Estimation for Entanglement Negativity of Free Fermions

0.59 MB·

by  Christopher P. Herzog

YITP-SB-15-17
Estimation for Entanglement Negativity of Free
Fermions
Christopher P. Herzog and Yihong Wang
6
1
0 C. N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy
2
Stony Brook University, Stony Brook, NY 11794

y
a
M
2 Abstract
1
In this letter we study the negativity of one dimensional free fermions. We derive the general

] formoftheZ symmetricterminmomentsofthepartialtransposed(reduced)densitymatrix,
h N
t which is an algebraic function of the end points of the system. Such a path integral turns out
-
p
to be a convenient tool for making estimations for the negativity.
e
h
[

1 Introduction
4
v
8
Measuresofquantumentanglementhavebecomeafocusofintenseresearchactivityattheboundaries
7
6 between quantum information, quantum field theory, condensed matter physics, general relativity
0
andstringtheory(seerefs.[1–3]forreviews). Onekeyquantity,theentanglemententropy,measures
0
.
1 thequantumentanglementbetweentwocomplementarypiecesofasysteminapurestate. However
0
the entanglement entropy is no longer a good measure of quantum entanglement if the initial state
6
1 of the system is mixed. Negative eigenvalues in the partial transpose of the density matrix ρT2
:
v
implies quantum entanglementeven in a (bipartite) mixed state scenario [4,5]. This observation led
i
X
to the proposal of the negativity [6,7], which was later demonstrated to be a good entanglement
r
a measure [8].
Like the entanglement entropy, the negativity in a quantum field theory can be computed by
employing the replica trick [9,10]. In this setting, the negativity is the N →1 limit of the partition
function of an N-sheeted spacetime. In practice, these partition functions can only be computed in
special cases [9,10]. For conformal field theories in 1+1 dimensions, the negativity of the single
intervalandthetwoadjacentintervalcasesisdeterminedbyconformalsymmetry.1 Anotherspecial
case where the negativity can be determined, at least for N > 1, is a massless free scalar field in
1+1 dimensions. In this case, the N-sheeted partition functions are known in terms of Riemann-
Siegel theta functions although it is not known in general how to continue the result away from
1Seeref.[11]foranextensiontothemassivecase.
integer N > 1 and in particular to N = 1. Since the partial transposed reduced density matrix is
Gaussian, the negativity for a free scalar can be checked through a lattice computation by using
Wick’s Theorem [10,12].
The case of free fermions in 1+1 dimensions appears to be more difficult than the case of free
scalars however. The partial transpose of the reduced density matrix is no longer Gaussian but a
sum of two, generically non-commuting, Gaussian matrices [13]:
1 (cid:16) (cid:17)
ρT2 = √ eiπ/4O +e−iπ/4O . (1)
+ −
2
(We will define O in section 2.) This fact brings additional complication to both the lattice and
±
field theoretical calculations. On the lattice side, eigenvalues of (ρT2)N cannot be simply derived
from eigenvalues of a covariance matrix as in the Gaussian case. In a field theory setting, one has
tosumoverpartitionfunctionswithdifferentspinstructure, correspondingtodifferenttermsinthe
expansion of (eiπ/4O +e−iπ/4O )N. Various efforts have been made to tame the difficulties in
+ −
deriving the negativity of free fermions: On the lattice side, algebraic simplification and numerical
diagonalization of products of these two Gaussian matrices yields the N >1 moments of negativity
forthetwodisjointintervalcase[13,14].2 (Monte-Carloandtensornetworkmethodshavealsobeen
used to calculate negativity for the Ising model [15–17] which, although not identical to the Dirac
fermion, is closely related.) The analytical form of such moments are derived by evaluation of the
corresponding path integrals [18,19]. However in the existing results the sheet number N does not
appearasacontinuousvariable; itremainsanopenproblemhowtotaketheN →1limittogetthe
negativity.3
InthisletterweshallintroduceaZ -symmetricfreefermionwithspecificchoiceofspinstructure.
N
Thisfermionhasseveralnicefeaturesthatwebelievewillhelpusexploreandunderstandthefeatures
of free fermion negativity. 1) The partition function explicitly reproduces the correct adjacent
interval limit. 2) The N → 1 limit of the N sheeted path integral can be easily derived. 3) There
existsanaturalgeneralizationtomultipleintervalcases,nonzerotemperature,andnonzerochemical
potential. 4)WhilesuchapartitionfunctionisnotanNth momentofρT2 (exceptinthespecialcase
N =2),itappearstobeausefulquantityforboundingtheseNth momentsincludingthenegativity
itself.
The rest of this letter is arranged as follows: In section 2 we review previous results. Section
3 contains a derivation of the partition function for the Z -symmetric free fermion system and in
N
particular tr(ON) and tr[(O O )N/2]. In section 4, we discuss bounds on the negativity and its
+ + −
Nth moments. We conclude in section 5 with remarks on possible generalizations of our results and
future directions. An appendix contains a discussion of a two-spin system.
2Seealsoref.[20]foranextensiontotwospatialdimensions.
3See[31]forrecentprogressonnegativityforfermionicsystems.
2
2 Review of Previous Results
We first review the definition of the negativity. For a state |Ψ(cid:105) in a quantum system with bipartite
(cid:78)
Hilbert space H=H H and density matrix ρ=|Ψ(cid:105)(cid:104)Ψ|, the reduced density matrix is defined
A B
(cid:78)
asρ =tr ρ. IfH isfactoredfurtherintoH =H H ,onecandefinethepartialtranspose
A B A A A1 A2
ofthereduceddensitymatrixρT2 astheoperatorsuchthatthefollowingidentityholdsforanye(1),
A i
(cid:68) (cid:12) (cid:12) (cid:69) (cid:68) (cid:69)
e(1) ∈ H and e(2), e(2) ∈ H : e(1)e(2)(cid:12)ρT2(cid:12)e(1)e(2) = e(1)e(2)|ρ |e(1)e(2) . The logarithmic
k A1 j l A2 i j (cid:12) A (cid:12) k l i l A k j
negativity is defined as the logarithm of the trace norm4 of ρT2. Since ρT2 is Hermitian, its trace
A A
norm can be written as the following limit
E ≡log|ρT2|=log lim tr(cid:16)ρT2(cid:17)Ne (2)
A A
Ne→1
where N is an even integer. This analytic continuation suggests the utility of also defining higher
e
moments of the partial transpose:
(cid:104) (cid:105)
R(N)≡tr (ρT2)N . (3)
A
We are interested in systems in one time and one spatial dimension. We will assume a factorization
of the Hilbert space corresponding to a partition of the real line with A and A each being the
1 2
union of a collection of disjoint intervals: A =∪p (s ,t ) and A =∪q (u ,v ).
1 i=1 i i 2 i=1 i i
Inthispaper,weareparticularlyinterestedinthecaseoffree,masslessfermionsin1+1dimension
with the continuum Hamiltonian
ˆ
L
H =∓i Ψ†(t,x)∂ Ψ(t,x)dx (4)
x
0
where {Ψ†(t,x),Ψ(t,x(cid:48))} = δ(x−x(cid:48)). The sign determines whether the fermions are left moving
or right moving. We will take one copy of each to reassemble a Dirac fermion. It will often be
convenient to consider the lattice version of this Hamiltonian as well
i (cid:88)(cid:16) (cid:17)
H =∓ Ψ†Ψ −Ψ† Ψ , (5)
2 j j+1 j+1 j
j
and anticommutation relation {Ψ†,Ψ } = δ , which suffers the usual fermion doubling problem.
j k jk
We choose as our vacuum the state annihilated by all of the Ψ .
j
The authors of ref. [13] were able to give a relatively simple expression for the negativity in the
discretecasebyworkinginsteadwithMajoranafermionsa = 1(Ψ†+Ψ )anda = 1(Ψ†−Ψ ).
2j−1 2 j j 2j 2i j j
Re-indexing, we can write the reduced density matrix as a sum over words made of the a :
j
2n
ρ =(cid:88)c (cid:89)aτj (6)
A τ j
τ j=1
where τ is either zero or one, depending on whether the word τ contains the Majorana fermion a ,
j j
and n is the length of region A. Consider now instead the matrices O constructed from ρ by
± A
4ThetracenormofamatrixM isdefinedasthesumofitssingularvalues: |M|≡tr(cid:104)(cid:0)M†M(cid:1)1/2(cid:105). ForHermitian
matrices,singularvaluesareabsolutevaluesoftheeigenvalues.
3
multiplying all the a in region A by ±i:
j 2
  
O± =(cid:88)cτ,σ2(cid:89)n1aτjj2n1(cid:89)+2n2(±iaj)σj . (7)
τ,σ j=1 j=2n1+1
Here n is the length of region A , and we have broken the sum into words τ involving region A
j j 1
and words σ involving region A . As we already described in eq. (1), the central result of ref. [13] is
2
that the partial transpose of the reduced density matrix can be written in terms of O .
±
While the spectrum of ρ is not simply related to the spectra of O , it is true that O and O
A ± + −
are not only Hermitian conjugates but are also related by a similarity transformation and so have
the same eigenvalue spectrum. Consider a product of all of the Majorana fermions in A ,
2
2(n(cid:89)1+n2)
S =in2 a , (8)
j
j=2n1+1
which squares to one, S2 = 1. This operator provides the similarity transformation between O
+
and O , i.e. O = SO S. This similarity transformation means, along with cyclicity of the trace,
− + −
that if we have a trace over a word constructed from a product of O and O , the trace is invariant
+ −
undertheswapO ↔O . Employingthissimilaritytransformation,thenegativityforthefirstfew
+ −
even N can be written thus
tr[(ρT2)2] = tr(O O ) , (9)
A + −
1 1
tr[(ρT2)4] = − tr(O4)+tr(O2O2)+ tr[(O O )2] , (10)
A 2 + + − 2 + −
3 1 3 3
tr[(ρT2)6] = − tr(O O5)+ tr[(O O )3]+ tr(O3O3)+ tr(O O O2O2) . (11)
A 2 + − 4 + − 4 + − 2 + − + −
Toobtainanalyticexpressionsfortr[(ρT2)N]fromthedecomposition(1)ofρT2,akeystep[14]is
A A
therelationbetweenmatrixelementsofρ andmatrixelementsofO . Considerarbitrarycoherent
A ±
states (cid:104)ζ(x)| and |η(x)(cid:105) that further break up into (cid:104)ζ (x ),ζ (x )| and |η (x ),η (x (cid:105) according to
1 1 2 2 1 1 2 )
the decomposition of A into A and A . Then the matrix elements of ρ and O are related via
1 2 A ±
(cid:104)ζ(x)|O |η(x)(cid:105)=(cid:104)ζ (x ),±η∗(x )|U†ρ U |η (x ),∓ζ∗(x )(cid:105) , (12)
± 1 1 2 2 2 A 2 1 1 2 2
where U is a unitary operator (whose precise form [14] does not concern us) that acts only on the
2
part of the state in region A .
2
In pursuit of an analytic expression, let us move now to a path integral interpretation of tr[ρN]
A
and tr[(ρT2)N]. The trace over ρN becomes a path integral over an N sheeted cover of the plane,
A A
branched over A. Now consider instead trON given the relation (12). Performing a change of
+
variables, we can replace U acting on ζ∗ and η∗ with ζ and η inside the trace, and we see that
2 2 2 2 2
tr(ρN) is related to trON by an orientation reversal of region A . In terms of the N sheets, fixing a
A + 2
direction, passing through an interval in A , we move up a sheet while passing through an interval
1
in A we move down a sheet. Indeed, the trace of any word constructed from the O and O has a
2 + −
similar path integral interpretation.
4
Given the sign flip relation O =SO S however, replacing some of the O by O in the word
− + + −
(cid:104) (cid:105)
will change the spin structure of the N sheeted cover. In particular, consider a word tr (cid:81)N O
i=1 si
where the nth and (n+1)th letters are both O . Now replace the (n+1)th letter with O . Any
+ −
cycle passing (once) through the corresponding cut in A between the nth and (n+1)th sheet will
2
now pick up a minus sign compared to the situation before the replacement. In figure 1, we show a
cycle that would pick up such a sign.
A1 A2
Figure 1
For simplicity, consider the case where A is a single interval bounded by s<t and A a single
1 2
interval bounded by u < v. The trace of a word constructed from O , up to an undetermined
±
over-all normalization c , can be written in terms of a Riemann-Siegel theta function [14]
N
tr(cid:34)(cid:89)N Osi(cid:35)=c2N(cid:18)(t−1s)−(vx−u)(cid:19)2∆N (cid:12)(cid:12)(cid:12)(cid:12)ΘΘ[e(]τ˜(τ(˜x(x))))(cid:12)(cid:12)(cid:12)(cid:12)2 , e= 0δ  , (13)
i=1
where 0 is a vector of N −1 zeros and δ is fixed by the word (cid:81)N O . In particular, if s (cid:54)=s ,
i=1 si i i+1
then δ =1/2 and δ =0 otherwise. The characteristic δ =0 is associated with having antiperiodic
i i i
boundary conditions around the corresponding fundamental cycle, while the characteristic δ =1/2
i
has periodic boundary conditions [18]. The exponent
(cid:18) (cid:19)
c 1
∆ = N − (14)
N 12 N
is the dimension of a twist operator field with c=1 for a Dirac fermion. The cross ratio is defined
to be
(s−t)(u−v)
x≡ ∈(0,1) . (15)
(s−u)(t−v)
(Thelimitinwhichtheintervalsbecomeadjacentcorrespondstox→1.) TheRiemann-Siegeltheta
function is defined as
 
Θ[e](z|M)≡ (cid:88) eiπ(m+(cid:15))t·M·(m+(cid:15))+2πi(m+(cid:15))t·(z+δ) , e≡ (cid:15)  , (16)
δ
m∈ZN−1
and further Θ(z|M)≡Θ[0](z|M). The (N −1)×(N −1) period matrix is then [10,21]
N−1
τ =i 2 (cid:88) sin(πk/N)2F1(k/N,1−k/N;1;1−x)cos[2π(k/N)(i−j)] , (17)
i,j N F (k/N,1−k/N;1;x)
2 1
k=1
5
andfurtherτ˜(x)=τ(x/(x−1)). ThereareRiemann-Siegelthetafunctionsthatonecanwritedown
for multiple interval cases as well, but we shall not need their explicit form.
Amongthewordsthatenterinthebinomialexpansionoftr[(ρT2)N],thetracestr(ON)=tr(ON)
A + −
andtr[(O O )N/2]arespecial. Eveninthemultipleintervalcase,thesetwotracescanbeexpressed
+ −
as rational functions of the endpoints of the intervals. Although we have no proof in general,
observationally it seems to be true that among the words of a fixed length tr(ON) is the smallest in
+
magnitudewhiletr[(O O )N/2]isthelargest. Thesetwoconsiderationssuggesttheutilityoftrying
+ −
to bound the negativity using the rational functions tr(O )N and tr[(O O )N/2], as we pursue in
+ + −
section 4.
Inthetwointervalcase,itfollowsfromtheresult(13)thattr(ON)andtr[(O O )N]arerational
+ + −
functions. That tr(ON) reduces to a rational function is obvious since δ =0. That tr[(O O )N/2]
+ + −
reduces as well follows from Thomae’s formula [22,23] that when δ =1/2 for all i.
i
(cid:12)(cid:12)(cid:12)Θ[e](τ˜)(cid:12)(cid:12)(cid:12)2 =|1−x|−N/4 . (18)
(cid:12) Θ(τ˜) (cid:12)
To see more generally that these words are rational functions of the endpoints, in the next section
we employ bosonization.5
3 Bosonization and Rationality
Consider the normalized partition function of the free Dirac field on the Z -curve defined by the
N
following set:
(cid:40) (cid:12) p q (cid:41)
X = (z,y)(cid:12)(cid:12)yN =(cid:89)z−si (cid:89) z−vi,(z,y)∈C2 . (19)
N (cid:12) z−t z−u
(cid:12) i i
i=1 i=1
One can see that X , as the set of all points in C2 satisfying the equation in the set, has N sheets
N
corresponding to N different roots of a nonzero complex number. These N copies of C are cut open
along intervals in A on the real axis. As we choose the ordering s <t and u <v , such open cuts
i i i i
are glued cyclicly if in A and anti-cyclicly if in A .
1 2
While the Riemann surface (19) has an explicit Z symmetry, to specify a partition function,
N
we also have to give the spin structure. The spin structure can generically break this symmetry, i.e.
we can associate relative factors of minus one to cycles that would otherwise be related by the Z
N
(cid:81)
shift symmetry. A generic word O will generically have a spin structure that does not respect
i si
this symmetry. However, a few words do, namely tr(ON)=tr(ON) and tr[(O O )N/2]. The word
+ − + −
tr(O )N preserves the natural anti-periodic boundary conditions, while the word tr[(O O )N/2]
+ + −
associates an additional −1 to fundamental cycles that intersect both A and A .
1 2
IfweassumetheZ symmetryispreservedbythespinstructure,thenthebosonizationprocedure
N
is especially simple. Denote the partition function on X by Z[N]. Rather than a path integral
N
of a single Dirac field on X in (19), Z[N] can be considered as a path integral of a vector valued
N
5ForanapplicationofThomae’sformulatoamultipleintervalR´enyientropycomputation,seeref.[24].
6
Dirac field Ψ(cid:126) (z) on C: Ψ(x) = (Ψ (z),··· ,Ψ (z)). Ψ (x) is the value of the original field Ψ at
1 N i
coordinate (z,y ) on X . When going anti-clockwise around a branch point w by a small enough
i N
circle C , Ψ(x) gets multiplied by a monodromy matrix T (w).
w
Define the matrix
 
0 ω
 
 ω 
 
 
T ≡ . .  (20)
 
 
 0 ω 
 
ω 0
where ω = e2πiNN−1. This value of ω is chosen so that T satises the overall boundary condition
TN =(−1)N−1id where id is the N ×N identity matrix. The reason for the factor (−1)N−1 comes
from considering a closed loop that circles one of the branch points N times. Such a loop should
be a trivial closed loop in the y coordinate and come with an overall factor of −1, standard from
performing a 2π rotation of a fermion.6
The matrix T is not the only Z symmetric matrix satisfying TN = (−1)N−1id. A relative
N
phase ei2πk/N, k = 1,2,...,N −1, between monodromy matrices at different branch points is also
allowed. Choose the basis of Ψ(x) so that T (s ) = T and take into account the constraint that
1
T (t )T (s )=id, T (v )T (u )=id. Then, the monodromy matrices are fixed to be
i i+1 i i+1
T (s )=T , T (t )=T−1 , (21)
i i
T (u )=exp(2πi(N −k)/N)T−1 , T (v )=exp(2πik/N)T , (22)
i i
For us, e2πik/N represents an extra phase, in addition to the conventional anti-periodic boundary
condition, when Ψ is transported around a cycle of the Riemann surface.
If we insist on the usual spin structure for fermions, that Ψ can only pick up an overall factor
of ±1 around any closed cycle, then two values of k are singled out, k = 0 for all N and k = N/2
for even N. The choice k = 0 will produce a partition function that computes tr(ON),while the
+
choice k =N/2 will produce a partition function that computes tr[(O O )N/2]. As we will discuss
+ −
below, thereareapairofadditionalspecialchoices, k =(N±1)/2foroddN, whichdonothavean
(cid:81)
interpretation as a tr[ O ], but which nevertheless have some nice properties. For now, we will
i si
keep the dependence on k arbitrary.
As introduced in refs. [10,25–27], a twist operator σk (w) is defined as the field that simulates
R
thefollowingmonodromybehavior: Ψ(cid:126) (x)σk (w)→exp(2πik/N)TRΨ(cid:126) (x)σk (w)whenxisrotated
R R
counter-clockwisearoundw. ThenZ[N]canbeexpressedasacorrelationfunctionoftwistoperators
on a single copy of C rather than as a partition function on X ,
N
(cid:42) p q  (cid:43)
(cid:89) (cid:89)
Z[N]∼  σ10(si)σ−01(ti) σ−k1(uj)σ1k(vj) . (23)
i=1 j=1
AO
6InordertopreserveanexplicitZN symmetry,wehavechosenaslightlydifferentmatrixthaninref.[25].
7
The subscript AO means the operators are in ascending order of coordinates. Such correlation
functionscanbecalculatedthroughbosonization(seee.g.ref.[25]). DiagonalizationofT leadstoN
decoupled fields, Ψ(cid:101)l. Each Ψ(cid:101)l is multivalued, picking up a phase e−iNl2π, eiNl2π, eil−Nk2πor e−il−Nk2π
when rotated counter-clockwise around s , t , u , or v respectively. Then one can factorize each
i i i i
multi-valued field Ψ(cid:101)l into a gauge factor that describes this multi-valuedness and a single valued
´
free Dirac field: Ψl =ei xx0dx(cid:48)µAlµ(x)ψl(x). The gauge field dependent part of the partition function
contains the branch point dependence of Z[N] and is moreover straightforward to evaluate. With
the notation [26],
(cid:26) (cid:27)
1−N lR+k+(N −1)/2
q (R,k)≡ + , (24)
l 2N N
wherethecurlybracesdenotethefractionalpartofanumberandl∈(cid:96)=(cid:8)−N−1,−N−1 +1,...,N−1(cid:9),
2 2 2
the gauge field Al (x) satisfies the contour integrals
µ
˛ ˛
2πl 2πl
dxµAl (x)=− , dxµAl (x)= , (25)
µ N µ N
˛Csi ˛Csi
dxµAl (x)=2πq (1,N −k) , dxµAl (x)=2πq (−1,k) . (26)
µ l µ l
Cui Cvi
The Lagrangian density7 in terms of ψl(x) becomes L = (cid:80)N ψ¯lγµ(cid:0)∂ +iAl(cid:1)ψl. From eqs. (25)
l=1 µ µ
and (26) and Green’s theorem we have:
p q (cid:20) (cid:21)
(cid:88)(cid:88) l
(cid:15)µν∂ Al (x)=2π (δ(x−s )−δ(x−t ))−q (1,N −k)(δ(x−u )−δ(x−v )) .
ν µ N i i l i i
i=1j=1
(27)
Since the ψl’s are decoupled, the partition function becomes a product of expectation values of
operators that depend on the gauge field A :
µ
T [N]≡ Z[N] =(cid:89)(cid:104)ei´Alµjlµd2x(cid:105) , (28)
(Z[1])N
l∈(cid:96)
where jµ is the Dirac current ψ¯lγµψl. After bosonization, it becomes jµ = 1 (cid:15)µν∂ φl. Then T [N]
l l 2π ν
can be written as a correlation function of free boson vertex operators Ve(w)=e−i2eφl(w),
N(cid:89)2−1 (cid:104)ei´Alµjlµd2x(cid:105)=(cid:42)(cid:89)p (cid:89)q V2l/N(si)V−2l/N(ti)V2ql(−1,k)(uj)V2ql(1,N−k)(vj)(cid:43) . (29)
l=−N−1 i=1j=1
2
To evaluate the correlation function of twist operators, we use
(cid:42) m (cid:43)
(cid:89) V (w ) =(cid:89)|w −w |−eiej(cid:15)−m (30)
ei i i j
li=1 i(cid:54)=j
where(cid:15)isaUVcut-offtotakeintoaccounttheeffectofcoincidentpointsinthecorrelationfunction.
7Our conventions for the Clifford algebra are that {γµ,γν}=2δµν. For example, we could choose γx =σ3 and
γt=σ1
8
We also need the sums
N−1
(cid:88)2 l2 N2−1
= , (31)
N2 12N
l=−N−1
2
N−1
(cid:88)2 lql(1,N −k) = N2−1 − (N −k)k . (32)
N2 12N 2N
l=−N−1
2
to get an explicit expression for T [N].
Toshortentheexpressions,weadoptthefollowingnotation: {s }=S;{t }=T;{u }=U;{v }=
i i i i
V along with
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:89) (cid:12) (cid:12) (cid:89) (cid:12)
[Y,Z]=(cid:12) (y−z)(cid:12) , [Y,Y]=(cid:12) (y1−y2)(cid:12) . (33)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)y∈Y,z∈Z (cid:12) (cid:12)y1,y2∈Y,y1(cid:54)=y2 (cid:12)
Then T [N] can be written as:
T[N]=L−N62N−1XN62N−1−(N−Nk)k , (34)
where we have defined
[S,T][U,V] [S,V][T,U]
L≡ , X ≡ . (35)
[S,S][T,T][U,U][V,V](cid:15)p+q [S,U][T,V]
Fixing the appropriate spin structures, we claim then that
(cid:18)L(cid:19)−N62N−1
tr(ON)=tr(ON) = , (36)
+ − X
(cid:18)L(cid:19)−N62N−1
tr[(O O )N/2] = X−N/4 . (37)
+ − X
Comparing with the two interval case (13), we can absorb c into the (cid:15) dependence of L. A nice
N
feature of these expressions is that it is straightforward to take the N →1 limit.
3.1 Adjacent Limits
Let us consider adjacent limits of the two-interval negativity. We call the single-interval negativity
the case when s = v and t = u, and there is only one length scale, say l = t−s. We call the
two-adjacent-interval negativity the case where t = u and we have two length scales, l = t−s
1
and l = v−u. The single-interval and two-adjacent-interval negativities are given by a two point
2
function and a three point function of twist fields respectively. They are therefore fully determined
by conformal symmetry [9,10]:
R(No)∼l−N6o2N−1 , R(Ne)∼l−N6e2N−4 , (38)
R(No)∼(l1l2(l1+l2))−N1o22N−1 , R(Ne)∼(l1l2)−N1e22N−4 (l1+l2)−N1e22N+2 . (39)
While tr(ON) simply vanishes in these coincident limits, we claim that tr[(O O )N/2] reproduces
+ + −
R(N ) for even N, in both the single-interval and two-adjacent-interval cases. This agreement
e
9
provokes the question is there a choice of k for odd N for which T[N] has the correct adjacent
interval limits? The answer is yes. If we choose k =(N ±1)/2, then
o
T [No]=L−N6o2N−o1X−N12o2N−o1 , (40)
and this expression reproduces R(N ) in the adjacent interval limits.
o
To see why the values k =N /2 and k =(N ±1)/2 are singled out, we consider the merging of
e o
twistoperatorsσk(w )σ0(w )→σk(w ). Thecorrespondingconstraintonthecorrelationfunction
1 i 1 i+1 2 i
is
(cid:68) (cid:69) (cid:68) (cid:69)
lim σk1 (w )···σk(w )σ0(w )··· |w −w |−γi(i+1) = σk1 (w )···σk(w )··· (41)
wi+1→wi R1 1 1 i 1 i+1 i i+1 R1 1 2 i
along with a corresponding constraint from considering σ0 (w )σk (w ). We have defined
−1 i −1 i+1
(cid:88)
γ ≡ q (R ,k )q (R ,k ) . (42)
ij l i i l j j
l∈(cid:96)
These constraints can only be satisfied if the following identities holds for all l∈(cid:96):
q (−2,k)=q (−1,0)+q (−1,k) , q (2,k)=q (1,0)+q (1,k) . (43)
l l l l l l
The k values (N −1)/2, N /2 and (N +1)/2 are the only solutions.
o e o
4 Bounds on the Negativity
WediscussthreetypesofboundsonR(N)inthefollowingsubsections. Thefirst,whichfollowsfrom
a triangle inequality on the Schatten p-norm, is an upper bound on the moments of the partially
transposed density matrix. The second two are conjectural. We are able to demonstrate these
conjectured bounds only for small N >1.
The Schatten p-norm, defined as
(cid:107)M(cid:107) ≡(cid:16)tr(cid:16)(cid:0)M†M(cid:1)p/2(cid:17)(cid:17)1/p , p∈[1,∞) , (44)
p
is a generalization of the trace norm. Indeed, the Schatten 1-norm is the trace norm.
Because tr[(ρT2)N]1/N is the Schatten N-norm of ρT2, for all even N we have by the triangle
A A
inequality that
(cid:16)(cid:13) (cid:13) (cid:17)N (cid:16)(cid:13) (cid:13) (cid:13) (cid:13) (cid:17)N
tr[(ρT2)N]= (cid:13)ρT2(cid:13) ≤2−N/2 (cid:13)eiπ/4O (cid:13) +(cid:13)e−iπ/4O (cid:13) =2N/2tr[(O O )N/2] . (45)
A (cid:13) A (cid:13) (cid:13) +(cid:13) (cid:13) −(cid:13) + −
N N N
The N →1 limit of (45) leads to an upper bound on the negativity in terms of tr[(O O )1/2]
+ −
R(1)=(cid:13)(cid:13)(cid:13)ρTA2(cid:13)(cid:13)(cid:13)T ≤(cid:13)(cid:13)(cid:13)(cid:13)1+2 iO+(cid:13)(cid:13)(cid:13)(cid:13)T +(cid:13)(cid:13)(cid:13)(cid:13)1−2 iO−(cid:13)(cid:13)(cid:13)(cid:13)T =√2tr[(O+O−)1/2]=√2X−1/4 . (46)
We have thus established that tr[(O O )N/2] provides a rigorous upper bound on the negativity
+ −
and its Nth moments, for free fermions.
10