1) β ,
Appendix D: Multipartite observable v,i w,i w,j v,i w,i v,j
i(cid:54)=j i(cid:54)=j
(D10)
We considered an interference operator Lˆ in Eq. (43) (cid:89)(cid:0)βw∗,iβv,i(cid:1) βv,j=g (cid:89)(cid:0)|βv,i|2+|βw,i|2(cid:1) βw,j,
whose expectation value is the real part of an off- i(cid:54)=j i(cid:54)=j
diagonal element of the density operator ρˆ, (cid:104)Lˆ(cid:105) =
whichhasthestructureoftheeigenvalueproblemin(D5)
2Re(ρ ). Local unitary operations al-
(1,...,N),(N+1,...,2N)
withthesolutioninEqs.(D6)and(D7). Henceforeachj
low the generalization to other off-diagonal elements or
wegetthesolutionforcomponentswith|β |=|β |=
phase shifts, Re(exp[iϕ]ρ(1,...,N),(N+1,...,2N)). √ v,j w,j
The injective transformations Tˆ of the orthonormal 1/ 2, yielding |βv,i|2+|βw,i|2 =1 and the eigenvalues
j
basis {|n(cid:105)}n∈N, g =±(cid:12)(cid:12)(cid:89)(cid:0)β∗ β (cid:1)(cid:12)(cid:12)=±(1/2)K−1. (D11)
(cid:12) w,i v,i (cid:12)
Tˆ |n(cid:105)=|nN+j(cid:105) for j =1,...,N, (D1)
j i(cid:54)=j
are constructed such that one can directly see that the Note for (cid:81) (cid:0)β∗ β (cid:1) = 0, we get the trivial SEvalue
for all j,j(cid:48) = 1,...,N and n,n(cid:48) ∈ N an orthogonality is i(cid:54)=j v,i w,i
given, (cid:104)nN +j|n(cid:48)N +j(cid:48)(cid:105) = δ δ . Therefore, we get g = 0 and, for example, the SEvector ˆI|b1,...,bK(cid:105) =
forˆI∈{ˆ1,Πˆ+,Πˆ−} the orthogno,nn(cid:48)alji,tjy(cid:48) relation ˆI|v1,...,vK(cid:105). Finally, the maximal SEvalue is
(cid:104)1,...,N|ˆI|N+1,...,2N(cid:105)=0 (D2) G=sup{g}=(1/2)K−1. (D12)
as well as the normalizations
Note that this result is independent of the particular K-
(cid:104)N+1,...,2N|ˆI|N+1,...,2N(cid:105) partition (N1,...,NK) and, due to especially chosen or-
thonormality in (D2), the result is also independent of
=(cid:104)1,...,N|ˆI|1,...,N(cid:105)=1/ν(ˆI), (D3)
the spin statistics.
10
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