Constraints on correlations in multiqubit systems
CConstraints on correlations in multiqubit systems
Nikolai Wyderka, Felix Huber, Otfried G¨uhne
Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany (Dated: April 10, 2019)The set of correlations between particles in multipartite quantum systems is larger than those inclassical systems. Nevertheless, it is subject to restrictions by the underlying quantum theory. Inorder to better understand the structure of this set, a possible strategy is to divide all correlationsinto two components, depending on the question of whether they involve an odd or an even numberof particles. For pure multi-qubit states we prove that these two components are inextricablyinterwoven and often one type of correlations completely determines the other. As an application,we prove that all pure qubit states with an odd number of qubits are uniquely determined amongall mixed states by the odd component of the correlations. In addition, our approach leads toinvariants under the time evolution with Hamiltonians containing only odd correlations and cansimplify entanglement detection.
PACS numbers: 03.65.Aa, 03.65.Ta, 03.65.Wj
Introduction.—
Correlations in quantum mechanics arestronger than their counterparts in the classical world.This fact is important for many applications in quantuminformation processing. Taking a closer look, however,the former statement sounds like a truism and one re-alizes that many insights in quantum theory stem fromthe fact that quantum mechanical correlations are lim-ited and not arbitrarily strong. For instance, the factthat in Bell experiments quantum correlations do notreach the values admissible by non-signaling theories hasled to insightful discussions about underlying physicalprinciples [1, 2]. To give another example, for three ormore particles monogamy relations bound the entangle-ment between different pairs of particles, and their studyis essential for the progress of entanglement theory [3–6].If one considers multiparticle systems, however, notonly correlations between pairs of particles are relevantbut also those between different sets of particles. Anymulti-qubit state can be expressed in terms of tensorproducts of Pauli matrices via the Bloch decomposition.Different terms act on different sets of qubits, describingthe correlations between just this set of particles. Con-sequently, one may ask whether there are any relationsbetween these components of the total correlations. Forinstance, for three particles, denoted by
A, B, and C ,three different contributions can be distinguished (seealso Fig. 1): First, there are single-body terms, actingon individual parties alone and determining the singleparty density matrices. Second, there are two-body cor-relations acting on the pairs AB , BC , and CA . Finally,there are three-body correlations acting on all three par-ticles ABC . So the question arises: Are these three con-tributions independent of each other or is one of themdetermined by the others?In this paper we present an approach to answer thisand more general questions for multi-qubit systems. Weidentify two components of the correlations, dependingon the question of whether they act on an odd or even
Figure 1. Visualization of the decomposition of a three-particle state ρ into even and odd correlations. A state ρ is expanded in Bloch representation as ρ ∝ + P + P + . . . ,where P j denotes all terms containing j -body correlations.We prove that the even correlations P e are determined by theodd correlations P o for pure states of an odd number of qubits,so the three qubit state is completely determined by P o . number of particles. We prove that the even correlationsand odd correlations obey strong relations, one compo-nent often completely determining the other one. Besidestheir fundamental interest, our results have several prac-tical applications: We prove that all pure qubit stateswith an odd number of qubits are uniquely determinedamong all mixed states by the odd component of the cor-relations. This generalizes a famous result by Woottersand co-workers for three particles [7]. In addition, ourapproach can be used to characterize ground states aris-ing from Hamiltonians having even or odd interactionsonly, and the time evolution under Hamiltonians havingthe odd component only. Finally, we apply our insightsto simplify the task of entanglement detection in certainscenarios. The Bloch representation.—
This representation is ob-tained by expanding an n -qubit quantum state ρ in terms a r X i v : . [ qu a n t - ph ] A p r of tensor products of Pauli operators. So we can write ρ = 12 n (cid:88) α ,...,α n c α ...α n σ α ⊗ . . . ⊗ σ α n , (1)where α i ∈ { , , , } , σ = , and σ , σ , σ are theusual Pauli matrices. The coefficients c α ...α n are givenby the expectation values c α ...α n = Tr( σ α ⊗ . . . ⊗ σ α n ρ ) . In our approach we will sort the terms in the Blochrepresentation according to the number of qubits theyact on. First, we can assign to any basis element σ α ⊗ . . . ⊗ σ α n its weight, wt( σ α ⊗ . . . ⊗ σ α n ) := |{ i | α i (cid:54) = 0 }| , as the number of non-trivial Pauli matrices. Then, wegroup the terms in the decomposition according to theirweight ρ = 12 n ( ⊗ n + n (cid:88) j =1 P j ) , (2)where P j denotes the sum over all contributions of weight j . We call P j also the j -body correlations, being deter-mined by the expectation values taken on groups of j particles. As an example, consider the two-qubit Bellstate | Ψ + (cid:105) = ( | (cid:105) + | (cid:105) ) / √
2, for which the correspond-ing density operator reads | Ψ + (cid:105)(cid:104) Ψ + | = ( ⊗ + σ x ⊗ σ x + σ y ⊗ σ y − σ z ⊗ σ z ) / , so we have P = 0 and P = σ x ⊗ σ x + σ y ⊗ σ y − σ z ⊗ σ z .As our main starting point, we further group the op-erators according to the parity of their weight and define P e := (cid:88) j even , j (cid:54) =0 P j , P o := (cid:88) j odd P j . (3)Note that P = ⊗ n is excluded from P e . Then we canwrite states in the even-odd-decomposition (see Fig. 1) ρ = 12 n ( + P e + P o ) . (4)The central point of our paper is that there are strongrelations between P e and P o , and in many cases one de-termines the other. State inversion.—
Our approach is based on the stateinversion map, which, for any qubit state, can be definedas follows [8, 9]: ˜ ρ := σ ⊗ ny ρ T σ ⊗ ny . (5)Physically, the state inversion is obtained by complexconjugation followed by a spin flip. This can be rep-resented by the anti-unitary inversion operator F :=( iσ y ) ⊗ n C [10]. Here, first the complex conjugation C is performed and then ( iσ y ) ⊗ n is applied to a pure state.We have that F † = ( − n F and for pure states we write | ˜ ψ (cid:105) = F | ψ (cid:105) . It follows that pure states remain pure un-der the state inversion. Note that on single-qubit Paulimatrices we have
F σ i F † = − σ i for i (cid:54) = 0. Thus, theaction of F in Bloch decomposition is to flip the sign in front of each term that has an odd weight. Starting fromEq. (4), we can also write˜ ρ = 12 n ( + P e − P o ) . (6)This allows for an easier representation of the even andodd correlations, namely, + P e = 2 n − ( ρ + ˜ ρ ) , P o = 2 n − ( ρ − ˜ ρ ) . (7)The key observation is that under the state inversion,pure states of an odd number of qubits are mapped toorthogonal states. This fact was known before [11–15],however, we give a proof that allows for generalization toqudit systems, for which the statement is new. Observation 1.
For pure n -qudit states ρ = | ψ (cid:105)(cid:104) ψ | with n odd we have that ρ ˜ ρ = 0 . (8) Proof.
Let ρ = | ψ (cid:105)(cid:104) ψ | be the pure quantum state and de-note the n parties by A , . . . , A n . Using the Schmidt de-composition one can verify that for any bipartition M | ¯ M of the parties one has for the reduced state ρ M := Tr ¯ M ( ρ )the relation ( ρ M ⊗ ¯ M ) ρ = ( ρ ¯ M ⊗ M ) ρ. (9)Let us denote by ρ ij... = ρ { A i ,A j ,... } ⊗ the reduced stateof parties A i , A j , . . . , padded by identities which are act-ing trivially on the other particles. Then, state inversioncan be written as a sum over its reductions [16]˜ ρ = − (cid:88) ≤ i ≤ n ρ i + (cid:88) ≤ i Throughout thissection, we consider pure qubit states of an odd numberof parties, denoted by | ψ odd (cid:105) . We can directly prove ourfirst main result. Observation 2. For pure n -qubit states | ψ odd (cid:105) , writtenin the even-odd decomposition as in Eq. (4), we have that(1) the even and odd components of the correlations com-mute: [ P e , P o ] = 0 ;(2) the odd correlations uniquely determine the even cor-relations via + P e = 12 n − P ; (11) (3) the eigenvalues Λ = ( λ , . . . , λ n ) of P e and P o are Λ( P e ) = (2 n − − , n − − , − , . . . , − , Λ( P o ) = (2 n − , − n − , , . . . , . (12) Proof. We use Eq. (7) to write P o = 2 n − ( | ψ odd (cid:105)(cid:104) ψ odd | − | ˜ ψ odd (cid:105)(cid:104) ˜ ψ odd | ) , + P e = 2 n − ( | ψ odd (cid:105)(cid:104) ψ odd | + | ˜ ψ odd (cid:105)(cid:104) ˜ ψ odd | ) . (13)From Observation 1 it follows that both + P e and P o are diagonal in the same basis and commute. The eigen-values then can be read off. Furthermore, Eq. (11) canbe directly verified in the common eigenbasis.The fact that P e is given by P o for pure states can be re-stated in the language of uniqueness: Pure qubit states ofan odd number of parties are uniquely determined amongpure states (UDP) by the odd correlations. This leads tothe converse question of whether these states are also de-termined by the even correlations P e . The answer to thisquestion is negative, but the set of compatible states israther small. Remark 3. Given the even correlations P e of a pure n -qubit state | ψ odd (cid:105) , the set of admissible odd correlations P o to retrieve a pure state again is a two-parameter fam-ily. The proof is given in Appendix A.Application I: Uniqueness among all states.— So far,we have shown that for an odd number of parties, theodd correlations uniquely determine the state among allpure states. This is already a generalization of previousresults [17, 18], but one can ask the more general ques-tion, whether a state is determined by the correlationsamong all states (UDA), pure or mixed [19]. For thatquestion, some results are known [7, 20], which we cangeneralize now. Corollary 4. Consider a pure qubit state | ψ (cid:105) of n partieswhere n is odd. Then the state is uniquely determinedamong all mixed states by P o . Proof. Recall that in the even-odd decomposition, thestate reads | ψ (cid:105)(cid:104) ψ | = ( + P e + P o ) / n . Suppose there werea mixed state ρ with the same odd correlations. Then wecould write it as a convex sum of pure states, ρ = (cid:88) i p i n ( + P ( i )e + P ( i )o ) , (14)where (cid:80) i p i = 1 and (cid:80) i p i P ( i )o = P o . From Observa-tion 2 we know that P o has two non-vanishing eigenval-ues λ o ± = ± n − , and the same holds for every P ( i )o as they originate from pure states. Because the largesteigenvalue of the sum equals the sum of all the maximaleigenvalues, all P i o must share the same correspondingeigenvector. The same is true for the second and low-est eigenvalue. Thus, P ( i )o = P ( j )o for all i, j follows. Asthe P ( i )e are uniquely determined by the P ( i )o , they alsocoincide and therefore ρ = | ψ (cid:105)(cid:104) ψ | . This result can be seen as a generalization of Ref. [7],where it was shown that almost all three-qubit states aredetermined among all states by P and P . Corollary 4shows that all three-qubit states are determined amongall states by P and P , and it is remarkable that thisgeneralizes to all odd numbers of parties. An immediate consequence of Corollary 4 is that allpure states of an odd number of parties are unique groundstates of odd-body Hamiltonians. More precisely, choos-ing H = − P o = 2 n − ( | ˜ ψ odd (cid:105)(cid:104) ˜ ψ odd | − | ψ odd (cid:105)(cid:104) ψ odd | ) yieldsa specific example of such a Hamiltonian. Results for an even number of qubits.— We now turnto the case of even n , and throughout this section, | ψ even (cid:105) denotes a pure state on an even number of qubits. Al-though in this case | ψ (cid:105) and | ˜ ψ (cid:105) do not need to be per-pendicular, one can gain some insight on the even andodd components of the correlations. We denote the over-lap by | (cid:104) ˜ ψ | ψ (cid:105) | = α with a positive number α such thatTr( ρ ˜ ρ ) = α .For pure states and n = 2, α is just the concurrence[8]. For pure states with n ≥ α is known as the n -concurrence of a state and is known to be an entangle-ment monotone [11]. For our purpose, we need to dis-tinguish three cases: The case where α = 0, the case of0 < α < α = 1.If α = 0, we recover the case of an odd number ofqubits and the same results are valid. Examples forsuch states are the W -state, | W (cid:105) = ( | . . . (cid:105) + . . . + | . . . (cid:105) ) / √ n , and all completely separable states. Inthis case, all the results from the previous sections applyand P o determines P e .If α = 1, | ψ (cid:105) ∝ | ˜ ψ (cid:105) , which means that there are onlyeven correlations present in | ψ (cid:105) and P o = 0. In thiscase, the even correlations are not determined by the oddcorrelations at all. One prominent example for such astate is the n -party Greenberger-Horne-Zeilinger (GHZ)state, | GHZ (cid:105) = ( | . . . (cid:105) + | . . . (cid:105) ) / √ < α < 1, even though the results from the previouschapter do not apply, the spectrum of P e is still ratherfixed, leading to the following: Observation 6. Let | ψ even (cid:105) be a pure qubit state with | (cid:104) ψ even | ˜ ψ even (cid:105) | = α (cid:54) = 0 . Write | ψ even (cid:105) in the even-odd decomposition as in Eq. (4). Then(1) the even correlations P e uniquely determine the oddcorrelations P o up to a sign; and(2) the family of pure states having the same odd corre-lations P o as | ψ even (cid:105) is one-dimensional. The even cor-relations can be parametrized in terms of P o . The proof of this Observation is given in Appendix B.The results of all the previous observations are summa-rized in Table I.A statement similar to Corollary 4 is not true for aneven number of parties with α (cid:54) = 0, as the family of mixedstates pρ + (1 − p )˜ ρ = [ + P e + (2 p − P o ] / n shares thesame even body correlations, unless α = 1, in which case P o = 0 and the state is determined.As a final remark, note that pure states mixed withwhite noise can be reconstructed as well from knowledgeof P o ( P e ) for n odd ( n even), as the noise parameter canbe deduced from the eigenvalues of the operators. Application II: Ground states.— Some of our findingscan be related to the Kramers theorem [21]. Consider a n even and 0 < α < n odd or α = 0 P o given One-dimensional P e is uniquelysolution space for P e determined P e given ± P o is uniquely de- Two-dimensionaltermined up to the sign solution space for P o Table I. Summary of the relations between the even and oddcomponents of pure state correlations as derived in Observa-tion 2, Remark 3 and Observation 6. The detailed relationscan be found in the corresponding proofs. Additionally, if n is even and α = 1, the state exhibits only even correlationsand given P e , only P o = 0 is compatible. Hamiltonian that contains even-body interactions only,such as the Ising model without external field or the t - J -model. A unique ground state of such a Hamiltonianmust have even correlations only. This, however, is notpossible if n is odd, in which case odd correlations mustbe present according to Eq. (11). On the other hand, if n is even, then the ground state must belong to the classof even states, i.e., α = 1. Second, consider Hamiltoni-ans with odd-body interactions only. The ground-stateenergy of such Hamiltonians is a function of P o only.Thus, a unique ground state for n even can only be astate which is determined uniquely by P o , which are ex-actly the states perpendicular to their inverted states,i.e., having α = 0 like the W -state or product states. Application III: Unitary time evolution.— Another ap-plication concerns the orbits of certain states under thetime evolution of Hamiltonians. Here, our approach al-lows one to re-derive and understand previous resultsfrom Ref. [12], where a completely different approach wasused. Consider a Hamiltonian H o consisting of odd-bodyinteractions only. Then, any operator P evolves in timeas P ( t ) = e − iH o t P e iH o t = ∞ (cid:88) m =0 ( − it ) m m ! [ H o , P ] m , (15)where [ H o , P ] m := [ H o , [ H o , P ] m − ] is the m -timesnested commutator with [ H o , P ] = P .Now, recall that we denote by wt( T ) the weight ofa tensor product of Pauli matrices. For these weights,Lemma 1 from Ref. [22], adapted for the case of com-mutators, can be used. It states that for the weight ofthe commutator of two tensor products S and T one hasthat: wt([ S, T ]) ≡ wt( S ) + wt( T ) + 1 (mod 2) , (16)provided that the commutator does not vanish. Thislemma encodes the commutator rules of the Pauli matri-ces. Therefore, by linearity, commuting two odd or twoeven Hermitian operators yields an odd operator, whilecommuting an even and an odd operator yields an evenoperator. Consider, for example, the three-qubit operators S = σ x ⊗ σ y ⊗ σ z + ⊗ ⊗ σ y and T = ⊗ σ x ⊗ σ z . Then, S has odd and T has even weight. Their commutator isgiven by [ S, T ] = − iσ x ⊗ σ z ⊗ + 2 i ⊗ σ x ⊗ σ x , whichhas even weight.Thus, if H and P are odd, all the nested commutatorsin Eq. (15) are odd too, and P ( t ) stays odd for all times t . On the other hand, if H is odd but P is even, then P ( t ) remains even. By Eqs. (4) and (6), ˜ ρ evolves tooas exp( − iHt )˜ ρ exp( iHt ), as the state inversion and thetime evolution commute in this case. Therefore, given aquantum state ρ , the overlap α = Tr( ρ ˜ ρ ) stays constantfor all times. This is also true for mixed states. In thatcase, the result also holds for the n -concurrence C n , givenby the convex roof construction for α , as the value ofTr( ρ ˜ ρ ) stays constant for any decomposition of ρ into asum over pure states [23]. So we have the following. Observation 7. Any quantum state ρ ( t ) , whose timeevolution is governed by an odd-body interacting Hamil-tonian has a constant value of α and C n . This result can be useful as follows: Recent experi-ments enabled the observation of the spreading of quan-tum correlations under interacting Hamiltonians for sys-tems out of thermal equilibrium [24, 25]. Observation 7shows that large classes of Hamiltonians preserve certainproperties of a quantum state and deviations thereof maybe used to characterize the actually realized Hamiltonian.For instance, Refs. [26, 27] proposed methods to engineerHamiltonians with three-qubit interactions only. Experi-mentally, the n -concurrence is not easy to measure; how-ever, bounds can be found with simple methods [28–30].A simple scheme that detects even-body terms in theHamiltonian is the following.Start with any state | ψ (0) (cid:105) with zero n -concurrenceand let it evolve under the Hamiltonian in question. Aftera fixed time t , the state can be decomposed as | ψ ( t ) (cid:105) = √ F | GHZ (cid:105) + √ − F | χ (cid:105) (17)with (cid:104) GHZ | χ (cid:105) = 0. The n -concurrence of the state isgiven by C n ( | ψ ( t ) (cid:105) ) = | (cid:104) ψ ( t ) | ˜ ψ ( t ) (cid:105) | = | F (cid:104) GHZ | GHZ (cid:105) + (1 − F ) (cid:104) χ | σ ⊗ ny | χ ∗ (cid:105) + (cid:112) F (1 − F )( (cid:104) GHZ | σ ⊗ ny | χ ∗ (cid:105) + H.c.) | = | F + (1 − F ) (cid:104) χ | ˜ χ (cid:105) | , (18)as (cid:104) GHZ | σ ⊗ ny | χ ∗ (cid:105) = (cid:104) GHZ | χ ∗ (cid:105) = (cid:104) GHZ | χ (cid:105) ∗ = 0. Theright hand side is always lower bounded by C n ( | ψ ( t ) (cid:105) ) ≥ F − (1 − F ) . (19)If F > Observation 8. If n is even, it is not possible to producea GHZ state from a W -state (or any state with C n = 0 )by unitary or adiabatic time evolution under Hamiltoni-ans with odd interactions only.Application IV: Entanglement detection.— The resultsof this paper yield insight into the structure of pure quan-tum states that is still subject to ongoing research [31].Consider a pure state of n qubits with n being odd.Suppose that the odd correlations P , P , . . . , P n − aregiven. If the state is biseparable, there are ( n − / n − n − 2, etc., up to( n − / n + 1) / P . The second case,namely, two qubits vs. n − n − P , P , . . . , P n − already, where noknowledge of the highest correlations P n is needed. Thisis in contrast to the case of mixed states. Possible generalizations.— While the results obtained in this paper are validfor qubit systems only, some extensions to higher-dimensional systems are possible, as we will discuss now.As stated in the main text, the state inversion can be gen-eralized to systems of internal dimension d , as discussedin [32], ˜ ρ := ⊗ n − n (cid:88) i =1 ρ ( i ) + (cid:88) i We introduced the decomposition ofmultipartite qubit states in terms of even and odd cor-relations. For pure states, we showed that the even andodd correlations are deeply connected, and often one typeof correlations determines the other. This allowed us toprove several applications, ranging from the unique de-termination of a state by its odd correlations to invariantsunder Hamiltonian time evolution and entanglement de-tection.For future work, it would be highly desirable to gener-alize the approach to higher-dimensional systems. Somefacts about state inversion are collected in the previoussection, but developing a general theory seems challeng-ing. Furthermore, it may be very useful if one can extendour theory to a quantitative theory, where the correla-tions within some subset of particles are measured withsome correlation measure and then monogamy relationsbetween the different types of correlations are developed. Note added.— In a previous version of this manuscript,we claimed that “[...] if a state | ψ (cid:105) is uniquely deter-mined among all states by certain sets of correlations(for example, odd-body correlations), then | ψ (cid:105) is theunique ground state of some Hamiltonian having inter-action terms from that set only.” This statement is notcorrect, as a recently found analytical counterexample onsix qubits shows [34].The error in the reasoning occured after Lemma 5:The conclusion “As R ( | ψ (cid:105)(cid:104) ψ | ) is extremal, there existsa linear witness L in the projected space R ( O ) of allHermitian operators O with (cid:104) ψ | L | ψ (cid:105) being minimal” iswrong. An explanation of the fallacy is already given inRef. [35] (cf. Fig. 1): an explicit two-dimensional set isconstructed, in which some extremal points can not beseparated from all other extremal points by any linearwitness. Thus, any such a witness cannot detect a singlestate uniquely. In our context, such a witness cannot beused as a Hamiltonian with non-degenerate ground statespace.We removed the wrong statement and the now unnec-essary Lemma 5, and inserted a correct argument for theclaim that “[...] all pure states of an odd number of par-ties are unique ground states of odd-body Hamiltonians.” Acknowledgments.— We thank Jens Siewert, GavinBrennen and Lorenza Viola for fruitful discussions. Thiswork was supported by the Swiss National Science Foun-dation (Doctoral Mobility Grant No. 165024), the DFG,the ERC (Consolidator Grant No. 683107/TempoQ),and the House of Young Talents Siegen. APPENDIXA: Proof of Remark 3 Remark 3. Given the even correlations P e of a pure n -qubit state | ψ odd (cid:105) , the set of admissible odd correlations P o to retrieve a pure state again is a two-parameter fam-ily.Proof. Let ρ = | ψ odd (cid:105)(cid:104) ψ odd | and ˜ ρ = | ˜ ψ odd (cid:105)(cid:104) ˜ ψ odd | ,and write + P e = 2 n − ( ρ + ˜ ρ ). Thus, the eigenvectorswith eigenvalue 2 n − of + P e are a superposition of | ψ odd (cid:105) and | ˜ ψ odd (cid:105) . Given only P e , one can choose anyof its eigenvectors | η (cid:105) from the two-dimensional subspaceof eigenvalue 2 n − − 1. As | ˜ η (cid:105) is orthogonal to | η (cid:105) , itfollows that + P e = 2 n − ( | η (cid:105)(cid:104) η | + | ˜ η (cid:105)(cid:104) ˜ η | ). Therefore,every choice of an eigenvector gives rise to compatiblecorrelations P ( r )o via P ( r )o = 2 n − ( | η (cid:105)(cid:104) η | − | ˜ η (cid:105)(cid:104) ˜ η | ) , (23)resulting in the total state ρ = | η (cid:105)(cid:104) η | . By fixing one of theeigenstates | η (cid:105) , one can parametrize all valid solutions by P ( r )o ( θ, φ ) = 2 n − [cos θ ( | η (cid:105)(cid:104) η | − | ˜ η (cid:105)(cid:104) ˜ η | )+ sin θ ( e iφ | ˜ η (cid:105)(cid:104) η | + e − iφ | η (cid:105)(cid:104) ˜ η | )] (24)for all real valued θ and φ . B: Proof of Observation 6 Observation 6. Let | ψ even (cid:105) be a pure qubit state with | (cid:104) ψ | ˜ ψ (cid:105) | = α (cid:54) = 0 . Write | ψ even (cid:105) in the even-odd de-composition as in Eq. (4). Then(1) the even correlations P e uniquely determine the oddcorrelations P o up to a sign; and(2) the family of pure states having the same odd corre-lations P o as | ψ even (cid:105) is one-dimensional. The even cor-relations can be parametrized in terms of P o .Proof. Let ρ = | ψ even (cid:105)(cid:104) ψ even | . Before proving thestatements, we investigate the eigenvectors and eigen-values of P e and P o . As + P e = 2 n − ( ρ + ˜ ρ ), it mustbe of rank two if α (cid:54) = 1. Thus, it has two non-vanishing eigenvalues, lying in the span of | ψ (cid:105) and | ˜ ψ (cid:105) . Calculating( + P e ) | ψ (cid:105) = 2 n − ( | ψ (cid:105) + αe iφ | ˜ ψ (cid:105) ) , ( + P e ) | ˜ ψ (cid:105) = 2 n − ( | ˜ ψ (cid:105) + αe − iφ | ψ (cid:105) ) (25)yields the two non-vanishing eigenvalues1 + λ e ± = 2 n − (1 ± α ) (26)and the corresponding orthonormal eigenvectors | e ± (cid:105) = 1 (cid:112) ± α ) ( | ψ (cid:105) ± e iφ | ˜ ψ (cid:105) ) . (27)We can also determine the action of P o on these eigen-vectors, which reveals that it is purely off-diagonal in theeigenbasis of P e , P o | e ± (cid:105) = 2 n − ( ρ − ˜ ρ ) | e ± (cid:105) = 2 n − (cid:112) − α | e ∓ (cid:105) . (28)Thus, the eigenvectors of P o are given by | o ± (cid:105) = 1 √ | e + (cid:105) ± | e − (cid:105) ) (29)and the eigenvalues are given by λ o ± = ± n − (cid:112) − α . (30)We are now in position to prove the claims. Let us provestatement two first:(2) By assumption, P o is known. The eigenvalues de-termine the overlap α by Eq. (30). Knowledge of α fixesthe eigenvalues of any admissible reconstructed P ( r )e . Theadmissible eigenvectors of P ( r )e can be obtained fromEq. (29) to read | e ± (cid:105) = 1 √ | o + (cid:105) ± | o − (cid:105) ). (31)However, the eigenvectors | o ± (cid:105) are only unique up to aphase. Taking into account this extra phase while omit-ting a global phase yields | e ± (cid:105) = 1 √ | o + (cid:105) ± e iϕ | o − (cid:105) ) . (32)This allows us to write all compatible even correlationsas + P ( r )e = (1 + λ e + ) | e + (cid:105)(cid:104) e + | + (1 + λ e − ) | e − (cid:105)(cid:104) e − | = 2 n − ( | o + (cid:105)(cid:104) o + | + αe − iϕ | o + (cid:105)(cid:104) o − | + | o − (cid:105)(cid:104) o − | + αe iϕ | o − (cid:105)(cid:104) o + | ) . (33)This is a one-dimensional space of admissible recon-structed even correlations, parametrized by ϕ .We now show the first statement:(1) Assume that now P e is given. Can we uniquelyreconstruct in the odd correlations P o from knowledge of P e ? Unfortunately, the eigenvectors | e ± (cid:105) are again onlydetermined up to a phase. Therefore, every reconstructedoperator P ( r ) o of the form P ( r )o = λ o + | o + (cid:105)(cid:104) o + | + λ − | o − (cid:105)(cid:104) o − | (34)= λ o + ( e iϕ | e + (cid:105)(cid:104) e − | + e − iϕ | e − (cid:105)(cid:104) e + | ) (35)for all ϕ ∈ R would be a valid operator, such that12 n ( + P e + P r o ) (36)is a pure state again. However, only certain choicesof ϕ recreate a P o which exhibits solely odd correla-tion in Bloch decomposition. This can be seen as fol-lows: As shown in Lemma 9 below, | e ± (cid:105)(cid:104) e ± | can onlyexhibit even correlations. This means that | e ± (cid:105) are eigen-vectors of the inversion operator F introduced above,i.e. F | e ± (cid:105) ∝ | e ± (cid:105) . Recall that for n even, F † = F . Thus, F | e + (cid:105)(cid:104) e − | F † = e i Λ | e + (cid:105)(cid:104) e − | for some Λ. The conditionthat P ( r )o contains only odd correlations can be writtenas P ( r )o + ˜ P ( r )o = P ( r )o + F P ( r )o F † = 0 . (37)Eq. (35) translates this to e iϕ + e − i ( ϕ − Λ) = 0 , (38)which exhibits exactly two solutions for ϕ . Thus, thereare only two possible reconstructions P ( r )o , correspondingto the original P o and its negation, − P o .All that is left is to show the used assumption that theeigenvectors | e ± (cid:105) exhibit only even correlations. Note,that this is a special case of Kramers theorem [21], stat-ing that the eigenstates of a Hamiltonian exhibiting evencorrelations only is either at least two-fold degenerate orexhibits itself only even correlations. Lemma 9. Let P = λ + | p + (cid:105)(cid:104) p + | + λ − | p − (cid:105)(cid:104) p − | be aHermitian operator which exhibits only even correlationsin the Bloch decomposition, (cid:104) p + | p − (cid:105) = 0 and λ − < λ + .Then | p + (cid:105)(cid:104) p + | and | p − (cid:105)(cid:104) p − | also exhibit only even corre-lations.Proof. We regard P as a Hamiltonian with the uniqueground state | p − (cid:105) . As P has even correlations only, F P F † = P . Thus λ − = Tr( P | p − (cid:105)(cid:104) p − | ) = Tr( F P F † | p − (cid:105)(cid:104) p − | )= Tr( P F | p − (cid:105)(cid:104) p − | F † ) , (39)as F † = F if n is even. Thus, also F | p − (cid:105) is a groundstate of P . As by assumption the ground state is unique, F | p − (cid:105) ∝ | p − (cid:105) must hold true and therefore, | p − (cid:105)(cid:104) p − | exhibits only even correlations. This implies that also | p + (cid:105)(cid:104) p + | has even correlations only. [1] M. Paw(cid:32)lowski, T. Paterek, D. Kaszlikowski, V. Scarani,A. Winter, and M. ˙Zukowski, Nature (London) , 1101(2009). [2] M. Navascu´es, Y. Guryanova, M. J. Hoban, and A. Ac´ın,Nature Commun. , 6288 (2015).[3] V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev.A ,220503 (2006).[5] C. Eltschka, and J. Siewert, Phys. Rev. Lett. , 140402(2015).[6] P. Kurzynski, T. Paterek, R. Ramanathan, W.Laskowski, and D. Kaszlikowski, Phys. Rev. Lett. ,180402 (2011).[7] N. Linden, S. Popescu, and W. K. Wootters, Phys. Rev.Lett. , 207901 (2002).[8] W. K. Wootters, Phys. Rev. Lett. , 2245 (1998).[9] S. Hill, and W. K. Wootters, Phys. Rev. Lett. , 5022(1997).[10] V. Buˇzek, M. Hillery, and R. F. Werner, Phys. Rev. A , R2626 (1999).[11] A. Wong, and N. Christensen, Phys. Rev. A , 044301(2001).[12] S. S. Bullock, G. K. Brennen, and D. P. O’Leary, J. Math.Phys. , 062104 (2005).[13] A. Osterloh, and J. Siewert, Phys. Rev. A , 012337(2005).[14] P. Butterley, A. Sudbery, and J. Szulc, Found. Phys. ,83 (2006).[15] S. Designolle, O. Giraud, and J. Martin, Phys. Rev. A , 032322 (2017).[16] W. Hall, Phys. Rev. A , 022311 (2005).[17] L. Di´osi, Phys. Rev. A , 010302 (2004).[18] N. Wyderka, F. Huber, and O. G¨uhne, Phys. Rev. A ,010102 (2017).[19] J. Chen, H. Dawkins, Z. Ji, N. Johnston, D. Kribs, F.Shultz, and B. Zeng, Phys. Rev. A , 012109 (2013).[20] N. S. Jones, and N. Linden, Phys. Rev. A , 012324(2005).[21] H. A. Kramers, Proc. R. Acad. Sci. Amsterdam , 959(1930).[22] F. Huber, O. G¨uhne, and J. Siewert, Phys. Rev. Lett. , 200502 (2017).[23] A. Uhlmann, Phys. Rev. A , 032307 (2000).[24] M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P.Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath,and S. Kuhr, Nature (London) , 484 (2012).[25] P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller,R. Blatt, and C. F. Roos, Nature (London) , 202(2014).[26] J. K. Pachos, and E. Rico, Phys. Rev. A , 053620(2004).[27] H.P. B¨uchler, A. Micheli, and P. Zoller, Nature Physics , 726 (2007).[28] C. Schmid, N. Kiesel, W. Wieczorek, H. Weinfurter,F. Mintert, and A. Buchleitner, Phys. Rev. Lett. ,260505 (2008).[29] S. J. van Enk, Phys. Rev. Lett. , 190503 (2009).[30] C. Zhang, S. Yu, Q. Chen, H. Yuan, and C. H. Oh, Phys.Rev. A , 042325 (2016).[31] D. Goyeneche, G. Ca˜nas, S. Etcheverry, E. S. G´omez, G.B. Xavier, G. Lima, and A. Delgado, Phys. Rev. Lett. , 090401 (2015).[32] C. Eltschka, and J. Siewert, Quantum , 64 (2018).[33] P. Rungta, V. Buˇzek, C. M. Caves, M. Hillery, and G. J.Milburn, Phys. Rev. A , 042315 (2001).[34] S. Karuvade, P.D. Johnson, F. Ticozzi, L. Viola, arXiv preprint arXiv:1902.09481 (2019). [35] J. Chen, Z. Ji, B. Zeng, D.L. Zhou, Phys. Rev. A86