Table Of ContentCharacterizing ground and thermal states of few-body Hamiltonians
Felix Huber and Otfried Gu¨hne
Naturwissenschaftlich-Technische Fakult¨at, Universita¨t Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany
(Dated: June 30, 2016)
Thequestionwhetheragivenquantumstateisagroundorthermalstateofafew-bodyHamilto-
niancanbeusedtocharacterizethecomplexityofthestateandisimportantforpossibleexperimen-
tal implementations. We provide methods to characterize the states generated by two- and, more
generally, k-body Hamiltonians as well as the convex hull of these sets. This leads to new insights
intothequestionwhichstatesareuniquelydeterminedbytheirmarginalsandtoageneralizationof
theconceptofentanglement. Finally,certificationmethodsforquantumsimulationcanbederived.
PACSnumbers: 03.65.Ud,03.67.Mn
6
1
Introduction.—Interactionsinquantummechanicsare
0 Q1
described by Hamilton operators. The study of their
2 2
Q
properties, such as their symmetries, eigenvalues, and
n
ground states, is central for several fields of physics.
u
J Physically relevant Hamiltonians, however, are often re-
stricted to few-body interactions, as the relevant inter-
9
2 action mechanisms are local. But the characterization
of generic few-body Hamiltonians is not well explored,
h] since in most cases one starts with a given Hamiltonian conv(Q2) state space c=onSvE(QP1)
p and tries to find out its properties.
- In quantum information processing, ground and ther- FIG. 1: Schematic view of the state space, the exponential
t
n malstatesoflocalHamiltoniansareofinterestforseveral families Q1 and Q2, and their convex hulls. While the whole
a reasons: First, if a desired state is the ground or ther- space of mixed states is convex, the exponential families are
u
non-convex low-dimensional manifolds. The convex hull of
mal state of a sufficiently local Hamiltonian, it might
q
Q are the fully separable states and our approach allows to
[ be experimentally prepared by engineering the required 1
interactions and cooling down or letting thermalise the characterize the convex hull for arbitrary Qk.
2
physical system [1]. For example, one may try to pre-
v
0 pare a cluster state, the resource for measurement-based
3 quantumcomputation,asagroundstateofalocalHamil- especially tailored to cluster and, more generally, graph
6 tonian [2]. Second, on a more theoretical side, ground states. In previous approaches it was only shown that
1
states of k-body Hamiltonians are completely charac- some special states are far away from the eigenstates of
0
. terized by their reduced k-body density matrices. The local Hamiltonians [12], but no general method for esti-
1 question which states are uniquely determined by their mating the distance is known.
0
marginals has been repeatedly studied and is a variation
6
Our approach leads to new insights in various direc-
oftherepresentabilityproblem,whichaskswhethergiven
1
tions. First, it has been shown that cluster and graph
: marginalscanberepresentedbyaglobalstate[3]. Ithas
v states can, in general, not be exact ground states of two-
turned out that many pure states have the property to
i
X be uniquely determined by a small set of their marginals body Hamiltonians [2], but it was unclear whether they
still can be approximated sufficiently well. Our method
r [4, 5], and for practical purposes it is relevant that often
a shows that this is not the case and allows to bound the
entanglement or non-locality can be inferred by consid-
distancetogroundandthermalstates. Second,asshown
ering the marginals only [6].
inRef.[4],almostallpurestatesofthreequbitsarecom-
In this paper we present a general approach to char-
pletely determined by their two-party reduced density
acterize ground and thermal states of few-body Hamil-
matrices. As we prove, for N 5 qubits or four qutrits
tonians. We use the formalism of exponential families, a ≥
this is not the case, but we present some evidence that
conceptfirstintroducedforclassicalprobabilitydistribu-
the fact might still be true for four qubits. Finally, our
tions by Amari [7] and extended to the quantum setting
methodresultsinwitnesses,whichcanbeusedinaquan-
in Refs. [8–11]. This offers a systematic characteriza-
tum simulation experiment to certify that a three-body
tionofthecomplexityofquantumstatesinaconception-
HamiltonianoraHamiltonianhavinglong-rangeinterac-
ally pleasing way. We derive two methods that can be
tions was generated.
usedtocomputevariousdistancestothermalstatesofk-
bodyHamiltonians: Thefirstmethodisgeneralanduses The setting.— A two-local (or two-body) Hamiltonian
semidefinite programming, while the second method is of a system consisting of N spin-1/2 particles can be
2
written as states inside a given convex set. Consequently, the ob-
servation of a negative expectation value proves that a
H =(cid:88)N (cid:88) λ(ij)σ(i) σ(j), (1) state is outside of the set. We will see below that such
i,j=1 αβ αβ α ⊗ β
witnesses can be used to certify quantum simulation.
where σ(i) denotes a Pauli matrix 1,σ ,σ ,σ acting Quantum exponential families.— We recall some re-
α { x y z} sults on the characterization of quantum exponential
on the i-th particle etc. Note that the identity matrix is
families [10, 11]. Given a state (cid:37), consider its distance
included,soH canalsocontainsingleparticleterms. We
from the exponential family in terms of the relative
denote the set of all possible two-local Hamiltonians by Q2
entropy (or divergence) S((cid:37) τ) = tr[(cid:37)(log((cid:37)) log(τ))].
Hitso2ntihaaenndsHibenyisaeHnnkba.enrAaglnomgeooxudasmelmphaleanvnfioenrrgathnteewasoree-tlsotoc-fnakle-iHglohacbmaolirlHtoiannmtieairln-- Ainsfotrhmeactlioosnesptrsotjaetcetitoon(cid:37)(cid:37)˜i2n. Q|I|t2,hoanseboebetnaisnhsotwhn−e stoh-actaltlehde
followingthreecharacterizationsfortheinformationpro-
actions. However, our approach generally ignores any
jection (cid:37)˜ are equivalent [10]:
geometrical arrangement of the particles. Finally, for an 2 ∈Q2
(a) (cid:37)˜ is the unique minimizer of the relative entropy of
arbitrary multi-qubit operator A we call the number of 2
(cid:37) from the set ,
qubits where it acts on non-trivially the weight of A. In Q2
practice,thiscanbedeterminedbyexpandingAinterms
(cid:37)˜ =argmin S((cid:37) τ). (3)
of tensor products of Pauli operators and looking for the 2 τ∈Q2 ||
largest non-trivial product. (b)Ofthesetofstateshavingthesametwo-bodyreduced
Thesetweaimtocharacterizeistheso-calledexponen- density matrices (2-RDMs) as (cid:37), denoted by ((cid:37)), (cid:37)˜
2 2
M
tial family 2, consisting of thermal states of two-local has a maximal von Neumann entropy
Q
Hamiltonians
(cid:37)˜ =argmax S(µ). (4)
(cid:8) (cid:12) e−βH (cid:9) 2 µ∈M2((cid:37))
Q2 = τ(cid:12)τ = tr[e−βH],H ∈H2 . (2) (c) Finally, (cid:37)˜2 is the unique intersection of 2 and
Q
((cid:37)). From (b) it follows that if for a state σ another
2
M
Ground states can be reached in the limit of infinite in- state (cid:37) of higher entropy but having the same 2-RDMs
verse temperature β. For any k, the exponential fami- can be found, then σ must lie outside of . A further
2
Q
lies k can be defined in a similar fashion. The set 1 discussion can be found in Appendix A [14].
Q Q
consists of mixed product states, the set of the full States not in are said to have irreducible correla-
N 2
Q Q
state space. The exponential families form the hierar- tions of order three or higher, because they contain in-
chy 1 2 N, and a suitable βH can be formation which is not already present in their 2-RDMs,
Q ⊆ Q ⊆ ··· ⊆ Q
seenasawayofparameterizingaspecificdensitymatrix if one wishes to reconstruct the global state from its
τ =e−βH/tr[e−βH]. Thequestionarises,whatstatesare marginalsaccordingtoJaynes’maximumentropyprinci-
in ? And for those which are not, what is their best ple [15]. This is conceptionally nice, but also has certain
k
Q
approximation by states in ? drawbacks. Importantly, the irreducible correlation as
k
Q
It turns out to be fruitful to consider the convex hull quantified by the relative entropy is not continuous, as
shown in Ref. [16]. In addition, the relative entropy is
(cid:8)(cid:88) (cid:88) (cid:9)
conv( 2)= piτi τi 2, pi =1, pi 0 , difficult to estimate experimentally without doing state
Q i | ∈Q i ≥
reconstruction, sootherdistances suchasthe fidelityare
and ask whether a state is in this convex hull or not (see preferable. These properties make the relative entropy
also Fig. 1). The convex hull has a clear physical inter- somewhat problematic and give further reasons why we
pretation as it contains all states that can be generated consider the convex hull.
by preparing thermal states of two-body Hamiltonians Characterization via semidefinite programming.—Our
stochastically with probabilities p . In this way, taking first method to estimate the distance of a given state
i
the convex hull can be seen as a natural extension of the to the convex hull of relies on semidefinite program-
2
Q
conceptofentanglement: Thethermalstatesofone-body ming [17]. This optimization method is insofar useful,
Hamiltoniansarejustthemixedproductstatesandtheir assemidefiniteprogramsareefficientlysolvableandtheir
convex hull are the fully separable states of N particles solutionscanbecertifiedtobeoptimal. Moreover,ready-
[13]. Inthisframework,theresultofLindenetal.[4]can to-use packages for their implementation are available.
be rephrased as stating that all three-qubit states are in As a first step we formulate a semidefinite program to
the closure of the convex hull conv( 2), since nearly all test if a given pure ψ state is outside of 2. From the
Q | (cid:105) Q
pure states are ground states of two-body Hamiltonians. characterizationinEq.(4)itfollowsthatitsufficestofind
Finally, the characterization of the convex hull leads a different state (cid:37) having the same 2-RDMs as ψ . If (cid:37)
| (cid:105)
to the concept of witnesses that can be used for the ex- is mixed, its entropy is higher than that of ψ , mean-
| (cid:105)
perimental detection of correlations [13]. Witnesses are ing that ψ cannot be its own information projection
| (cid:105)
observables which have positive expectation values for andthereforeliesoutsideof . If(cid:37)ispure,considerthe
2
Q
3
4
convexcombination(ψ ψ +(cid:37))/2,againhavingahigher 5 4
| (cid:105)(cid:104) |
entropy. To simplify notation we define for an arbitrary
5 3
N-qubit operator X the operator Rk(X) as the projec- 6 3
tionofX ontothoseoperators,whichcanbedecomposed
into terms having at most weight k. In practice, Rk(X) 1 2 1 2
canbecomputedbyexpandingX inPaulimatrices, and
removing all terms of weight larger than k. Note that FIG.2: Examplesofgraphsdiscussedinthispaper. Left: The
R ((cid:37)) may have negative eigenvalues. five-qubit ring-cluster graph. The corresponding ring-cluster
k
The following semidefinite program finds a state with state |C5(cid:105) has a finite distance to the exponential family Q2.
Middle: Themaximallyentangledsix-qubit|M (cid:105)stateisnot
the same k-body marginals as a given state ψ : 6
| (cid:105) intheconvexhullofQ . Right: The2Dperiodic5×5cluster
3
state |C (cid:105) is not in conv(Q ).
min: tr[(cid:37)ψ ψ ] 5×5 4
(cid:37) | (cid:105)(cid:104) |
subject to: R ((cid:37))=R (ψ ψ ),
k k
| (cid:105)(cid:104) |
tr[(cid:37)]=1, (cid:37)=(cid:37)†, (cid:37) δ1. (5) whether the results of Ref. [4] can be generalized. Re-
≥ call that in this reference it has been shown that nearly
While this program can be run with δ = 0, it is useful all pure states of three qubits are uniquely determined
to choose δ to be strictly positive. Then, a strictly pos- (amongallmixedstates)bytheirreducedtwo-bodyden-
itive (cid:37) may be found, which is guaranteed to be distant sity matrices. This means that they are ground states of
from the state space boundary. Consequently, if ψ is two-bodyHamiltonians. Consequently,theclosureofthe
| (cid:105)
disturbed, one can still expect to find a state with the convex hull conv( ) contains all pure states and there-
2
Q
same reduced density matrices in the vicinity of (cid:37). This fore also all mixed states, and the semidefinite program
can be used to prove that the distance to is finite, in Eq. (5) will not be feasible for δ strictly positive. The
2
Q
and will allow us to construct witnesses for proving irre- questioniswhetherthisresultholdsformorequbitstoo.
ducible correlations in ψ . We make this rigorous in the Concerning pure five-qubit states, we numerically
| (cid:105)
followingObservation. Forthat, let (ψ )betheballin foundafractionof40%tobeoutsideofconv( ). Inthe
trace distance Dtr(µ,η)= 21tr(|µ−ηB|)|ce(cid:105)ntered at |ψ(cid:105). caseofpurefour-qubitstateshowever, notestQed2 random
Observation 1. Consider a pure state ψ and a state has been found to lie outside of conv( ). Given
mixed state (cid:37) δ1 with Rk((cid:37)) = Rk(ψ ψ ).|T(cid:105)hen, for the fact that the test works well in the casesQo2f five and
≥ | (cid:105)(cid:104) |
anystateσ intheball δ(ψ )avalidstate(cid:37)˜in δ((cid:37))can sixqubits,thisleadsustoconjecturethatnearlyallpure
B | (cid:105) B
befound, suchthattheirk-partyreduceddensitymatrices four qubit states are in conv( ), and hence also in .
2 2
match. Moreover, the entropy of (cid:37)˜is larger than or equal This would imply that a simQilar result as the one Qob-
to the entropy of σ. This implies that the ball δ(ψ ) tainedbyRef.[4]holdsinthecaseoffourqubits: almost
B | (cid:105)
contains no thermal states of k-body Hamiltonians. every pure state of four qubits is completely determined
The proof is given in Appendix B [14]. by its two-particle reduced density matrix. More details
In the Observation, we considered the trace distance, are given in Appendix C [14].
but a ball in fidelity instead of trace distance can be ob-
Characterization via the graph state formalism.—
tained: Consider a state σ near ψ , having the fidelity
| (cid:105) The family of graph states includes cluster states and
F(σ,ψ) = α 1 δ2, where F((cid:37),ψ) = tr[(cid:37)ψ ψ ] =
≥ − | (cid:105)(cid:104) | GHZ states and has turned out to be important for
ψ (cid:37)ψ . Then from the Fuchs-van-de-Graaf inequality
(cid:104) | | (cid:105) (cid:112) measurement-basedquantumcomputationandquantum
follows D (σ, ψ ψ ) 1 F(σ,ψ) δ, and Obser-
tr | (cid:105)(cid:104) | ≤ − ≤ error correction [19]. Due to their importance, the ques-
vation 1 is applicable [18].
tion whether graph states can be prepared as ground
The usage of the fidelity as a distance measure has a
states of two-body Hamiltonians has been discussed be-
clear advantage from the experimental point of view, as
fore[2]. Generally,graphstateshaveshowntonotbeob-
it allows the construction of witnesses for multiparticle
tainable as unique non-degenerate ground states of two-
correlations. Indeed the observable
local Hamiltonians. Further, any ground state of a k-
=(1 δ2)1 ψ ψ (6) local Hamiltonian H can only be (cid:15)-close to a graph state
W − −| (cid:105)(cid:104) | G with m(G )>k at the cost of H having an (cid:15)-small
| (cid:105) | (cid:105)
has a positive expectation value on all states in and, energy gap relative to the total energy in the system [2].
k
duetothelinearityofthefidelity,alsoonallstateQswithin Here m(G ) is the minimal weight of any element in the
| (cid:105)
the convex hull conv( ). So, a negative expectation stabilizer S of state G (see also below). But as pointed
value signals the preseQncke of k-body correlations. Wit- outinRef.[2],thisd|oe(cid:105)snotimplythatgraphstatescan-
nesses for entanglement have already found widespread not be approximated in general, as (cid:15) is a relative gap
applications in experiments [13]. only.
Equipped with a method to test whether a pure state Let us introduce some facts about graph states. A
is in conv( ) or not we are able to tackle the question graph consists of vertices and edges (see Fig. 2). This
2
Q
4
defines the generators can therefore serve as a witness =α1 C C
5×5 5×5
W −| (cid:105)(cid:104) |
(cid:89) forconv( 4),whereα=(225 1)/225. Itshouldbenoted
g =σ(a) σ(b), (7) Q −
a x b∈N(a) z thatthiswitnesscanalsobeusedforconv(Q2),forwhich
thevalueαmightbeimproved[23]. Finally,theminimal
where the product of the σ(b) runs over all vertices con- distanceD intermsoftherelativeentropyfrom can
z k k
Q
nectedtovertexa,calledneighborhoodN(a). Thegraph belowerboundedbythefidelitydistancefromitsconvex
state G can be defined as the unique eigenstate of all hull conv( ), see Appendix D for details [14].
k
| (cid:105) Q
the g , that is G = g G . This can be rewritten with
a a Quantum simulation as an application.— The aim of
| (cid:105) | (cid:105)
the help of the stabilizer. The stabilizer S is the com-
quantum simulation is to simulate a physical system of
mutative group consisting of all possible 2N products of
interest by another well-controllable one. Naturally, it is
(cid:81)
g , that is S = s = g . Then, the graph state
caan be written a{s iG G a=∈I2−aN} (cid:80) s [19]. This for- crucialtoascertainthattheinteractionsreallyperformas
| (cid:105)(cid:104) | si∈S i intended. Differentproposalshaverecentlycomeforward
mula allows to determine the reduced density matrices
to engineer sizeable three-body interactions in systems
of graph states easily, since one only has to look at the
of cold polar molecules [24], trapped ions [25], ultracold
products of the generators g .
a atomsintriangularlattices[26],Rydbergatoms[27]and
For instance, all stabilizer elements of the five-qubit
circuit QED systems [28]. Using the ring cluster state
ring cluster state C5 have at least weight three, and witness =α1 C C derivedabove,itispossible
| (cid:105) N N
therefore the 2-RDMs of C are maximally mixed. By W −| (cid:105)(cid:104) |
5 to certify that three- or higher-body interactions have
| (cid:105)
choosing δ = 2−5 in Observation 1, the maximum over-
beenengineered. Thisisdonebylettingthesystemunder
laptoconv( )isboundedbyF (C ,τ) 1 δ2
Q2 τ∈Q2 | 5(cid:105) ≤ − ≈ controlthermalise. Ifthen <0ismeasured,onehas
0.99902. Note that Ref. [20] has demonstrated a slightly (cid:104)W(cid:105)
certified that interactions of weight three or higher are
(cid:112)
better bound F(C ,τ) 1/32+ 899/960 0.99896.
5 present. At least five qubits are generally required for
| (cid:105) ≤ ≈
However,bothboundsarebyfarnotreachableincurrent
this, but by further restricting the interaction structure,
experiments. In fact, one can do significantly better. In
four qubits can be enough for demonstration purposes.
the following, we will formulate a stricter bound by first
Thiscanalreadybedonewithafidelityof93.75%,which
considering and the ring cluster state C for an ar-
2 N is within reach of current technologies. Further details
Q | (cid:105)
bitrarynumberofqubitsN 5,buttheresultisgeneral.
can be found in Appendix E [14].
≥
Observation 2. The maximum overlap between the
Asanoutlook,onemaytrytoextendthisideaofinter-
N-qubit ring cluster state C and an N-qubit state τ
N
| (cid:105) ∈ action certification to the unitary time evolution under
is bounded by
2
Q local Hamiltonians. For instance, digital quantum sim-
D 1 ulation can efficiently approximate the time evolution of
sup C τ C − , (8)
N N
τ∈Q2(cid:104) | | (cid:105)≤ D a time-independent local Hamiltonian and in Ref. [29]
aneffective6-particleinteractionhasbeenengineeredby
whereD =2N isthedimensionofthesystem. Moregen-
applying a stroboscopic sequence of universal quantum
erally, for an arbitrary pure state with maximally mixed
gates. Theprocessfidelitywasquantifiedusingquantum
reduced k-party states in a d⊗N-system, the overlap with
process tomography, however it would be of interest to
is bounded by (dN 1)/dN.
k prove that the same time evolution cannot be generated
Q −
The proof is given in Appendix D [14].
by 5-particle interactions only.
In the case of five qubits, F (C ,τ) 31/32
τ∈Q2 | 5(cid:105) ≤ ≈ Conclusion.—Wehaveprovidedmethodstocharacter-
0.96875, which improves the bound on the distance to
izethermalandgroundstatesoffew-bodyHamiltonians.
conv( ) by more than two orders of magnitude [21].
2
Q Our results can be used to test experimentally whether
From Observation 2, we can construct the witness
three-body or higher-order interactions are present. For
= D−11 C C , (9) futurework,itwouldbedesirabletocharacterizetheen-
N N
W D −| (cid:105)(cid:104) | tanglement properties of , e.g. to determine whether
2
Q
the entanglement in these states is bounded, or whether
which detects states outside of conv( ). In a similar
2
Q they can be simulated classically in an efficient manner.
fashion, any state having the maximally mixed state as
Furthermore,itisofsignificantexperimentalrelevanceto
k-particle RDMs can be used to construct a witness for
develop schemes to certify that a unitary time evolution
conv( ). First, there is a four-qutrit state with maxi-
k
Q was generated by a k-body Hamiltonian.
mally mixed 2-RDMs [22], which can be used to derive
a witness for conv( ). The highly entangled six-qubit Acknowledgement.— We thank Tobias Galla, S¨onke
2
Q
state M (seethegraphinFig.2)hasmaximallymixed Niekamp, and Marcin Paw(cid:32)lowski for discussions. This
6
3-RDM| s,(cid:105)so = 631 M M isawitnesstoexclude work was supported by the SNSF, the COST Action
W 64 −| 6(cid:105)(cid:104) 6|
thermal states of three-body Hamiltonians. Third, con- MP1209, the FQXi Fund (Silicon Valley Community
sidera5 52Dclusterstatewithperiodicboundarycon- Foundation), the DFG, and the ERC (Consolidator
×
ditions(seeFig.2). Thisstatehasm(C )=5[2],and Grant 683107/TempoQ).
5×5
| (cid:105)
5
%
APPENDIX
A. Further discussion of the marginal set M ((cid:37)) and
k τ
its relation to Q
k
%˜
k
The marginal set ((cid:37)) consists of quantum states
k
M
having the same k-party reduced density matrices (k-
RDMs) as (cid:37)
((cid:37))= µ µ =(cid:37) for all A k , (10) (%)
Mk { | A A | |≤ } k
M
k
where µ is the reduced state obtained by tracing out Q
A
all subsystems not contained in A. This set is convex,
as its states stay in the marginal family under convex FIG. 3: The information projection (cid:37)˜k lies in the unique in-
tersection of Q and M ((cid:37)). It is also the minimizer of the
combination. k k
relative entropy S(ρ||·) in Q .
The exponential family consists of thermal states k
k
Q
of k-local Hamiltonians
(cid:8) (cid:12) eH (cid:9) The information projection (cid:37)˜k of (cid:37) is the element in
Q2 = τ(cid:12)τ = tr[eH],H ∈H2 . (11) k((cid:37)) having the largest von Neumann entropy. Given
M
(cid:37), its information projection (cid:37)˜ , and a τ , the
k k
∈ Q
In contrast to the marginal set, the exponential families Pythagorean relation then simplifies to
are,apartfrom ,notconvex. Toseethis,notethat
k n
cQonv(Q1) is the setQof separable states, having a volume S((cid:37)||τ)=S((cid:37)||ρ˜k)+S((cid:37)˜k||τ). (16)
and a number of free parameters corresponding to the
The above definition of the information projection is
dimension of the state space. In addition, conv( ) is
Qk equivalent to (cid:37)˜ being in the unique intersection of
larger than the set of separable states for k 2. How-
≥ the exponential family with ((cid:37)), and to (cid:37)˜ =
emveears,uQrek zhearso.noTthaussmQakny(cid:40)frceoenvp(aQrakm),etaenrds aQnkdciasnansoettboef argWmeinpτr∈oQvkidSe(ρt|w|τo).exTahmiQspiklsesi.llusFtirrMastte,kdcionnsFidige.r3t.he kfive-
convex.
qubit ring cluster state C and its information pro-
The relations between the marginal set and the | 5(cid:105)
jection onto . The state C has maximally mixed
exponential family originates in two special ways to Q2 | 5(cid:105)
2-body marginals, and of the set (C ), the max-
parametrize a quantum state [8]. These are the affine M2 | 5(cid:105)
imally mixed state has the highest entropy. Second,
(also called mixed) and the exponential representations
consider the one-parameter family of states GHZ =
α
| (cid:105)
(cid:37)aff =1/D+ηiAi, η ( 1,1)D2−1, (|000(cid:105)+eiα|111(cid:105))/√2. Allofitstwo-partyreducedstates
∈ − are equal to (00 00 + 11 11)/2. Also,
(cid:37) =exp[θ A ψ(θ)], θ RD2−1, (12) | (cid:105)(cid:104) | | (cid:105)(cid:104) |
exp i i− ∈ (cid:90)
γ = GHZ GHZ dα
where D is the dimension of the system, {Ai} is a suit- α| α(cid:105)(cid:104) α|
able orthonormal basis of the operator space, and we
=(000 000 + 111 111)/2 (17)
sumoverrepeatedindices. TheMassieufunctionψ(θ)= | (cid:105)(cid:104) | | (cid:105)(cid:104) |
logtr[expH] is not only required for normalization, but has the same 2-RDMs and is thus an element of the
alsodefines, togetherwiththepotentialφ(η)= S(η)= marginal set (GHZ ). Additionally, γ is the infor-
− M2 | α(cid:105)
tr[(cid:37)log(cid:37)], a Legendre transform ψ(θ)+φ(η) θ η =0. mationprojectionof GHZ onto ,asitistheelement
− i i | α(cid:105) Q2
The relations of maximum entropy in (GHZ ). As known from
2 α
M | (cid:105)
Ref. [4], almost all three qubit states are determined by
∂ψ(θ) ∂φ(η)
ηi = =tr[(cid:37)A ], θ = =tr[HA ], (13) their 2-RDMs, and thus the irreducible three-body cor-
i i i
∂θ ∂η
i i relationisdiscontinuousat GHZ . Moreexamplescan
α
| (cid:105)
follow. For any two states (cid:37)(η) and (cid:37)(cid:48)(θ(cid:48)), the following be found in Ref. [11].
Pythagorean relation for the relative entropy holds As a last part in this section, we relate the expo-
nential family conv( ) to the sets of ground and ex-
k
S((cid:37)||(cid:37)(cid:48))=φ(η)+ψ(θ(cid:48))−ηiθi(cid:48), (14) citedstatesoflocalHQamiltoniansrespectively. Asargued
above, nondegenerate ground states of k-local Hamilto-
and its repeated application yields
nians H are determined by their k-RDMs and be-
k k
∈H
long to the closure of . Nondegenerate excited states
S((cid:37) (cid:37)(cid:48)(cid:48)) Qk
|| of k-local Hamiltonians are completely determined by
=S((cid:37) (cid:37)(cid:48))+S((cid:37)(cid:48) (cid:37)(cid:48)(cid:48))+(η η(cid:48)) (θ(cid:48) θ(cid:48)(cid:48)). (15)
|| || i− i · i− i their 2k-RMDs [30], and are therefore ground states of
6
suitable 2k-local Hamiltonians H . The argu-
2k 2k
∈ H
ment rests on the fact that any eigenstate of a Hamil-
tonian Hk will also be the ground state of (Hk λ1)2, ψ σ
−
where λ is the corresponding eigenvalue. A similar ar- | i
X
gument also holds for nondegenerate ground and eigen-
states. But as can be seen by parameter counting, there
exist 2k-local Hamiltonians which cannot be written as
H2k =(Hk−λ1)2 with Hk ∈Hk. Thus the set of eigen- Bδ(ψ)
states of k-local Hamiltonians ES( ) is a proper sub-
k
H
set of the set of ground states of 2k-local Hamiltonians
GS( ), ES( )(cid:40)GS( ). Finally, thermal states of
2k k 2k
H H H
kan-ldociatlfoHllaomwislttohnaitancsonavr(e in)t(cid:40)hecocnonvv(GexS(hull )o)f.ES(Hk), Bδ(%) X
k 2k %˜
Q H %
B. Proof of Observation 1 FIG. 4: Illustration of Observation 1: If a strictly positive
(cid:37) can be found, then for a given perturbation σ of |ψ(cid:105) one
can find a corresponding (cid:37)˜in the vicinity of (cid:37), such that the
Observation 1. Consider a pure state ψ and a
mixed state (cid:37) δ1 with R ((cid:37)) = R (ψ ψ ).| T(cid:105)hen for reduced density matrices of σ and (cid:37)˜are the same.
k k
≥ | (cid:105)(cid:104) |
anystateσ intheball (ψ )avalidstate(cid:37)˜in ((cid:37))can
δ δ
B | (cid:105) B
befound, suchthattheirk-partyreduceddensitymatrices equality holds, then again σ / due to the uniqueness
k
∈Q
match. Moreover, the entropy of (cid:37)˜is larger than or equal of the information projection and because of σ =(cid:37)˜.
(cid:54)
to the entropy of σ. This implies that the ball Bδ(|ψ(cid:105)) First, note that if ρ fulfills the condition S((cid:37)) ≥ 2Cδ,
contains no thermal states of k-body Hamiltonians. where C = δlog( δ ) (1 δ)log(1 δ), then also
δ − D−1 − − −
as required S((cid:37)˜) S(σ). This follows from the sharp
Proof. Any σ in the trace ball (ψ ) can be written as ≥
Bδ | (cid:105) Fannes-Audenaert inequality [31]
σ = ψ ψ +X, with a traceless X. The trace can be
decom| p(cid:105)o(cid:104)se|d as tr(X) = ψ X ψ +(cid:80) ψ⊥ X ψ⊥ = 0, (cid:0) d (cid:1)
(cid:104) | | (cid:105) i(cid:104) i | | i (cid:105) S(η) S(µ) dlog (1 d)log(1 d), (20)
where the ψ⊥ are orthogonal to ψ . The second term | − |≤− D 1 − − −
| i (cid:105) | (cid:105) −
of this expression is positive, since where d=D (η,µ) and D =2N is the dimension of the
tr
(cid:88) ψ⊥ X ψ⊥ =(cid:88) ψ⊥ (X+ ψ ψ )ψ⊥ system. Recall that σ ∈ Bδ(ψ) and (cid:37)˜∈ Bδ((cid:37)). Thus the
(cid:104) i | | i (cid:105) (cid:104) i | | (cid:105)(cid:104) | | i (cid:105) entropyofσcanbeatmostCδ,andtheentropyof(cid:37)˜must
i i be at least S((cid:37)) C . Requiring S((cid:37)) 2C therefore
=(cid:88) ψ⊥ σ ψ⊥ 0. (18) ensures that the−entrδopy of (cid:37)˜is higher t≥han oδr equal to
(cid:104) i | | i (cid:105)≥
i that of σ.
Itremainstoshowthat(cid:37)indeedfulfillsthiscondition.
Sowemust have ψ X ψ 0. Furthermore, X canonly
(cid:104) | | (cid:105)≤ For that, note that the eigenvalues of (cid:37) are larger than
have one negative eigenvalue λ , otherwise there would
− δ but smaller than 1/N due to the normalization of (cid:37).
be also ψ⊥ with ψ⊥ X ψ⊥ < 0, which is in contra-
| i (cid:105) (cid:104) i | | i (cid:105) Furthermore assume D 8, since we are considering at
diction to σ 0. From tr(X) = 0 it follows that λ has ≥
≥ − least three qubits. From the bounds on the eigenvalues
the largest modulus of all eigenvalues and consequently
it follows that the entropy of (cid:37) is bounded by
tr(X ) = 2λ . Since D (ψ ψ ,σ) = trX /2 δ, it
− tr
| | | | | (cid:105)(cid:104) | | | ≤
follows that λ δ. S((cid:37)) [1 (D 1)δ]log[1 (D 1)δ]
| −|≤ ≥− − − − −
For σ δ(ψ ) we choose (cid:37)˜= (cid:37)+X as a candidate (D 1)δlog(δ) Γ. (21)
having th∈eBk-R|D(cid:105)Ms of σ. We have − − ≡
So,weconsiderthefunction (δ,D)=Γ 2C andhave
δ
F −
R (σ)=R (ψ ψ +X)=R ((cid:37)+X)=R ((cid:37)˜). (19) to show its positivity. Let us first fix D. Taking the sec-
k k k k
| (cid:105)(cid:104) | ond derivative of with respect to δ one directly finds
F
Furthermore, (cid:37)˜is a positive semidefinite density matrix, that this second derivative is strictly negative. This im-
because of (cid:37)˜= (cid:37)+X (δ λ )1 0. Thus, for any plies that assumes only one maximum in the interval
−
≥ −| | ≥ F
state σ in (ψ ) there exists a state (cid:37)˜ in ((cid:37)), such [0,1/D] and that the minima are assumed at the bor-
δ δ
B | (cid:105) B
that the k-RDMs of σ and (cid:37)˜match. ders. We have (0,D)=0 and it remains to prove that
F
Now we show that the entropy of (cid:37)˜ is larger than or (D) = (1/D,D) is positive. For D = 8 one can di-
G F
equal to the entropy of σ, as this ensures that σ is not rectly check that as well as its derivative is positive.
G
in . Namely, if the entropy of (cid:37)˜is larger, a state with Furthermore, the second derivative of (D) with respect
k
Q G
thesamek-RDMsbutofhigherentropythanσ hasbeen to D is strictly positive for any D 8, which proves the
≥
found, and σ is outside of . If on the other hand claim.
k
Q
7
δ 10−3 10−4 10−5 10−6 10−7 generalization is then straightforward. For N 5, the
≥
5qb 0.0040 0.1325 0.2976 0.3729 0.4000 ring cluster state CN has m(CN )=3, that is, all the
| (cid:105) | (cid:105)
two-bodyreduceddensitymatricesaremaximallymixed
6qb 0.7680 0.8872 0.8897 1.0000 1.0000
[2]. Since the family of thermal states is invariant under
TABLEI: Fractionofpurefiveandsixqubitstateswhichare the addition of the identity τ(H) τ(H +θ1), we can
(cid:55)→
outsideoftheconvexhullsofQ ,asdetectedbythesemidef- choose H to be traceless when maximizing the overlap.
2
inite program from Eq. (5). See the text for further details. So tr[H] = 0 and tr[H C C ] = 0 follows. Note that
N N
| (cid:105)(cid:104) |
thiswastheonlypartintheproofwherethepropertyof
C having maximally mixed 2-RDMs was required.
N
C. Numerical results | (cid:105)
We write H and C C in the eigenbasis η of
N N i
| (cid:105)(cid:104) | {| (cid:105)}
H,
Let us first consider states of five and six qubits. We
(cid:88)
report in Table I numerical results for the fraction of H = η η η
i i i
| (cid:105)(cid:104) |
pure states lying outside of conv( ), with the condi-
2 i
Q
tionofpositivedefinitenessδ rangingfrom10−3 to10−7. (cid:88)
C C = c c η η , (23)
Wetested300(cid:48)000(30(cid:48)000)randomfive-qubit(six-qubit) | N(cid:105)(cid:104) N| i j| i(cid:105)(cid:104) j|
ij
states distributed to the Haar measure [32] with our
semidefinite program using the solver MOSEK [33]. As andobtainfollowingconditions,wherethesecondresults
can be seen from the Table, at least 40% of all tested from the normalization of the ring cluster state:
five-qubitstatesand100%ofalltestedsix-qubitstateslie
(cid:88)
outside of conv( 2). Thus, a similar result as in Ref. [4] f1 = ηi =0, (24)
Q
does not hold in the cases of five and six qubits. i
(cid:88)
Concerning conv(Q3), a single five-qubit state and no f2 = pi−1=0, pi =|ci|2 ≥0, (25)
six-qubit state has been detected to lie outside. We as-
i
cribethelatterresulttoaratherweakstatistics,asstates (cid:88)
f = p η =0. (26)
3 i i
in the vicinity of M are easily detected by our semi-
6
| (cid:105) i
definite program (cf. Fig. 2).
Let us now turn to the case of four qubits. Here, none Under these conditions, we have to maximize
toohfuit8ssiidmseialolifgoencnoernravan(ldQfoe2ma).tupTruehreeofnsutfoamuteersr-qichuaablvirteessbyueslettnesmufogsu.gneWsdtsetotahlbsaoet F = (cid:80)(cid:80)iipeieηiηi. (27)
tested special examples of highly entangled four-qubit
If H is nontrivial, it must have both some positive and
states, such as the cluster state, classes of hypergraph
negative eigenvalues. Then at least two of the p must
i
states [34], the Higuchi-Sudbery M state [35] or the
| 4(cid:105) be nonzero. We use the method of Lagrange multipliers
χ -state [13, 36]. While many of theses states can be
| (cid:105) and consider
shown to be outside of , we were not able to prove
2
athnaatlytthiceayllhyaovrewaifithnittheedhiQsetlapnocfetthoeQse2m. Tidheifisniimtepplireosgtrhaamt Λ= (cid:80)(cid:80)iipeieηiηi +λ1f1+λ2f2+λ3f3. (28)
they might be approximated by thermal states of two-
If the maximum is attained for some value p which is
body Hamiltonians. k
notattheborderofthedomain[0,1],wethenmusthave
∂Λ eηk
D. Proof of Observation 2 ∂pk = (cid:80)ieηi +λ2+λ3ηk =0. (29)
Observation 2. The maximum overlap between the
For a given spectrum of H, η =(η ,...,η ), Eq. (29)
1 D
N-qubit ring cluster state C and an N-qubit state τ { }
N has a solution for at most two values, η and η . For
| (cid:105) ∈ + −
is bounded by
2 any η not equal to η or η , the corresponding variable
Q i + −
(cid:2) eH (cid:3) D 1 pi has to lie at the boundary of the domain [0,1], which
τs∈uQp2(cid:104)CN|τ|CN(cid:105)=Hs∈uHp2tr tr[eH]|CN(cid:105)(cid:104)CN| ≤ D− , iηmpalineds tηhatcapni =be0l aifnηdil(cid:48)∈/fo{lηd+d,ηe−ge}n.erTathee, weiigtehnvcaolrurees-
(22) sp+onding−pl ,pl(cid:48). But then, it is easy to see that it is
whereD =2N isthedimensionofthesystem. Moregen- optimal to +max−imize one of those by taking p =(cid:80) pl
erally, for an arbitrary pure state with maximally mixed + l +
reduced k-party states in a d⊗N-system, the overlap with and p− =(cid:80)l(cid:48)pl−(cid:48) and setting the others to zero. Second,
k is bounded by (dN 1)/dN. consideringthesetofηi ∈/ {η+,η−}wherepi =0onecan
Q − further see with Jensen’s inequality that it is optimal to
Proof. We consider first only the ring cluster state, the take all of the η equal, that is (D 2)η = (η +η ).
i i + −
− −
8
So, the whole problem reduces to a problem with four
variables,
p+eη+ +p−eη−
max = max .
pi,ηiF p±,η± eη+ +eη− +(D 2)e−(η++η−)/(D−2) 1 2 3 4
−
(30)
Fromtheconditionsitfollowsthatwecanchooseη >0, FIG. 5: Graph of the linear cluster state η with particles
+
2 and 3 exchanged. This state cannot be approximated by
which implies that η = η p /p < 0. We have to
− + + −
− Hamiltonians with nearest-neighbor interactions only.
prove that the upper bound is is (D 1)/D. Rewriting
p = η+ , we aim to show that −
− η+−η−
Note that the minimal fidelity distance fidelity dis-
(cid:16) (cid:17)
1− η+η−+η− eη+ + η+η−+η−eη− D−1. (31) ttahneceprfersoemncteheocfoinrvreedxuhcuiblllecocnovr(rQelka)ticoanns.beTuhseedmtoinsihmoawl
eη+ +eη− +(D 2)e−(η++η−)/(D−2) ≤ D
− distance D to in terms of the relative entropy is
k k
Q
This can be rewritten to bounded by
(D 1)(η η )(cid:2)eη+ +eη− +(D 2)e−(η++η−)/(D−2)(cid:3) D ((cid:37)) min S((cid:37) σ) logmaxF((cid:37),σ). (37)
+ − k
D−(η eη−− η eη+) 0. − (32) ≥σ∈conv(Q2) || ≥− σ∈Q2
+ −
− − ≥
This follows from a recent result on α-R´enyi relative en-
Regrouping terms leads to
tropies [37],
(cid:18) (cid:19)
η +η
(D−1)(D−2)(η+−η−)exp − D+ 2− S((cid:37)||σ)≥S1/2((cid:37)||σ)=−logF((cid:37),σ). (38)
−
(cid:124) (cid:123)(cid:122) (cid:125)
Therefore, the divergence of the five qubit ring cluster
t1
state from is bounded by D (C ) 0.0317.
[η +(D 1)η ]exp(η ) 2 2 5
+ − − Q | (cid:105) ≥
− −
(cid:124) (cid:123)(cid:122) (cid:125)
t2
+[η +(D 1)η ]exp(η ) 0. (33)
− + + E. Interaction certification
− ≥
(cid:124) (cid:123)(cid:122) (cid:125)
t3
Tocertifythathigherthantwo-bodyinteractionshave
The term t is always positive, while the signs of t and
1 2 been engineered, a four qubit state can be used by fur-
t dependuponthechoiceofη andη . Soconsiderthe
3 + − ther restricting the possible interaction structure of the
following three cases:
system. As an example, consider an ion chain of four
1. Case: (D 1)η < η : Then t 0, but t < 0. qubits in a linear trap, where the only two-body inter-
+ − 2 3
However, −we have t |+t| 0 beca≥use of actions allowed are of the nearest-neighbor type. Then
1 3
≥ the four-qubit linear cluster state η, which is a usual lin-
(η +η )= η + η (D 2)η (34) ear cluster state with a permutation of particles 2 and
+ − + − +
− − | |≥ −
3 (see Fig. 5), cannot be obtained as a ground or ther-
and mal state but only be approximated up to a fidelity of
α=(N 1)/N =15/16=93.75%. This value is within
(D 1)(D 2)(η+ η−) reach of−current technologies.
− − −
(D 1)(D 2)η 2η To see why this state cannot be obtained, note that it
− −
≥ − − | |≥ | |
η +(D 1)η η +(D 1)η . (35) has the generator G = XIZI,IXZZ,ZZXI,IZIX ,
≥| −| − + ≥| − − +| where X,Y,Z,I stand f{or the Pauli matrices and th}e
identity respectively. The stabilizer is then given by
2. Case: (D 1)−1η η (D 1)η : This case
+ − +
− ≤| |≤ −
directly leads to t 0 and t 0.
2 ≥ 3 ≥ S = IIII,IXZZ,IYZY,IZIX,
{
3. Case: η <(D 1)−1η : Thent 0,butt <0. XIZI,XXIZ,XYIY,XZZX,
− + 3 2
| | − ≥
However,wehavet2+t3 0,becauseofeη+ >eη− YIYX,YXXY, YYXZ,YZYI,
≥
and −
ZIXX, ZXYY,ZYYZ,ZZXI . (39)
− }
(D 1)η +η 3η +η
+ − + −
− ≥ The nearest-neighbor marginals of the graph state
2η η +(D 1)η . (36)
+ + −
≥ ≥ −
(cid:88)
η =2−4 s , (40)
i
This finishes the proof.
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