AA family of multipartite entanglement measures
P´eter Vrana
Institute of Mathematics, Budapest University of Technology and Economics,Egry J´ozsef u. 1., 1111 Budapest, Hungary MTA-BME Lend¨ulet Quantum Information Theory Research GroupAugust 26, 2020
Abstract
We construct a family of additive entanglement measures for pure multipartitestates. The family is parametrised by a simplex and interpolates between the R´enyientropies of the one-particle reduced states and the recently-found universal spectralpoints (Christandl, Vrana, and Zuiddam, STOC 2018) that serve as monotones fortensor degeneration.
The two main approaches to quantifying entanglement (and more generally, to informationquantities) are the operational and the axiomatic one. Operational entanglement measuresaim at directly characterising the performance in quantum information processing proto-cols such as entanglement assisted quantum communication or secret key distillation. Incontrast, the axiomatic approach starts with a list of properties (see e.g. [PV07PV07] or [Chr06Chr06,Table 3.2] for examples), desirable from a mathematical point of view or believed to graspsome physical aspect of nonlocality, and then seeks for quantities satisfying them. How-ever, such a list of requirements tends to be subjective and also reflects the intendedrange of applications (e.g. some properties are only relevant in an asymptotic context)and it has been argued that the only requirement for an entanglement measure shouldbe monotonicity under local operations and classical communication (LOCC) [Vid00Vid00]. Inaddition, constructing entanglement measures that satisfy a given set of axioms is oftena challenging task. For example, additivity is a valuable property in asymptotic settings,but there are only a handful of additive entanglement measures known, especially in themultipartite setting [VW02VW02, CW04CW04, YHW08YHW08, YHH + + P . When ap-plying their measurement to the marginals of a k -partite state | ψ (cid:105) , we obtain a sequenceof probability distributions on the space of k -tuples of ordered spectra (i.e. the possibleoutcomes of the measurements). This sequence satisfies a large deviation principle with arate function whose value at λ ∈ P k will be denoted by I ψ ( λ ) (this is a special case of the µ -capacity from [FW20FW20] and the rate function from [BCV20BCV20], when the compact group is1 a r X i v : . [ m a t h - ph ] A ug product of unitary groups). It will also be convenient to introduce a notation for theweighted average of a collection of entropies. For a probability distribution θ on the set[ k ] = { , , . . . , k } we set H θ ( λ ) = k (cid:88) j =1 θ ( j ) H ( λ j ) . (1)We consider the quantity E α,θ ( | ψ (cid:105) ) = sup λ ∈P k (cid:2) (1 − α ) H θ ( λ ) − αI ψ ( λ ) (cid:3) (2)where α ∈ (0 , α = 0 we extend by continuity: in this limit the value of the rate func-tion is not relevant anymore, but the supremum is restricted to the set (cid:110) λ ∈ P k (cid:12)(cid:12)(cid:12) I ψ ( λ ) (cid:54) = 0 (cid:111) ,which is the entanglement polytope [WDGC13WDGC13] of the state | ψ (cid:105) . Therefore the α → E α,θ ( | ψ (cid:105) ⊗ | ϕ (cid:105) ) = E α,θ ( | ψ (cid:105) ) + E α,θ ( | ϕ (cid:105) ) and that if there is an LOCC channel mapping | ψ (cid:105) to | ϕ (cid:105) then E α,θ ( | ψ (cid:105) ) ≥ E α,θ ( | ϕ (cid:105) ).In fact, we prove the stronger statement that F α,θ ( | ψ (cid:105) ) = 2 E α,θ ( | ψ (cid:105) ) is an element ofthe asymptotic spectrum of LOCC transformations as introduced in [JV19JV19]. As the proofrelies on the theory developed there, we provide here a brief overview and explain theconnection to [CVZ18CVZ18] and the asymptotic restriction of tensors.Asymptotic tensor restriction can be viewed as a weak notion of asymptotic entangle-ment transformation of pure states [CDS08CDS08], where we require the target to be reachedexactly with an arbitrarily low but nonzero probability of success, as proposed in [BPR + + ψ and ϕ are tensors of order k (which we may interpret as statevectors of k -partite states) and a ≥ b means that a restricts to b (i.e. can be convertedvia SLOCC [DVC00DVC00]), thensup (cid:110) R ∈ R ≥ (cid:12)(cid:12)(cid:12) ψ ⊗ n ≥ ϕ ⊗ Rn + o ( n ) for large n (cid:111) = inf f ∈ ∆( T k ) log f ( ψ )log f ( ϕ ) , (3)where ∆( T k ) is the asymptotic spectrum of tensors, defined as the set of real valuedfunctions on (equivalence classes of) tensors that are(S1) monotone under restriction(S2) multiplicative under the tensor product(S3) additive under the direct sum(S4) normalised to r on the unit tensor of rank r (unnormalised r -level GHZ state | . . . (cid:105) + | . . . (cid:105) + · · · + | rr . . . r (cid:105) , denoted (cid:104) r (cid:105) ).Tensor restrictions or stochastic entanglement transformations provide no control onthe probabilities and do not offer any meaningful notion of approximate transformations.As a refinement of the problem, one may investigate the trade-off between the transfor-mation rate and the exponential rate at which the error approaches zero or one on eitherside of the optimal rate. This has been answered in [HKM + +
02] for bipartite entanglementconcentration, allowing either an error (i.e. nonzero distance from the target state) or a2robability of failure (i.e. nonzero chance of not reaching the target state). They find that,as with many information processing tasks, these trade-off relations can be characterisedin terms of R´enyi information quantities, in this case R´enyi entanglement entropies.The main result of [JV19JV19] provides a characterisation like (33) of the error exponentsfor probabilistic entanglement transformations of pure multipartite states in the converseregime. This means that for each large n we require an LOCC protocol that, when runon ψ ⊗ n as the input state, declares success with probability at least 2 − rn , in which casethe resulting state is exactly ϕ ⊗ Rn + o ( n ) . For a given r > R isinf f ∈ ∆( S k ) rα ( f ) + log f ( | ψ (cid:105) )log f ( | ϕ (cid:105) ) , (4)where this time ∆( S k ) is the set of functions f on pure unnormalised states that in additionto (S2)(S2), (S3)(S3) and (S4)(S4) satisfy for some α = α ( f ) ∈ [0 , f ( √ p | ψ (cid:105) ) = p α f ( | ψ (cid:105) )(S1’) monotonicity under LOCC, or equivalently [JV19JV19, Theorem 3.1] f ( | ψ (cid:105) ) ≥ (cid:16) f (Π j | ψ (cid:105) ) /α + f (( I − Π j ) | ψ (cid:105) ) /α (cid:17) α (5)(in a limit sense for α = 0) for every local orthogonal projecion Π j at party j .Note that the exponent can be recovered as α = α ( f ) = log f ( √ | . . . (cid:105) ). We call ∆( S k )the asymptotic spectrum of LOCC transformations.To provide some intuition on the relevant functions, it is helpful to study the bipartitecase, where an explicit description of ∆( S ) is available [JV19JV19, Section 4.]. In this case forevery α ∈ [0 ,
1] there is precisely one monotone f α ∈ ∆( S ) with α ( f α ) = α , and its valueon (cid:80) i √ p i | ii (cid:105) is (cid:80) i p αi . Another way to write this is f α ( | ψ (cid:105) ) = 2 (1 − α ) H α (Tr | ψ (cid:105)(cid:104) ψ | ) . (6)For this reason we regard the monotones f ∈ ∆( S k ) as generalisations of the R´enyi entan-glement entropies and α ( f ) as the generalisation of the order of the R´enyi entropy.An appealing feature of results like this is that they provide a bridge between theaforementioned operational and axiomatic approaches: they single out a list of axioms,and not only show that there exist monotones satisfying them simultaneously, but that infact there are sufficiently many of them to characterise operational quantities (in this casetransformation rates). In the context of ordered commutative monoids (as a mathematicalmodel for general resource theories), a similar result is presented in [Fri17Fri17]. It should beemphasised that the proofs of both (33) and (44) are non-constructive in the sense that theydo not provide any explicit monotones in ∆( T k ) (respectively ∆( S k )). It is a nontrivialtask to construct explicit monotones satisfying the respective axioms. A simple way toobtain new monotones from old ones is to compose a (possibly partial) flattening (i.e.grouping parties together) with an element of ∆( S k (cid:48) ) for some k (cid:48) < k . For k (cid:48) = 2 thisconstruction gives the R´enyi entropies (exponentiated as in (66)) of the reduced densitymatrices. We will refer to these as trivial points of ∆( S k ).∆( T k ) can be identified with a subset of ∆( S k ), namely α − (0). The first examples ofnontrivial points in the asymptotic spectrum of tensors have been found in [CVZ18CVZ18]. In thepresent work we extend that construction and obtain nontrivial points in the asymptoticspectrum of LOCC transformations with α (cid:54) = 0. The new family of monotones interpolatesbetween the quantum functionals of [CVZ18CVZ18] and the (exponentiated) R´enyi entropies ofthe single-party marginals. We prove the following properties of F α,θ ( | ψ (cid:105) ) = 2 E α,θ ( | ψ (cid:105) ) .3 heorem 1.1. Let α ∈ [0 , and θ ∈ P ([ k ]) . For every p ≥ , r ∈ N , | ψ (cid:105) ∈ H ⊗ · · · ⊗ H k and | ϕ (cid:105) ∈ K ⊗ · · · ⊗ K k the followings hold:(i) F α,θ ( √ p | ψ (cid:105) ) = p α F α,θ ( | ψ (cid:105) ) ,(ii) F α,θ ( (cid:104) r (cid:105) ) = r ,(iii) F α,θ ( | ψ (cid:105) ⊗ | ϕ (cid:105) ) = F α,θ ( | ψ (cid:105) ) F α,θ ( | ϕ (cid:105) ) ,(iv) F α,θ ( | ψ (cid:105) ⊕ | ϕ (cid:105) ) = F α,θ ( | ψ (cid:105) ) + F α,θ ( | ϕ (cid:105) ) ,(v) if there exists a trace-nonincreasing LOCC channel Λ such that Λ( | ψ (cid:105)(cid:104) ψ | ) = | ϕ (cid:105)(cid:104) ϕ | then F α,θ ( | ψ (cid:105) ) ≥ F α,θ ( | ϕ (cid:105) ) . That is, F α,θ is a point in the asymptotic LOCC spectrum with α ( F α,θ ) = α .Before turning to the proof, let us attempt to argue why the existence of such aninterpolating family is at least plausible. As mentioned above, for α = 0 these functionalsreduce to the logarithmic quantum functionals defined in [CVZ18CVZ18] as the supremum H θ ( λ )over the entanglement polytope. Remarkably, the R´enyi entropies also admit a variationalexpression, namely [Ari96Ari96, MA99MA99, Sha11Sha11](1 − α ) H α ( P ) = sup Q ∈P ( X ) [(1 − α ) H ( Q ) − α D ( Q (cid:107) P )] , (7)where D ( Q (cid:107) P ) is the relative entropy between the probability distributions Q and P .If we suspect that a common generalisation might exist, then the most straighforwardidea is that inside the supremum the first term should become (1 − α ) H θ , whereas therelative entropy should be replaced with a function that is infinite outside the entanglementpolytope and vanishes at the marginal spectra. These properties are satisfied by the ratefunction I ψ ( λ ).In addition, when α is nonzero and θ is concentrated on site j , we claim that (22) reducesto the exponentiated R´enyi entropy of the j th marginal. To see this, note that in this casethe first term in the supremum only depends on the j th component of λ , therefore theoptimisation over the remaining components can be carried out separately on the secondterm. According to the contraction principle, the infimum of the rate function over all butthe j th argument is the rate function for the ordinary Keyl–Werner estimation at site j ,i.e. the relative entropy distance from the ordered spectrum of the j th marginal.For α → θ necessarily collapse to a single one, thenorm squared [JV19JV19, Proposition 3.6]. However, as also suggested by (66), the interestinglimit at this point is (for normalised | ψ (cid:105) )lim α → − α E α,θ ( | ψ (cid:105) ) = k (cid:88) j =1 θ ( j ) H ( | ψ (cid:105)(cid:104) ψ | j ) , (8)where | ψ (cid:105)(cid:104) ψ | j is the j th marginal of the state. This is because in this limit the coefficientof the rate function goes to −∞ , penalising every point other than its unique zero, thecollection of the marginal spectra. For this reason − α E α,θ ( | ψ (cid:105) ) may be regarded as theR´enyi generalisations of the limit (88).The paper is structured as follows. In Section 22 we introduce our notations and collectsome known results about the representation theory of symmetric and unitary groups. Ourmain focus here is on the asymptotic dimension of irreducible representations and vanishingconditions for multiplicities. These can be expressed in terms of limits of rescaled inte-ger partitions (normalised decreasing nonnegative real sequences) and moment polytopes.Section 33 studies the rate function of a multiparty version of the spectrum estimationscheme of Keyl and Werner [KW01KW01]. In Section 44 we prove our main result, Theorem 1.11.1.4 Notations and preliminaries
Most of what follows is standard material in the representation theory of classical groupsand can be found in many textbooks, see e.g. [FH91FH91]. For an introduction aimed at quan-tum information theorists and emphasising the asymptotic aspects we refer to Hayashi’sbook [Hay17Hay17].We will denote by P n the set of partitions of the integer n , i.e. nonincreasing non-negative integer sequences summing to n . Partitions will serve as labels of irreduciblerepresentations of the unitary and the symmetric groups. In particular, if H is a finitedimensional Hilbert space then H ⊗ n has the Schur–Weyl decomposition H ⊗ n (cid:39) (cid:77) α ∈P n S α ( H ) ⊗ [ α ] , (9)which is an isomorphism of U ( H ) × S n -representations. Here S α ( H ) is the irreduciblerepresentation of U ( H ) with highest weight α if α has at most dim H parts and the zerorepresentation otherwise, while [ α ] is an irreducible representation of S n . The orthogonalprojection onto the isotypic component corresponding to α will be denoted by P H α or P α if the Hilbert space is clear from the context. The number of partitions of n into at mostdim H parts is upper bounded by ( n + 1) dim H . We need the following dimension estimatesdim S α ( H ) ≤ ( | α | + 1) d ( d − / (10)dim[ α ] ≥ | α | + d ) ( d +2)( d − / | α | H ( | α | α ) (11)where d = dim H .We denote by P the set of those functions N → R ≥ that are nonincreasing, finitelysupported and that sum to 1. In particular, when α ∈ P n , then we can consider itsnormalisation n α ∈ P , where the multiplication is understood entrywise. P is equippedwith the metric induced by the (cid:96) norm, the open ball of radius (cid:15) around α ∈ P is B (cid:15) ( α ).An element α ∈ P can be viewed as a probability distribution and we can consider itsShannon entropy H ( α ). When ρ is a quantum state on a finite dimensional Hilbert space,then its spectrum (with multiplicities and ordered nonincreasingly) is an element of P .Let α, β, γ ∈ P n . The Kronecker coefficient is defined as g αβγ = dim([ α ] ⊗ [ β ] ⊗ [ γ ]) S n .We define Kron to be the closure in P of (cid:26)(cid:18) n α, n β, n γ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | α | = | β | = | γ | = n, g αβγ (cid:54) = 0 (cid:27) . (12)Kron is also the set of triples that arise as marginal spectra of tripartite pure states.( α, β, γ ) ∈ Kron implies H ( α ) ≤ H ( β ) + H ( γ ) [Kly04Kly04, CM06CM06, Chr06Chr06].The Kronecker coefficients also appear in decompositions for the unitary groups. Let H and K be Hilbert spaces and α ∈ P n . Then there is a map U ( H ) × U ( K ) → U ( H ⊗ K )(given by the tensor product of the unitary operators) and we have the isomorphism S α ( H ⊗ K ) (cid:39) (cid:77) β,γ ∈P n C g αβγ ⊗ S β ( H ) ⊗ S γ ( K ) (13)as U ( H ) × U ( K )-representations. In terms of the projections P H⊗K α , P H β ⊗ id K ⊗ n andid H ⊗ n ⊗ P K γ on ( H ⊗ K ) ⊗ n we have the vanishing condition (here and below assuming thatthe individual projections do not vanish) P H⊗K α ( P H β ⊗ P K γ ) = 0 ⇐⇒ g αβγ = 0 . (14)5ote that the projections P H⊗K α and P H β ⊗ P K γ commute.Let 0 ≤ m ≤ n and α ∈ P n , β ∈ P m , γ ∈ P n − m . The Littlewood–Richardson coefficientis defined as c αβγ = dim(([ β ] ⊗ [ γ ]) ⊗ Res S n S m × S n − m [ α ]) S m × S n − m . We define LR • to be theclosure in [0 , × P of (cid:26)(cid:18) mn , n α, m β, n − m γ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | α | = n, | β | = m > , | γ | = n − m > , c αβγ (cid:54) = 0 (cid:27) (15)and LR q asLR q = (cid:8) ( α, β, γ ) (cid:12)(cid:12) ( q, α, β, γ ) ∈ LR • (cid:9) (16)for q ∈ [0 , q is also the set of triples ( α, β, γ ) such that there exist quantum states ρ, σ on a finite dimensional Hilbert space such that the spectra of qρ + (1 − q ) σ, ρ, σ are α, β, γ , respectively, which is a form of Horn’s problem [Hor62Hor62, Lid82Lid82, Kly98Kly98, Chr06Chr06].( α, β, γ ) ∈ LR q implies qH ( β ) + (1 − q ) H ( γ ) ≤ H ( α ) ≤ qH ( β ) + (1 − q ) H ( γ ) + h ( q )[CVZ18CVZ18].The Littlewood–Richardson coefficients also appear in decompositions for the unitarygroups in two ways. Let H and K be Hilbert spaces and α ∈ P n . Then U ( H ) × U ( K ) ≤ U ( H ⊕ K ) (as block diagonal operators) and we have the isomorphism S α ( H ⊕ K ) (cid:39) (cid:77) β,γ C c αβγ ⊗ S β ( H ) ⊗ S γ ( K ) (17)as U ( H ) × U ( K )-representations, where the sum is over partitions with | β | + | γ | = n = | α | .Consider the representation on ( H ⊕ K ) ⊗ n (cid:39) (cid:76) nm =0 C ( nm ) ⊗ H ⊗ m ⊗ K ⊗ n − m . In the directsummands the first factor corresponds to the possible m -element subsets of the n factorswhere the space H is chosen. In general, to specify a vector or an operator on this spacein terms of this isomorphism, one needs to choose a bijection between the basis elementsof C ( nm ) and the subsets, and in addition an ordering of the m and n − m factors amongthemselves. However, we will only encounter instances where the vectors or operatorsare S m × S n − m -invariant and we sum over all the m -element subsets, which eliminatesthe need for these choices. In particular, we consider the commuting projections P H⊕K α ,id C ( nm ) ⊗ P H β ⊗ id K ⊗ n − m and id C ( nm ) ⊗ id H ⊗ m ⊗ P K γ , in terms of which we have the vanishingcondition P H⊕K α (id C ( nm ) ⊗ P H β ⊗ P K γ ) = 0 ⇐⇒ c αβγ = 0 . (18)The second decomposition involving the Littlewood–Richardson coefficients is that ofthe tensor product of representations of a unitary group U ( H ). For β ∈ P m and γ ∈ P n − m we have the isomorphism S β ( H ) ⊗ S γ ( H ) (cid:39) (cid:77) α ∈P n S α ( H ) (19)as U ( H )-representations. Choosing a factorisation H ⊗ n (cid:39) H ⊗ m ⊗H ⊗ n − m we may considerthe projections P H α , P H β ⊗ id H ⊗ n − m and id H ⊗ m ⊗ P H γ . These projections commute andsatisfy the vanishing condition P H α ( P H β ⊗ P H γ ) = 0 ⇐⇒ c αβγ = 0 . (20)Next we consider Hilbert spaces of multipartite systems and introduce a compactnotation in order to simplify the formulas later. k denotes the number of subsystems6tensor factors), which can be considered fixed throughout. We use λ, µ, ν to denote k -tuples of partitions of some natural number n and write λ = ( λ , . . . , λ k ) when referringto the individual partitions (we will not need to label the parts of the partitions). If H = H ⊗ · · · ⊗ H k , then we can decompose each factor in H ⊗ n using the Schur–Weyldecomposition( H ⊗ · · · ⊗ H k ) ⊗ n (cid:39) (cid:77) λ ∈P kn k (cid:79) j =1 S λ j ( H j ) ⊗ [ λ j ] . (21)This is an isomorphism of U ( H ) × · · · × U ( H k ) × S kn -representations. A tuple of partitions λ can be identified with a dominant weight for the compact Lie group U ( H ) × · · · × U ( H k )(there are other dominant weights, but these are the ones that we shall encounter). P H λ (also P λ if the Hilbert space is understood) will denote the orthogonal projection onto thedirect summand corresponding to λ . It satisfies P H λ = P H λ ⊗ · · · ⊗ P H k λ k . (22)When λ ∈ P kn , we will also write n λ ∈ P k for ( n λ , . . . , n λ k ), meaning that everyelement of every partition in the k -tuple is rescaled. On P k we consider the distanceinduced by the maximum of the 1-norms. For λ = ( λ , . . . , λ k ) ∈ P k and a probabilitydistribution θ ∈ P ([ k ]) we will frequently use the weighted average of their entropies H θ ( λ ) = k (cid:88) j =1 θ ( j ) H ( λ j ) . (23)When λ, µ, ν ∈ P k , we will write ( λ, µ, ν ) ∈ Kron k to mean ∀ j ∈ [ k ] : ( λ j , µ j , ν j ) ∈ Kron. By taking convex combinations we can see that ( λ, µ, ν ) ∈ Kron k implies H θ ( λ ) ≤ H θ ( µ ) + H θ ( ν ) . (24)Similarly, when λ, µ, ν ∈ P kn we will use g λµν (cid:54) = 0 as an abbreviation for ∀ j ∈ [ k ] : g λ j µ j ν j (cid:54) =0 (one could define g λµν = (cid:81) kj =1 g λ j µ j ν j but this value will not play a role, the only thingthat matters is if it is zero or not).When λ, µ, ν ∈ P k , we will write ( λ, µ, ν ) ∈ LR kq to mean ∀ j ∈ [ k ] : ( λ j , µ j , ν j ) ∈ LR q .( λ, µ, ν ) ∈ LR kq implies qH θ ( µ ) + (1 − q ) H θ ( ν ) ≤ H θ ( λ ) ≤ qH θ ( µ ) + (1 − q ) H θ ( ν ) + h ( q ) . (25)Similarly, when λ ∈ P kn , µ ∈ P km , ν ∈ P kn − m we will use c λµν (cid:54) = 0 as an abbreviation for ∀ j ∈ [ k ] : c λ j µ j ν j (cid:54) = 0 (again, one may think c λµν = (cid:81) kj =1 c λ j µ j ν j but the value itself will not beused). In [ARS88ARS88] Alicki, Rudicki and Sadowski and in [KW01KW01] Keyl and Werner proposed anestimator for the spectrum of a density matrix, based on measuring the Schur–Weyl projec-tors P α on n identical copies of a quantum state. The measurement outcomes are labelledwith the partitions α ∈ P n and n α ∈ P is the estimate for the ordered spectrum. They7howed that as n → ∞ , the distributions converge weakly to the Dirac measure on thespectrum and satisfy a large deviation principle with rate function given by the relativeentropy. For our purposes the following formulation will be convenient: if r denotes theordered spectrum of ρ and α ∈ P thenlim (cid:15) → lim n →∞ − n log (cid:88) α ∈P n n α ∈ B (cid:15) ( α ) Tr( P α ρ ⊗ n ) = D ( α (cid:107) r ) . (26)Suppose that | ϕ (cid:105)(cid:104) ϕ | ∈ S ( H ⊗ H ) is a purification of ρ ∈ S ( H ). If we perform themeasurement independently on H ⊗ n and on H ⊗ n then it is easy to see that the outcomeswill be perfectly correlated and of course both sides see the same exponential behaviouras in (2626).In this section we will study the rate function for the same estimator in a multipartitesetting. Let ϕ ∈ H = H ⊗· · ·⊗H k be a unit vector and suppose that the k parties performa measurement with the local Schur–Weyl projectors P H j λ j on ϕ ⊗ n . Note that this is equiv-alent to measuring P H λ (see (2222)). The rescaled outcome n λ ∈ P k serves as the estimateof the k -tuple of marginal spectra. As in the bipartite case, each marginal estimate lookslike (2626) but this time the correlation between the estimates is more complicated. Weregard the resulting rate function as a multipartite generalisation of the relative entropyand it will play a central role in our construction of the entanglement monotones. Recallthat the classical relative entropy satisfies D ( Q (cid:107) P ⊗ P ) ≥ D ( Q (cid:107) P ) + D ( Q (cid:107) P ) (27)where Q and Q are the marginals of Q , and D ( qQ ⊕ (1 − q ) Q (cid:107) P ⊕ P ) = q D ( Q (cid:107) P ) + (1 − q ) D ( Q (cid:107) P ) − h ( q ) , (28)where Q ∈ P ( X ), Q ∈ P ( X ), ⊕ is the direct sum resulting in a distribution on X ∪ X and h ( p ) = − p log p − (1 − p ) log(1 − p ). The results in this section can be viewed asanalogous properties satisfied by the rate function in the simultaneous spectrum estimationproblem and may be of independent interest.We make the following definition: Definition 3.1.
Let λ ∈ P k and ϕ ∈ H = H ⊗ · · · ⊗ H k . The rate function is defined as I ϕ ( λ ) = lim (cid:15) → lim n →∞ − n log (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) (29)To see that this quantity is well defined (possibly ∞ ), we only need to show that thelimit as n → ∞ exists, which is then clearly monotone in (cid:15) . We postpone the proof ofthis fact (Proposition A.1A.1, see also [BCV20BCV20]) and of the following technical lemmas toSection AA. Lemma 3.2.
Let ϕ ∈ H , ( n l ) l ∈ N a sequence of natural numbers such that lim l →∞ n l = ∞ and ( λ ( n l ) ) l ∈ N a sequence such that λ ( n l ) ∈ P kn l and lim l →∞ n l λ ( n l ) = λ . Then lim inf l →∞ − n l log (cid:13)(cid:13) P λ ( nl ) ϕ ⊗ n l (cid:13)(cid:13) ≥ I ϕ ( λ ) . (30)8 emma 3.3. For every ϕ ∈ H and λ ∈ P k there exists a sequence ( λ ( n ) ) n ∈ N such that λ ( n ) ∈ P kn , lim n →∞ n λ ( n ) = λ (31) and lim n →∞ − n log (cid:13)(cid:13) P λ ( n ) ϕ ⊗ n (cid:13)(cid:13) = I ϕ ( λ ) . (32)We mention that the rate function can be expressed via a single-letter formula as aspecial case of [FW20FW20, BCV20BCV20]: I ϕ ( λ ) = inf U sup A,N (cid:104) λ, α (cid:105) − log Tr N ∗ A/ U ∗ | ϕ (cid:105)(cid:104) ϕ | U A/ N, (33)where the infimum is over tensor product unitaries U , the supremum is over A ∈ R dim H ⊕· · · ⊕ R dim H k , identified with diagonal matrices as A ⊗ I ⊗ · · · ⊗ I + I ⊗ A ⊗ I ⊗ · · · ⊗ I + · · · and over tensor products N of upper triangular unipotent matrices. We will not make useof this expression in the proofs below, but we expect it to be useful for computations.First we derive some immediate properties of the rate function that will be used inSection 44. Proposition 3.4 (Basic properties of the rate function) . Let ϕ ∈ H = H ⊗ · · · ⊗ H k .(i) I √ pϕ ( λ ) = I ϕ ( λ ) − log p for every λ ∈ P k and p > .(ii) If ψ = ( A ⊗ · · · ⊗ A k ) ϕ where ∀ j : A j ∈ B ( H j ) and A ∗ j A j ≤ I then I ϕ ( λ ) ≤ I ψ ( λ ) for every λ ∈ P k (iii) If (cid:107) ϕ (cid:107) = 1 and λ is the collection of its marginal spectra then I ϕ ( λ ) = 0 (see also[BCV20BCV20, Corollary 3.25.]).Proof. (i)(i): This follows from Definition 3.13.1 and the equality − n log (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ( √ pϕ ) ⊗ n (cid:13)(cid:13) = − n log (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) − log p. (34)(ii)(ii): ( A ⊗ · · · ⊗ A k ) ⊗ n commutes with P λ , therefore (cid:13)(cid:13) P λ (( A ⊗ · · · ⊗ A k ) ϕ ) ⊗ n (cid:13)(cid:13) = (cid:13)(cid:13) ( A ⊗ · · · ⊗ A k ) ⊗ n P λ ϕ ⊗ n (cid:13)(cid:13) = (cid:104) P λ ϕ ⊗ n , ( A ∗ A ⊗ · · · ⊗ A ∗ k A k ) ⊗ n P λ ϕ ⊗ n (cid:105)≤ (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) . (35)From this the statement follows using Definition 3.13.1.(iii)(iii): Let ρ j be the j th marginal of | ϕ (cid:105)(cid:104) ϕ | . (2626) implies that for every (cid:15) > j wehave lim n →∞ (cid:88) α ∈P n ∩ nB (cid:15) ( λ j ) Tr P α ρ ⊗ nj = 1 , (36)9hich implieslim n →∞ (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) = 1 . (37)Therefore the limit in Definition 3.13.1 is 0 (even without dividing by n ).The following inequality is analogous to (2727) and will be used in the proof of submul-tiplicativity in Proposition 4.24.2. Proposition 3.5 (Rate function and tensor product) . Let ψ ∈ H = H ⊗ · · · ⊗ H k and ϕ ∈ K = K ⊗ · · · ⊗ K k . For every λ ∈ P k the inequality I ψ ⊗ ϕ ( λ ) ≥ inf µ,ν ∈P k ( λ,µ,ν ) ∈ Kron k I ψ ( µ ) + I ϕ ( ν ) (38) holds.Proof. Let d = (cid:80) kj =1 (dim H j + dim K j ) and λ ∈ P kn . Using that the sum of Schur–Weylprojections is the identity, the estimate on the number of partitions with bounded lengthand the vanishing condition (1414) we have the inequality (cid:13)(cid:13)(cid:13) P H⊗K λ ( ψ ⊗ ϕ ) ⊗ n (cid:13)(cid:13)(cid:13) = (cid:88) µ,ν ∈P kn g λµν (cid:54) =0 (cid:13)(cid:13)(cid:13) P H⊗K λ ( P H µ ψ ⊗ n ⊗ P K ν ϕ ⊗ n ) (cid:13)(cid:13)(cid:13) ≤ (cid:88) µ,ν ∈P kn g λµν (cid:54) =0 (cid:13)(cid:13) P H µ ψ ⊗ n ⊗ P K ν ϕ ⊗ n (cid:13)(cid:13) ≤ ( n + 1) d max µ,ν ∈P kn g λµν (cid:54) =0 (cid:13)(cid:13) P H µ ψ ⊗ n (cid:13)(cid:13) (cid:13)(cid:13) P K ν ϕ ⊗ n (cid:13)(cid:13) (39)By Lemma 3.33.3 we can choose a sequence λ (1) , λ (2) , . . . such that λ ( n ) ∈ P kn , lim n →∞ n λ ( n ) = λ and lim n →∞ − n log (cid:13)(cid:13)(cid:13) P H⊗K λ ( ψ ⊗ ϕ ) ⊗ n (cid:13)(cid:13)(cid:13) = I ψ ⊗ ϕ ( λ ) . (40)For every n choose µ ( n ) , ν ( n ) ∈ P kn such that g λ ( n ) µ ( n ) ν ( n ) (cid:54) = 0 and attaining the maximumin (3939). Choose a subsequence n l such that both n l µ ( n l ) and n l ν ( n l ) converge (possiblesince finite dimensional slices of P are compact), and let their limits be µ, ν . Kron k isclosed, therefore ( λ, µ, ν ) ∈ Kron k . I ψ ⊗ ϕ ( λ ) = lim n →∞ − n log (cid:13)(cid:13)(cid:13) P H⊗K λ ( n ) ( ψ ⊗ ϕ ) ⊗ n (cid:13)(cid:13)(cid:13) = lim l →∞ − n l log (cid:13)(cid:13)(cid:13) P H⊗K λ ( nl ) ( ψ ⊗ ϕ ) ⊗ n l (cid:13)(cid:13)(cid:13) ≥ lim inf l →∞ − n l log (cid:13)(cid:13)(cid:13) P H µ ( nl ) ψ ⊗ n l (cid:13)(cid:13)(cid:13) − n l log (cid:13)(cid:13)(cid:13) P K ν ( nl ) ϕ ⊗ n l (cid:13)(cid:13)(cid:13) ≥ I ψ ( µ ) + I ϕ ( ν ) . (41)The first inequality follows from (3939) and the second one from Lemma 3.23.2.10he following two inequalities should be compared with (2828). In Section 44 the firstone (Proposition 3.63.6) will be used in the proof of additivity of our monotones, while thesecond one (Proposition 3.73.7) is needed in the proof of monotonicity. Proposition 3.6 (Rate function and direct sum) . Let ψ ∈ H = H ⊗ · · · ⊗ H k and ϕ ∈ K = K ⊗ · · · ⊗ K k . For every λ ∈ P k the inequality I ψ ⊕ ϕ ( λ ) ≥ inf q ∈ [0 , inf µ,ν ∈P k ( λ,µ,ν ) ∈ LR kq qI ψ ( µ ) + (1 − q ) I ϕ ( ν ) − h ( q ) (42) holds.Proof. Let d = 1+ (cid:80) kj =1 (dim H j +dim K j ) and λ ∈ P kn . Using that the sum of Schur–Weylprojections is the identity, the estimate on the number of partitions with bounded lengthand the vanishing condition (1818) we have the inequality (cid:13)(cid:13)(cid:13) P H⊕K λ ( ψ ⊕ ϕ ) ⊗ n (cid:13)(cid:13)(cid:13) = n (cid:88) m =0 (cid:88) µ ∈P km ν ∈P kn − m c λµν (cid:54) =0 (cid:13)(cid:13)(cid:13)(cid:13) P H⊕K λ (id C ( nm ) ⊗ P H µ ⊗ P K ν ) (cid:18) (cid:104) (cid:18) nm (cid:19) (cid:105) ⊗ ψ ⊗ m ⊗ ϕ ⊗ n − m (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n (cid:88) m =0 (cid:88) µ ∈P km ν ∈P kn − m c λµν (cid:54) =0 (cid:13)(cid:13)(cid:13)(cid:13) (id C ( nm ) ⊗ P H µ ⊗ P K ν ) (cid:18) (cid:104) (cid:18) nm (cid:19) (cid:105) ⊗ ψ ⊗ m ⊗ ϕ ⊗ n − m (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = n (cid:88) m =0 (cid:18) nm (cid:19) (cid:88) µ ∈P km ν ∈P kn − m c λµν (cid:54) =0 (cid:13)(cid:13) P H µ ψ ⊗ m (cid:13)(cid:13) (cid:13)(cid:13) P K ν ϕ ⊗ n − m (cid:13)(cid:13) ≤ ( n + 1) d max ≤ m ≤ nµ ∈P km ν ∈P kn − m c λµν (cid:54) =0 (cid:18) nm (cid:19) (cid:13)(cid:13) P H µ ψ ⊗ m (cid:13)(cid:13) (cid:13)(cid:13) P K ν ϕ ⊗ n − m (cid:13)(cid:13) (43)By Lemma 3.33.3 we can choose a sequence λ (1) , λ (2) , . . . such that λ ( n ) ∈ P kn , lim n →∞ n λ ( n ) = λ and lim n →∞ − n log (cid:13)(cid:13)(cid:13) P H⊕K λ ( ψ ⊕ ϕ ) ⊗ n (cid:13)(cid:13)(cid:13) = I ψ ⊕ ϕ ( λ ) . (44)For every n choose 0 ≤ m ( n ) ≤ n , µ ( n ) ∈ P km ( n ) , ν ( n ) ∈ P kn − m ( n ) such that c λ ( n ) µ ( n ) ν ( n ) (cid:54) = 0 andattaining the maximum in (4343). Choose a subsequence n l such that n l m ( n l ) , m ( nl ) µ ( n l ) and n l − m ( nl ) ν ( n l ) converge, and let their limits be q, µ, ν . LR k • is closed, therefore ( λ, µ, ν ) ∈ kq . I ψ ⊕ ϕ ( λ ) = lim n →∞ − n log (cid:13)(cid:13)(cid:13) P H⊕K λ ( n ) ( ψ ⊕ ϕ ) ⊗ n (cid:13)(cid:13)(cid:13) = lim l →∞ − n l log (cid:13)(cid:13)(cid:13) P H⊕K λ ( nl ) ( ψ ⊕ ϕ ) ⊗ n l (cid:13)(cid:13)(cid:13) ≥ lim inf l →∞ − n l log (cid:18) n l m ( n l ) (cid:19) − m ( n l ) n l m ( n l ) log (cid:13)(cid:13)(cid:13) P H µ ( nl ) ψ ⊗ m ( nl ) (cid:13)(cid:13)(cid:13) − n l − m ( n l ) n l n l − m ( n l ) log (cid:13)(cid:13)(cid:13) P K ν ( nl ) ϕ ⊗ n l − m ( nl ) (cid:13)(cid:13)(cid:13) ≥ − h ( q ) + qI ψ ( µ ) + (1 − q ) I ϕ ( ν ) . (45)The first inequality follows from (4343) and the second one from Lemma 3.23.2. Proposition 3.7 (Rate function and local projections) . Let ψ ∈ H = H ⊗ · · · ⊗ H k and Π = Π = Π ∗ ∈ B ( H j ) for some j ∈ [ k ] . Consider the vectors ψ = ( I ⊗ · · · ⊗ I ⊗ Π ⊗ I ⊗· · · I ) ψ and ψ = ( I ⊗ · · · ⊗ I ⊗ ( I − Π) ⊗ I ⊗ · · · I ) ψ . For every µ, ν ∈ P k and q ∈ [0 , the inequality qI ψ ( µ ) + (1 − q ) I ψ ( ν ) − h ( q ) ≥ inf λ ∈P ( λ,µ,ν ) ∈ LR kq I ψ ( λ ) (46) holds.Proof. If q = 1 (or q = 0) then the right hand side is I ψ ( µ ) (or I ψ ( ν )) and the inequalityfollows from part (ii)(ii) of Proposition 3.43.4, therefore we can assume q ∈ (0 ,
1) and that ψ and ψ are nonzero. Let d = (cid:80) kj =1 dim H j , 0 ≤ m ≤ n , µ ∈ P km , ν ∈ P kn − m . We define thedisjoint projections Π , Π , . . . , Π n via the generating function n (cid:88) i =0 Π i t i = I ⊗ · · · ⊗ I ⊗ (( I − Π) + t Π) ⊗ n ⊗ I ⊗ · · · ⊗ I. (47)Consider the S n -action on H ⊗ n that permutes the factors and permutations of thevector ( P µ ψ ⊗ m ) ⊗ ( P ν ψ ⊗ n − m ). Π m is S n -invariant, therefore it commutes with P λ . ψ and ψ are supported in the orthogonal subspaces Π H j and ( I − Π) H j at site j and P µ ⊗ P ν commutes both with S m × S n − m and with Π ⊗ m ⊗ ( I − Π) n − m . Therefore everyelement σ ∈ S n either fixes this product (this happens iff σ ∈ S m × S n − m ) or sends itto an orthogonal one. The same is true for ψ ⊗ m ⊗ ψ ⊗ n − m , and the sum of its distinctpermutations is Π m ψ ⊗ n . Using this and the vanishing condition (2020) we get (summingover distinct permutations σ , i.e. over a set of representatives of the cosets S n /S m × S n − m )12 nm (cid:19) (cid:13)(cid:13) P µ ψ ⊗ m (cid:13)(cid:13) (cid:13)(cid:13) P ν ψ ⊗ n − m (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) σ σ · ( P µ ψ ⊗ m ⊗ P ν ψ ⊗ n − m ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:88) λ ∈P kn c λµν (cid:54) =0 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P λ (cid:88) σ σ · ( P µ ψ ⊗ m ⊗ P ν ψ ⊗ n − m ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:88) λ ∈P kn c λµν (cid:54) =0 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P λ (cid:88) σ σ · ( ψ ⊗ m ⊗ ψ ⊗ n − m ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:88) λ ∈P kn c λµν (cid:54) =0 (cid:13)(cid:13) P λ Π m ψ ⊗ n (cid:13)(cid:13) ≤ (cid:88) λ ∈P kn c λµν (cid:54) =0 (cid:13)(cid:13) P λ ψ ⊗ n (cid:13)(cid:13) ≤ ( n + 1) d max λ ∈P kn c λµν (cid:54) =0 (cid:13)(cid:13) P λ ψ ⊗ n (cid:13)(cid:13) (48)Let m (1) , m (2) , . . . , µ (1) , µ (2) , . . . , ν (1) , ν (2) , . . . be sequences such that 1 ≤ m ( n ) ≤ n − µ ( n ) ∈ P km ( n ) , ν ( n ) ∈ P kn − m ( n ) andlim n →∞ m ( n ) n = q (49)lim n →∞ m ( n ) µ ( n ) = µ (50)lim n →∞ − m ( n ) log (cid:13)(cid:13)(cid:13) P µ ( n ) ψ ⊗ m ( n ) (cid:13)(cid:13)(cid:13) = I ψ ( µ ) (51)lim n →∞ n − m ( n ) ν ( n ) = ν (52)lim n →∞ − n − m ( n ) log (cid:13)(cid:13)(cid:13) P ν ( n ) ψ ⊗ n − m ( n ) (cid:13)(cid:13)(cid:13) = I ψ ( µ ) . (53)For every n choose λ ( n ) ∈ P kn such that c λ ( n ) µ ( n ) ν ( n ) (cid:54) = 0 and attaining the maximum in (4848).Choose a subsequence n l such that n l λ ( n l ) converges and let its limit be λ . LR k • is closed,therefore ( λ, µ, ν ) ∈ LR kq . qI ψ ( µ ) + (1 − q ) I ψ ( ν ) − h ( q ) = lim n →∞ − n log (cid:13)(cid:13)(cid:13) P µ ( n ) ψ ⊗ m ( n ) (cid:13)(cid:13)(cid:13) − n log (cid:13)(cid:13)(cid:13) P ν ( n ) ψ ⊗ n − m ( n ) (cid:13)(cid:13)(cid:13) − n log (cid:18) nm ( n ) (cid:19) = lim l →∞ − n l log (cid:13)(cid:13)(cid:13) P µ ( nl ) ψ ⊗ m ( nl ) (cid:13)(cid:13)(cid:13) − n l log (cid:13)(cid:13)(cid:13) P ν ( nl ) ψ ⊗ n l − m ( nl ) (cid:13)(cid:13)(cid:13) − n l log (cid:18) n l m ( n l ) (cid:19) ≥ lim l →∞ − n l log (cid:13)(cid:13) P λ ( nl ) ψ ⊗ n l (cid:13)(cid:13) ≥ I ψ ( λ ) . (54)13he first inequality follows from (4848) and the second one from Lemma 3.23.2. In this section we construct our family of functionals on pure unnormalised states whichare monotone under trace-nonincreasing local operations and classical communication,additive under the direct sum and multiplicative under the tensor product, i.e. are LOCCspectral points in the terminology of [JV19JV19]. This famlily is parametrised by a number α ∈ [0 ,
1] and a point θ in the simplex P ([ k ]). For α = 0 they reduce to the quantumfunctionals introduced in [CVZ18CVZ18], while for α = 1 they collapse to a single function, thenorm squared.Like the quantum functionals of [CVZ18CVZ18] (and also the support functionals of Strassen[Str91Str91]), our functionals come in two flavours, an “upper” and a “lower” family. In Sec-tion 4.14.1 we define the upper version in terms of asymptotic representation theoretical data:the Shannon entropy, which measures the growth rate of the dimensions of the represen-tations of the symmetric group and the rate function studied in Section 33, and prove thatit is submultiplicative, subadditive and satisfies (55). In Section 4.24.2 we define the lowercounterpart in terms of the SLOCC orbit and prove that it is supermultiplicative. When α = 0 both functionals reduce to the ones defined in [CVZ18CVZ18] and are equal to each other.Below we will assume that α >
0. In this case we do not know if the lower and theupper functionals coincide. Nevertheless, in Section 4.34.3 we show that the regularisationof the lower one equals the upper one and is superadditive, which implies that the upperfunctionals are LOCC spectral points.
We start by defining the “upper” family of the functionals. After the definition we prove itsalgebraic (Proposition 4.24.2) and monotonicity (Proposition 4.34.3) properties. The main toolsin this section are the inequalities satisfied by the rate function, proved in Section 33, andthe entropy inequalities related to the Kronecker and Littlewood–Richardson coefficients,as explained in Section 22.
Definition 4.1 (Upper functionals) . Let α ∈ (0 ,
1] and θ ∈ P ([ k ]). The logarithmic upperfunctional is E α,θ ( | ψ (cid:105) ) = sup λ ∈P k (cid:2) (1 − α ) H θ ( λ ) − αI ψ ( λ ) (cid:3) (55)and the upper functional is F α,θ ( | ψ (cid:105) ) = 2 E α,θ ( | ψ (cid:105) ) .We remark that I ψ ( λ ) is lower semicontinuous and infinite outside a compact set,therefore for every α ∈ (0 ,
1] the supremum in (5555) is attained.
Proposition 4.2 (Submultiplicativity and subadditivity of the upper functional) . For any α ∈ (0 , , p ∈ [0 , ∞ ) , θ ∈ P ([ k ]) and vectors | ψ (cid:105) ∈ H ⊗ · · · ⊗ H k and | ϕ (cid:105) ∈ K ⊗ · · · ⊗ K k the followings hold:(i) F α,θ ( √ p | ψ (cid:105) ) = p α F α,θ ( | ψ (cid:105) ) .(ii) F α,θ ( | ψ (cid:105) ⊗ | ϕ (cid:105) ) ≤ F α,θ ( | ψ (cid:105) ) F α,θ ( | ϕ (cid:105) ) (iii) F α,θ ( | ψ (cid:105) ⊕ | ϕ (cid:105) ) ≤ F α,θ ( | ψ (cid:105) ) + F α,θ ( | ϕ (cid:105) )14 roof. (i)(i): The first term inside the supremum in (5555) does not change if we replace | ψ (cid:105) with √ p | ψ (cid:105) , while the second term acquires an α log p term by the scaling property of therate function (Proposition 3.43.4, (i)(i)). Therefore E α,θ ( √ p | ψ (cid:105) ) = E α,θ ( | ψ (cid:105) ) + α log p , whichis equivalent to the statement.(ii)(ii): We use Proposition 3.53.5 to bound E α,θ ( | ψ (cid:105) ⊗ | ϕ (cid:105) ) as E α,θ ( | ψ (cid:105) ⊗ | ϕ (cid:105) ) = sup λ ∈P k (cid:2) (1 − α ) H θ ( λ ) − αI ψ ⊗ ϕ ( λ ) (cid:3) ≤ sup λ ∈P k sup µ,ν ∈P k ( λ,µ,ν ) ∈ Kron k (cid:2) (1 − α ) H θ ( λ ) − αI ψ ( µ ) − αI ϕ ( ν ) (cid:3) ≤ sup λ ∈P k sup µ,ν ∈P k ( λ,µ,ν ) ∈ Kron k [(1 − α ) ( H θ ( µ ) + H θ ( ν )) − αI ψ ( µ ) − αI ϕ ( ν )]= sup µ,ν ∈P k [(1 − α ) ( H θ ( µ ) + H θ ( ν )) − αI ψ ( µ ) − αI ϕ ( ν )]= E α,θ ( | ψ (cid:105) ) + E α,θ ( | ϕ (cid:105) ) , (56)where the second inequality uses that ( λ, µ, ν ) ∈ Kron k implies H θ ( λ ) ≤ H θ ( µ ) + H θ ( ν ).(iii)(iii): We use Proposition 3.63.6 to bound E α,θ ( | ψ (cid:105) ⊕ | ϕ (cid:105) ) as E α,θ ( | ψ (cid:105) ⊕ | ϕ (cid:105) ) = sup λ ∈P k (cid:2) (1 − α ) H θ ( λ ) − αI ψ ⊕ ϕ ( λ ) (cid:3) ≤ sup λ ∈P k max q ∈ [0 , sup µ,ν ∈P k ( λ,µ,ν ) ∈ LR kq (cid:2) (1 − α ) H θ ( λ ) − α ( qI ψ ( µ ) + (1 − q ) I ϕ ( ν ) − h ( q )) (cid:3) ≤ sup λ ∈P k max q ∈ [0 , sup µ,ν ∈P k ( λ,µ,ν ) ∈ LR kq (cid:2) (1 − α ) ( qH θ ( µ ) + (1 − q ) H θ ( ν ) + h ( q )) − α ( qI ψ ( µ ) + (1 − q ) I ϕ ( ν ) − h ( q )) (cid:3) = max q ∈ [0 , sup µ,ν ∈P k (cid:2) (1 − α ) ( qH θ ( µ ) + (1 − q ) H θ ( ν ) + h ( q )) − α ( qI ψ ( µ ) + (1 − q ) I ϕ ( ν ) − h ( q )) (cid:3) = max q ∈ [0 , qE α,θ ( | ψ (cid:105) ) + (1 − q ) E α,θ ( | ϕ (cid:105) ) + h ( q )= log (cid:16) F α,θ ( | ψ (cid:105) ) + F α,θ ( | ϕ (cid:105) ) (cid:17) . (57)The second inequality uses that ( λ, µ, ν ) ∈ LR kq implies H θ ( λ ) ≤ qH θ ( µ ) + (1 − q ) H θ ( ν ) + h ( q ). Proposition 4.3. If Π is a projection on H j and | ψ (cid:105) ∈ H , ⊗ · · · ⊗ H k , then F α,θ ( | ψ (cid:105) ) /α ≥ F α,θ (Π j | ψ (cid:105) ) /α + F α,θ (( I − Π) j | ψ (cid:105) ) /α . (58) Proof.
Let | ψ (cid:105) = Π j | ψ (cid:105) and | ψ (cid:105) = ( I − Π) j | ψ (cid:105) . Choose µ, ν ∈ P k and q ∈ [0 , (cid:15) >
0, there is a k -tuple λ ∈ P k such that I ψ ( λ ) − (cid:15) ≤ qI ψ ( µ ) + (1 − q ) I ψ ( ν ) − h ( q ) (59)15nd ( λ, µ, ν ) ∈ LR kq .By definition,log F α,θ ( | ψ (cid:105) ) /α = 1 α E α,θ ( | ψ (cid:105) ) ≥ − αα H θ ( λ ) − I ψ ( λ ) ≥ − αα ( qH θ ( µ ) + (1 − q ) H θ ( ν )) − qI ψ ( µ ) − (1 − q ) I ψ ( ν ) + h ( q ) − (cid:15). (60)The inequality also holds if we take (cid:15) → µ, ν , thereforelog F α,θ ( | ψ (cid:105) ) /α ≥ q log F α,θ ( | ψ (cid:105) ) /α + (1 − q ) log F α,θ ( | ψ (cid:105) ) /α + h ( q ) . (61)Finally, the maximum of the right hand side over q islog (cid:16) F α,θ (Π j | ψ (cid:105) ) /α + F α,θ (( I − Π) j | ψ (cid:105) ) /α (cid:17) . (62) Next we define the lower versions of our functionals and show basic properties such asmonotonicity and supermultiplicativity (Proposition 4.54.5). The latter property impliesthat the regularisation of the lower functional exists. The regularised functional inheritsthe algebraic properties and in addition satisfies superadditivity under the direct sum(Proposition 4.74.7).In the following we will use the notation ψ (cid:31) ϕ to mean that there exist linear operators A , . . . , A k ( A j ∈ B ( H j )) satisfying A ∗ j A j ≤ I for all j ∈ [ k ] such that ϕ = ( A ⊗· · ·⊗ A k ) ψ .In addition, when ϕ ∈ H \ { } we set H θ ( ϕ ) = k (cid:88) j =1 θ j H (cid:32) | ϕ (cid:105)(cid:104) ϕ | j (cid:107) ϕ (cid:107) (cid:33) , (63)where the subscript j refers to the j th marginal. Definition 4.4 (Lower functionals) . The logarithmic lower functional is E α,θ ( | ψ (cid:105) ) = sup | ϕ (cid:105)∈H ψ (cid:31) ϕ (cid:104) (1 − α ) H θ ( ϕ ) + α log (cid:107) ϕ (cid:107) (cid:105) . (64)The lower functional is F α,θ ( | ψ (cid:105) ) := 2 E α,θ ( | ψ (cid:105) ) . Proposition 4.5 (Basic properties of the lower functional) . For any α ∈ [0 , ∞ ) , p ∈ [0 , ∞ ) , θ ∈ P ([ k ]) and vectors | ψ (cid:105) ∈ H ⊗ · · · ⊗ H k and | ϕ (cid:105) ∈ K ⊗ · · · ⊗ K k the followingshold:(i) F α,θ ( √ p | ψ (cid:105) ) = p α F α,θ ( | ψ (cid:105) ) .(ii) F α,θ ( (cid:104) r (cid:105) ) = r .(iii) If ψ (cid:31) ψ then F α,θ ( | ψ (cid:105) ) ≥ F α,θ ( | ψ (cid:105) ) . iv) F α,θ ( | ψ (cid:105) ⊗ | ψ (cid:105) ) ≥ F α,θ ( | ψ (cid:105) ) F α,θ ( | ψ (cid:105) ) Proof. (i)(i): If we replace | ψ (cid:105) with √ p | ψ (cid:105) , then the allowed | ϕ (cid:105) also get rescaled by √ p . Thefirst term in the supremum is not sensitive to this, while the second gets an additional α log p term. Therefore E α,θ ( √ p | ψ (cid:105) ) = E α,θ ( | ψ (cid:105) ) + α log p , which is equivalent to thestatement.(ii)(ii): The entropies in the supremum are upper bounded by log r for any choice of | ϕ (cid:105) since the local ranks cannot increase under separable operations. The norm is alsononincreasing, therefore log (cid:107) ϕ (cid:107) ≤ log (cid:107)(cid:104) r (cid:105)(cid:107) = log r . This proves that E α,θ ( (cid:104) r (cid:105) ) ≤ log r .On the other hand, | ϕ (cid:105) = (cid:104) r (cid:105) is feasible and achieves this upper bound.(iii)(iii): If ψ (cid:31) ϕ then by assumption also ψ (cid:31) ϕ , therefore the supremum for ψ istaken over a larger set than for ψ , which implies E α,θ ( | ψ (cid:105) ) ≥ E α,θ ( ψ ).(iv)(iv): If ψ i (cid:31) ϕ i ( i = 1 ,
2) then ψ ⊗ ψ (cid:31) ϕ ⊗ ϕ , therefore E α,θ ( | ψ (cid:105) ⊗ | ψ (cid:105) ) ≥ (1 − α ) H θ ( ϕ ⊗ ϕ ) + α log (cid:107) ϕ ⊗ ϕ (cid:107) = (1 − α ) H θ ( ϕ ) + α log (cid:107) ϕ (cid:107) + (1 − α ) H θ ( ϕ ) + α log (cid:107) ϕ (cid:107) . (65)Now take the supremum over the admissible | ϕ (cid:105) and | ϕ (cid:105) to get E α,θ ( | ψ (cid:105) ⊗ | ψ (cid:105) ) ≥ E α,θ ( | ψ (cid:105) ) + E α,θ ( | ψ (cid:105) ).Part (iv)(iv) of Proposition 4.54.5 says that E α,θ is superadditive under tensor product. Wealso have the additive upper bound E α,θ ( | ψ (cid:105) ) ≤ (1 − α ) (cid:80) kj =1 θ j log dim H j + α log (cid:107) ψ (cid:107) .Therefore the regularised lower functional exists and is finite, so the following definitionis meaningful. Definition 4.6 (Asymptotic lower functional) . The asymptotic logarithmic lower func-tional is E (cid:101) α,θ ( | ψ (cid:105) ) = lim n →∞ n E α,θ ( | ψ (cid:105) ⊗ n ) (66)and the asymptotic lower functional is F (cid:101) α,θ ( | ψ (cid:105) ) = lim n →∞ n (cid:113) F α,θ ( | ψ (cid:105) ⊗ n ) = 2 E (cid:101) α,θ ( | ψ (cid:105) ) . (67) Proposition 4.7 (Basic properties of the asymptotic lower functional) . For any α ∈ [0 , ∞ ) , p ∈ [0 , ∞ ) , θ ∈ P ([ k ]) and vectors | ψ (cid:105) ∈ H ⊗ · · · ⊗ H k and | ϕ (cid:105) ∈ K ⊗ · · · ⊗ K k thefollowings hold:(i) F (cid:101) α,θ ( √ p | ψ (cid:105) ) = p α F (cid:101) α,θ ( | ψ (cid:105) ) .(ii) F (cid:101) α,θ ( (cid:104) r (cid:105) ) = r .(iii) F (cid:101) α,θ ( | ψ (cid:105) ⊗ | ψ (cid:105) ) ≥ F (cid:101) α,θ ( | ψ (cid:105) ) F (cid:101) α,θ ( | ψ (cid:105) ) (iv) F (cid:101) α,θ ( | ψ (cid:105) ⊕ | ψ (cid:105) ) ≥ F (cid:101) α,θ ( | ψ (cid:105) ) + F (cid:101) α,θ ( | ψ (cid:105) ) Proof. (i)(i): By part (i)(i) of Proposition 4.54.5 we have F (cid:101) α,θ ( √ p | ψ (cid:105) ) = lim n →∞ n (cid:113) F α,θ (( √ p | ψ (cid:105) ) ⊗ n )= lim n →∞ n (cid:113) p nα F α,θ ( | ψ (cid:105) ) = p α F (cid:101) α,θ ( √ p | ψ (cid:105) ) . (68)17ii)(ii): By part (ii)(ii) of Proposition 4.54.5 we have F (cid:101) α,θ ( (cid:104) r (cid:105) ) = lim n →∞ n (cid:113) F α,θ ( (cid:104) r (cid:105) ⊗ n ) = lim n →∞ n (cid:113) F α,θ ( (cid:104) r n (cid:105) ) = lim n →∞ n √ r n = r. (69)(iii)(iii): By part (iv)(iv) of Proposition 4.54.5 we have F (cid:101) α,θ ( | ψ (cid:105) ⊗ | ψ (cid:105) ) = lim n →∞ n (cid:113) F α,θ (( | ψ (cid:105) ⊗ | ψ (cid:105) ) ⊗ n ) ≥ lim n →∞ n (cid:113) F α,θ ( | ψ (cid:105) ⊗ n ) F α,θ ( | ψ (cid:105) ⊗ n )= F (cid:101) α,θ ( | ψ (cid:105) ) F (cid:101) α,θ ( | ψ (cid:105) ) (70)(iv)(iv): Let q ∈ [0 , n ∈ N we expand the tensor power of the direct sum as( | ψ (cid:105) ⊕ | ψ (cid:105) ) ⊗ n = n (cid:77) m =0 (cid:104) (cid:18) nm (cid:19) (cid:105) ⊗ | ψ (cid:105) ⊗ m ⊗ | ψ (cid:105) ⊗ ( n − m ) . (71)Keep only the m = (cid:98) qn (cid:99) term. This can be done by a local projection, which is a separableoperation with one Kraus operator, i.e. ( | ψ (cid:105) ⊕ | ψ (cid:105) ) ⊗ n (cid:31) (cid:104) (cid:0) nm (cid:1) (cid:105) ⊗ | ψ (cid:105) ⊗ m ⊗ | ψ (cid:105) ⊗ ( n − m ) .Therefore, using (iii)(iii) and (iv)(iv) of Proposition 4.54.5 we have E α,θ (( | ψ (cid:105) ⊕ | ψ (cid:105) ) ⊗ n ) ≥ E α,θ (cid:18) (cid:104) (cid:18) n (cid:98) qn (cid:99) (cid:19) (cid:105) ⊗ | ψ (cid:105) ⊗(cid:98) qn (cid:99) ⊗ | ψ (cid:105) ⊗ n −(cid:98) qn (cid:99) (cid:19) ≥ log (cid:18) n (cid:98) qn (cid:99) (cid:19) + E α,θ ( | ψ (cid:105) ⊗(cid:98) qn (cid:99) ) + E α,θ ( | ψ (cid:105) ⊗ n −(cid:98) qn (cid:99) ) . (72)Divide both sides by n and take the limit n → ∞ : E (cid:101) α,θ ( | ψ (cid:105) ⊕ | ψ (cid:105) ) = lim n →∞ n E α,θ (( | ψ (cid:105) ⊕ | ψ (cid:105) ) ⊗ n ) ≥ lim n →∞ n log (cid:18) n (cid:98) qn (cid:99) (cid:19) + lim n →∞ (cid:98) qn (cid:99) n (cid:98) qn (cid:99) E α,θ ( | ψ (cid:105) ⊗(cid:98) qn (cid:99) )+ lim n →∞ n − (cid:98) qn (cid:99) n n − (cid:98) qn (cid:99) E α,θ ( | ψ (cid:105) ⊗ n −(cid:98) qn (cid:99) )= h ( q ) + qE (cid:101) α,θ ( | ψ (cid:105) ) + (1 − q ) E (cid:101) α,θ ( | ψ (cid:105) ) . (73)To finish the proof, take the maximum of the right hand side over q . The aim of this section is to prove that E (cid:101) α,θ = E α,θ , from which our main theorem imme-diately follows. The inequality E (cid:101) α,θ ( | ψ (cid:105) ) ≤ E α,θ ( | ψ (cid:105) ) relies on the spectrum estimationtheorem (similarly to [CVZ18CVZ18, Theorem 3.24]) and basic properties of the rate function.The reverse inequality is an application of the dimension estimates (1010) and (1111). Proposition 4.8. E α,θ ( | ψ (cid:105) ) ≤ E α,θ ( | ψ (cid:105) ) Proof.
Let | ϕ (cid:105) = ( A ⊗ · · · ⊗ A k ) | ψ (cid:105) where A ∗ j A j ≤ I for all j . Let λ = ( λ , . . . , λ k ) ∈ P k be the decreasingly ordered marginal spectra of | ϕ (cid:105)(cid:104) ϕ | / (cid:107) ϕ (cid:107) . Then we have I ψ ( λ ) ≤ I ϕ ( λ )= I ϕ/ (cid:107) ϕ (cid:107) ( λ ) + log (cid:107) ϕ (cid:107) − = − log (cid:107) ϕ (cid:107) (74)18y Proposition 3.43.4. It follows from Definition 4.14.1 that E α,θ ( | ψ (cid:105) ) ≥ (1 − α ) H θ ( λ ) − αI ψ ( λ ) ≥ (1 − α ) H θ ( λ ) + α log (cid:107) ϕ (cid:107) = (1 − α ) H θ ( ϕ ) + α log (cid:107) ϕ (cid:107) . (75)The supremum of the right hand side over the possible | ϕ (cid:105) is E α,θ ( | ψ (cid:105) ). Corollary 4.9. E (cid:101) α,θ ( | ψ (cid:105) ) ≤ E α,θ ( | ψ (cid:105) ) .Proof. E α,θ is subadditive under tensor product, therefore E (cid:101) α,θ ( | ψ (cid:105) ) = lim n →∞ n E α,θ ( | ψ (cid:105) ⊗ n ) ≤ lim n →∞ n E α,θ ( | ψ (cid:105) ⊗ n ) ≤ E α,θ ( | ψ (cid:105) ) . (76) Proposition 4.10. E (cid:101) α,θ ( | ψ (cid:105) ) ≥ E α,θ ( | ψ (cid:105) ) .Proof. Let ψ ∈ H = H ⊗ · · · ⊗ H k , choose λ ∈ P k . Let ( λ ( n ) ) n ∈ N be a sequence such that λ ( n ) ∈ P kn , n λ ( n ) → λ andlim n →∞ − n log (cid:13)(cid:13) P λ ( n ) ψ ⊗ n (cid:13)(cid:13) = I ψ ( λ ) , (77)as in Lemma 3.33.3. By Definition 4.44.4, we can estimate E α,θ ( | ψ (cid:105) ⊗ n ) using ψ ⊗ n (cid:31) P λ ( n ) ψ ⊗ n =: ϕ n . The j th marginal of ϕ n / (cid:107) ϕ n (cid:107) is a state on S λ ( n ) j ( H j ) ⊗ [ λ ( n ) j ], invariant under the actionof S n on the second factor, thus the reduced state on that factor is maximally mixed. Weuse the triangle inequality for the von Neumann entropy and the dimension estimates (1010)and (1111) to get H θ ( ϕ n ) ≥ k (cid:88) j =1 θ j (cid:18) log dim[ λ ( n ) j ] − log dim S λ ( n ) j ( H j ) (cid:19) ≥ n k (cid:88) j =1 θ j (cid:20) H (cid:18) n λ ( n ) j (cid:19) − log( n + 1) d j ( d j − / ( n + d j ) ( d j +2)( d j − / (cid:21) ≥ nH θ (cid:18) n λ ( n ) (cid:19) − log( n + d ) d − , (78)where d j = dim H j and d = dim H .This leads to the lower bound E (cid:101) α,θ ( | ψ (cid:105) ) ≥ lim sup n →∞ n E α,θ ( | ψ (cid:105) ⊗ n ) ≥ lim sup n →∞ n (cid:104) (1 − α ) H ( ϕ n ) + α log (cid:107) ϕ n (cid:107) (cid:105) ≥ lim sup n →∞ (cid:20) (1 − α ) H θ (cid:18) n λ ( n ) (cid:19) − n (1 − α ) log( n + d ) d − + α n log (cid:107) ϕ n (cid:107) (cid:21) = (1 − α ) H θ ( λ ) − αI ψ ( λ ) . (79)To finish the proof we take the supremum over λ so that the right hand side becomes E α,θ ( | ψ (cid:105) ). 19 roof of Theorem 1.11.1. By Corollary 4.94.9 and Proposition 4.104.10 we see that E (cid:101) α,θ = E α,θ .The scaling property (i)(i) follows from part (i)(i) of Proposition 4.24.2 (or from part (i)(i) of Propo-sition 4.74.7), the normalisation (ii)(ii) is the same as part (ii)(ii) of Proposition 4.74.7. The multiplica-tivity (iii)(iii) follows from part (iii)(iii) of Proposition 4.74.7 and part (ii)(ii) of Proposition 4.24.2. Theadditivity (iv)(iv) follows from part (iv)(iv) of Proposition 4.74.7 and part (iii)(iii) of Proposition 4.24.2.The monotonicity property (v)(v) is a consequence of the scaling property, Proposition 4.34.3and the equivalent condition (55) (from the characterisation [JV19JV19, Theorem 3.1.] of LOCCspectral points). Acknowledgements
I thank Matthias Christandl for numerous discussions on the simultaneous spectrum esti-mation problem. This research was supported by the J´anos Bolyai Research Scholarshipof the Hungarian Academy of Sciences and the National Research, Development and In-novation Fund of Hungary within the Quantum Technology National Excellence Program(Project Nr. 2017-1.2.1-NKP-2017-00001) and via the research grants K124152, KH129601.Part of this work was done while the author was with QMATH.
A Proofs of technical statements
Proposition A.1.
Let λ ∈ P k , (cid:15) > and ϕ ∈ H = H ⊗ · · · ⊗ H k . Then the limit lim n →∞ n log (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) (80) exists (possibly −∞ , with log 0 interpreted as −∞ ).Proof. If P λ ϕ ⊗ n = 0 for every n and λ with n λ ∈ B (cid:15) ( λ ) then the sequence is constant −∞ . We can thus assume P λ ϕ ⊗ n (cid:54) = 0 for some λ . In particular, ϕ (cid:54) = 0.Fix a maximal torus and a choice of positive Weyl chamber for U ( H ) × · · · × U ( H k ).Let W λ ∈ B ( H ⊗ n ) be the orthogonal projection onto the subspace of highest weight vectorswith weight λ . Then W λ ≤ P λ = dim S λ ( H ) (cid:90) U ( H ) ×···× U ( H k ) ( U ⊗ n ) ∗ W λ U ⊗ n d U, (81)where the integration is with respect to the Haar probability measure. dim S λ ( H ) ≤ ( n + 1) d where d = (cid:80) kj =1 dim H j (dim H j − .The only k -tuple for n = 1 is ((1) , . . . , (1)), therefore P ((1) ,..., (1)) ϕ = ϕ (cid:54) = 0. By (8181)there is an U ∈ U ( H ) × · · · × U ( H k ) such that W ((1) ,..., (1)) U ϕ (cid:54) = 0. In fact, the set (cid:8) U ∈ U ( H ) × · · · × U ( H k ) (cid:12)(cid:12) W ((1) ,..., (1)) U ϕ (cid:54) = 0 (cid:9) (82)is dense since for any U one can find A ∈ u ( H ) × · · · × u ( H k ) with U U − = e A and thenthe function t (cid:55)→ (cid:13)(cid:13) e tA U ϕ (cid:13)(cid:13) (83)is real analytic and not identically zero, therefore arbitrarily small values of t exist with e tA U ϕ (cid:54) = 0. 20et n ∈ N such that (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) (cid:54) = 0 . (84)The number of nonzero terms is at most ( n + 1) d where d = (cid:80) kj =1 H j , therefore thereis a λ ∈ P kn such that n λ ∈ B (cid:15) ( λ ) and (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) ≥ ( n + 1) − d (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) . (85)(8181) implies that (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) = (cid:104) ϕ ⊗ n , P λ ϕ ⊗ n (cid:105)≤ ( n + 1) d (cid:90) U ( H ) ×···× U ( H k ) (cid:104) ( U ϕ ) ⊗ n , W λ ( U ϕ ) ⊗ n (cid:105) d U, (86)therefore there exists U ∈ U ( H ) × · · · × U ( H k ) such that (with d = d + d ) (cid:13)(cid:13) W λ ( U ϕ ) ⊗ n (cid:13)(cid:13) ≥ ( n + 1) − d (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) (87)The left hand side of (8787) is continuous in U and the set (8282) is dense, therefore for every δ > U ∈ U ( H ) × · · · × U ( H k ) such that W ((1) ,..., (1)) U ϕ (cid:54) = 0 and (cid:13)(cid:13) W λ ( U ϕ ) ⊗ n (cid:13)(cid:13) ≥ (1 − δ )( n + 1) − d (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) . (88)Let n ∈ N be arbitrary and write n = qn + r with q, r ∈ N , r < n . Then µ := qλ + r ((1) , (1) , . . . , (1)) ∈ P kn and1 n µ − n λ = rn (cid:18) ((1) , (1) , . . . , (1))) − n λ (cid:19) , (89)which goes to 0 as n → ∞ , therefore n µ ∈ B (cid:15) ( λ ) for every large enough n .The tensor product of two highest weight vectors is itself a highest weight vector withthe sum of the two weights, therefore P µ ≥ W µ ≥ W ⊗ qλ ⊗ W ⊗ r ((1) , (1) ,..., (1)) . From this we getlim inf n →∞ n log (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) ≥ lim inf n →∞ n log (cid:13)(cid:13) W λ ( U ϕ ) ⊗ n (cid:13)(cid:13) q (cid:13)(cid:13) W ((1) , (1) ,..., (1)) ( U ϕ ) (cid:13)(cid:13) r = lim inf n →∞ (cid:16) − rn (cid:17) n log (cid:13)(cid:13) W λ ( U ϕ ) ⊗ n (cid:13)(cid:13) + rn log (cid:13)(cid:13) W ((1) , (1) ,..., (1)) ( U ϕ ) (cid:13)(cid:13) = 1 n log (cid:13)(cid:13) W λ ( U ϕ ) ⊗ n (cid:13)(cid:13) ≥ n log(1 − δ ) − dn log( n + 1) + 1 n (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) . δ >
0, therefore also for δ = 0. In particular, (8484) is not onlytrue for n , but also for every large enough n . Therefore we can take lim sup n →∞ of bothsides, and the resulting inequality means that the limit exists. Proof of Lemma 3.23.2.
Let (cid:15) > n l λ ( nl ) ∈ B (cid:15) ( λ ) for every large enough l , therefore (cid:13)(cid:13) P λ ( nl ) (cid:13)(cid:13) ≤ (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) . (91)Apply the logarithm to both sides, multiply by − n l and take the limit inferior:lim inf n →∞ − n l (cid:13)(cid:13) P λ ( nl ) (cid:13)(cid:13) ≥ lim inf n →∞ − n l (cid:88) λ ∈P knl nl λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n l (cid:13)(cid:13) = lim n →∞ − n (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) , (92)in the last step using Proposition A.1A.1. The claim follows after taking the limit (cid:15) → Proof of Lemma 3.33.3.
For every (cid:15) > I (cid:15) = lim n →∞ − n log (cid:88) λ ∈P kn n λ ∈ B (cid:15) ( λ ) (cid:13)(cid:13) P λ ϕ ⊗ n (cid:13)(cid:13) (93)so that lim (cid:15) → I (cid:15) = I ϕ ( λ ).Choose a sequence ( (cid:15) m ) ∞ m =1 such that (cid:15) ≥ (cid:15) -ball in P k is equalto P k ), lim m →∞ (cid:15) m = 0 and for each m a sequence ( λ ( m,n ) ) n such that λ ( m,n ) ∈ P kn , n λ ( m,n ) ∈ B (cid:15) m ( λ ) and (cid:107) P λ ( m,n ) ϕ ⊗ n (cid:107) is maximal subject to these conditions (such a choicemay not be possible for small n , we only define the sequence for large enough n ). Thenlim n →∞ − n log (cid:13)(cid:13) P λ ( m,n ) ϕ ⊗ n (cid:13)(cid:13) = I (cid:15) m . (94)Let n ( m ) be the smallest integer such that for all n ≥ n ( m ) the inequality − n log (cid:13)(cid:13) P λ ( m,n ) ϕ ⊗ n (cid:13)(cid:13) ≤ I (cid:15) m + (cid:15) m . (95)For every n ∈ N let M n = max { m ∈ [ n ] | n ≥ n ( m ) } . Then ( M n ) n ∈ N is an increasingsequence that is clearly unbounded. We claim that the sequence λ ( n ) := λ ( M n ,n ) satisfiesthe requirements. Indeed, (cid:13)(cid:13)(cid:13)(cid:13) n λ ( n ) − λ (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) n λ ( M n ,n ) − λ (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:15) M n → n ≥ n ( M n ), thereforelim sup n →∞ − n log (cid:13)(cid:13) P λ ( n ) ϕ ⊗ n (cid:13)(cid:13) = lim sup n →∞ − n log (cid:13)(cid:13) P λ ( Mn,n ) ϕ ⊗ n (cid:13)(cid:13) ≤ lim sup n →∞ ( I (cid:15) Mn + (cid:15) M n ) = I ϕ ( λ ) . (97)The opposite bound on the limit inferior follows from Lemma 3.23.2.22 eferences [Ari96] Erdal Arikan. An inequality on guessing and its application to sequentialdecoding. IEEE Transactions on Information Theory , 42(1):99–105, 1996. doi:10.1109/18.481781doi:10.1109/18.481781 .[ARS88] Robert Alicki, S(cid:32)lawomir Rudnicki, and S(cid:32)lawomir Sadowski. Symmetry prop-erties of product states for the system of
N n -level atoms.
Journal of mathe-matical physics , 29(5):1158–1162, 1988. doi:10.1063/1.527958doi:10.1063/1.527958 .[BCV20] Alonso Botero, Matthias Christandl, and P´eter Vrana. Large deviation prin-ciple for moment map estimation. 2020. arXiv:2004.14504arXiv:2004.14504 .[BPR +
00] Charles H Bennett, Sandu Popescu, Daniel Rohrlich, John A Smolin,and Ashish V Thapliyal. Exact and asymptotic measures of multipar-tite pure-state entanglement.
Physical Review A , 63(1):012307, 2000. arXiv:quant-ph/9908073arXiv:quant-ph/9908073 , doi:10.1103/PhysRevA.63.012307doi:10.1103/PhysRevA.63.012307 .[CDS08] Eric Chitambar, Runyao Duan, and Yaoyun Shi. Tripartite entanglementtransformations and tensor rank. Physical review letters , 101(14):140502,2008. arXiv:0805.2977arXiv:0805.2977 , doi:10.1103/PhysRevLett.101.140502doi:10.1103/PhysRevLett.101.140502 .[Chr06] Matthias Christandl. The structure of bipartite quantum states-insights fromgroup theory and cryptography . PhD thesis, 2006. arXiv:quant-ph/0604183arXiv:quant-ph/0604183 .[CM06] Matthias Christandl and Graeme Mitchison. The spectra of density op-erators and the Kronecker coefficients of the symmetric group.
Com-mun. Math. Phys. , 261(3):789–797, 2006. arXiv:quant-ph/0409016arXiv:quant-ph/0409016 , doi:10.1007/s00220-005-1435-1doi:10.1007/s00220-005-1435-1 .[CVZ18] Matthias Christandl, P´eter Vrana, and Jeroen Zuiddam. Universal points inthe asymptotic spectrum of tensors. In Proceedings of the 50th Annual ACMSIGACT Symposium on Theory of Computing , pages 289–296. ACM, 2018. arXiv:1709.07851arXiv:1709.07851 , doi:10.1145/3188745.3188766doi:10.1145/3188745.3188766 .[CW04] Matthias Christandl and Andreas Winter. “Squashed entanglement”: an ad-ditive entanglement measure. Journal of mathematical physics , 45(3):829–840,2004. arXiv:quant-ph/0308088arXiv:quant-ph/0308088 , doi:10.1063/1.1643788doi:10.1063/1.1643788 .[DVC00] Wolfgang D¨ur, Guifre Vidal, and J Ignacio Cirac. Three qubits can be en-tangled in two inequivalent ways. Physical Review A , 62(6):062314, 2000. arXiv:quant-ph/0005115arXiv:quant-ph/0005115 , doi:10.1103/PhysRevA.62.062314doi:10.1103/PhysRevA.62.062314 .[FH91] William Fulton and Joe Harris. Representation Theory: A First Course ,volume 129. Springer Science & Business Media, 1991.[Fri17] Tobias Fritz. Resource convertibility and ordered commutative monoids.
Mathematical Structures in Computer Science , 27(6):850–938, 2017. arXiv:1504.03661arXiv:1504.03661 , doi:10.1017/S0960129515000444doi:10.1017/S0960129515000444 .[FW20] Cole Franks and Michael Walter. Minimal length in an orbit closure as asemiclassical limit. 2020. arXiv:2004.14872arXiv:2004.14872 .[Hay17] Masahito Hayashi. A Group Theoretic Approach to Quantum Information .Springer, Cham, 2017. doi:10.1007/978-3-319-45241-8doi:10.1007/978-3-319-45241-8 .23HKM +
02] Masahito Hayashi, Masato Koashi, Keiji Matsumoto, Fumiaki Morikoshi,and Andreas Winter. Error exponents for entanglement concentra-tion.
Journal of Physics A: Mathematical and General , 36(2):527, 2002. arXiv:quant-ph/0206097arXiv:quant-ph/0206097 , doi:10.1088/0305-4470/36/2/316doi:10.1088/0305-4470/36/2/316 .[Hor62] Alfred Horn. Eigenvalues of sums of Hermitian matrices. Pacific Journal ofMathematics , 12(1):225–241, 1962.[JV19] Asger Kjærulff Jensen and P´eter Vrana. The asymptotic spectrum of LOCCtransformations.
IEEE Transactions on Information Theory , 66(1):155–166,Jan 2019. arXiv:1807.05130arXiv:1807.05130 , doi:10.1109/TIT.2019.2927555doi:10.1109/TIT.2019.2927555 .[Kly98] Alexander A Klyachko. Stable bundles, representation theory and Her-mitian operators. Selecta Mathematica, New Series , 4(3):419–445, 1998. doi:10.1007/s000290050037doi:10.1007/s000290050037 .[Kly04] Alexander Klyachko. Quantum marginal problem and representations of thesymmetric group. 2004. arXiv:quant-ph/0409113arXiv:quant-ph/0409113 .[KW01] M Keyl and RF Werner. Estimating the spectrum of a density opera-tor.
Physical Review A , 64(5):052311, 2001. arXiv:quant-ph/0102027arXiv:quant-ph/0102027 , doi:10.1103/PhysRevA.64.052311doi:10.1103/PhysRevA.64.052311 .[Lid82] Boris Viktorovich Lidskii. Spectral polyhedron of a sum of two hermi-tian matrices. Functional Analysis and Its Applications , 16:139–140, 1982. doi:10.1007/BF01081633doi:10.1007/BF01081633 .[MA99] Neri Merhav and Erdal Arikan. The Shannon cipher system with a guess-ing wiretapper.
IEEE Transactions on Information Theory , 45(6):1860–1866,1999. doi:10.1109/18.782106doi:10.1109/18.782106 .[PV07] Martin B Plenio and Shashank Virmani. An introduction to entangle-ment measures.
Quantum Information & Computation , 7(1):1–51, Jan-uary 2007. URL: http://dl.acm.org/citation.cfm?id=2011706.2011707http://dl.acm.org/citation.cfm?id=2011706.2011707 , arXiv:quant-ph/0504163arXiv:quant-ph/0504163 , doi:10.26421/QIC7.1-2doi:10.26421/QIC7.1-2 .[Sha11] Ofer Shayevitz. On R´enyi measures and hypothesis testing. In , pages 894–898.IEEE, 2011. doi:10.1109/ISIT.2011.6034266doi:10.1109/ISIT.2011.6034266 .[Str88] Volker Strassen. The asymptotic spectrum of tensors. Jour-nal f¨ur die reine und angewandte Mathematik , 384:102–152, 1988. doi:10.1515/crll.1988.384.102doi:10.1515/crll.1988.384.102 .[Str91] Volker Strassen. Degeneration and complexity of bilinear maps:some asymptotic spectra.
J. Reine Angew. Math , 413:127–180, 1991. doi:10.1515/crll.1991.413.127doi:10.1515/crll.1991.413.127 .[Vid00] Guifr´e Vidal. Entanglement monotones.
Journal of Mod-ern Optics , 47(2-3):355–376, 2000. arXiv:quant-ph/9807077arXiv:quant-ph/9807077 , doi:10.1080/09500340008244048doi:10.1080/09500340008244048 .[VW02] Guifr´e Vidal and Reinhard F Werner. Computable measure of entangle-ment. Physical Review A , 65(3):032314, 2002. arXiv:quant-ph/0102117arXiv:quant-ph/0102117 , doi:10.1103/PhysRevA.65.032314doi:10.1103/PhysRevA.65.032314 .24WDGC13] Michael Walter, Brent Doran, David Gross, and Matthias Christandl.Entanglement polytopes: multiparticle entanglement from single-particleinformation. Science , 340(6137):1205–1208, 2013. arXiv:1208.0365arXiv:1208.0365 , doi:10.1126/science.1232957doi:10.1126/science.1232957 .[YHH +
09] Dong Yang, Karol Horodecki, Michal Horodecki, Pawel Horodecki, JonathanOppenheim, and Wei Song. Squashed entanglement for multipartite statesand entanglement measures based on the mixed convex roof.
IEEE Trans-actions on Information Theory , 55(7):3375–3387, 2009. arXiv:0704.2236arXiv:0704.2236 , doi:10.1109/TIT.2009.2021373doi:10.1109/TIT.2009.2021373 .[YHW08] Dong Yang, Micha(cid:32)l Horodecki, and ZD Wang. An additive and oper-ational entanglement measure: conditional entanglement of mutual infor-mation. Physical review letters , 101(14):140501, 2008. arXiv:0804.3683arXiv:0804.3683 , doi:10.1103/PhysRevLett.101.140501doi:10.1103/PhysRevLett.101.140501doi:10.1103/PhysRevLett.101.140501doi:10.1103/PhysRevLett.101.140501