Almost-iid information theory

Almost-iid information theory Giulia Mazzola 1 , Da vid Sutter 2 , and Renato Renner 1 1 Institute for The or etic al Physics, ETH Zurich 2 IBM Rese ar ch Eur op e – Zurich Abstract Information-theoretic tec hniques are based on the assumption that resources are w ell c haracter- ized by indep enden t and identically distributed (iid) states. This assumption cannot b e justified op erationally , since, for example, correlations betw een subsequen t systems emitted b y a source cannot be detected b y any practical tomographic proto col. Op erationally motiv ated symmetry assumptions still imply , via de Finetti theorems, that the resources are describ ed by almost-iid states. This raises the question: Are almost-iid resources as effectiv e as p erfect iid resources for information-pro cessing tasks? Here w e address this question and prov e that the conditional en- trop y of almost-iid states asymptotically coincides with that of iid states. As an application, this implies that squashed entanglemen t is robust for almost-iid states, asymptotically matc hing its v alue on iid states. 1 In tro duction A common assumption in physics is that the same exp eriment can b e rep eated many times indep en- den tly . More precisely , one often assumes that the outcomes X 1 , . . . , X n + k of running the exp eriment n + k times are described b y indep endent and iden tically distributed (iid) random v ariables. In tech- nical terms, this means that their joint distribution P X n + k 1 is factorized, i.e., P X n + k 1 = ( P X ) n + k . The Italian mathematician Bruno de Finetti cautioned that the “most common and misleading error” in probabilit y theory is treating the iid assumption as fundamen tal [ 19 ]. He pro vided a mathematically con vincing argument to op erationally justify the iid assumption, relating it to the p erm utation inv ari- ance of the outcomes [ 18 , 30 ]. F or many realistic systems, p ermutation in v ariance follows from natural assumptions suc h as the indistinguishabilit y of subsystems (see [ 34 ] for a more detailed discussion). 1 Supp ose that we hav e a source that generates random v ariables X n + k 1 , where we assume that the underlying joint distribution P X n + k 1 is p ermutation-in v arian t and that each random v ariable tak es v alues in the set X . When selecting n of these n + k random v ariables, i.e. ignoring k random v ariables, de Finetti’s theorem [ 20 ] shows that there exists a probability measure µ on the set of distributions on X such that     P X n 1 − Z ( Q X ) n µ (d Q )     1 ≤ 2 dn n + k , (1) where ∥·∥ 1 denotes the ℓ 1 -norm and d = |X | . This result justifies that when ignoring k random v ariables the remaining ones are approximately a conv ex combination of iid random v ariables. De Finetti theorems hav e b een generalized to the quan tum case. In [ 12 ] a quantum de Finetti theorem has b een obtained as an implication of the classical result in the case where n = ∞ . This yields a result that is applicable under the assumption of having an infinite num ber of samples. Later in [ 25 , 13 ] quantum de Finetti theorems hav e b een deriv ed that work for a finite num ber of samples — analogous to Equation (1) . F or an y state | Φ ( n + k ) ⟩ on H ⊗ n + k that is symmetric (i.e. inv ariant under 1 Alternatively , p ermutation inv ariance can also b e enforced by randomly p erm uting the subsystems. 1 p erm utations of the subsystems), there exists a probabilit y measure µ on the unit sphere B ( H ) := {| θ ⟩ ∈ H : ∥ θ ∥ = 1 } suc h that     tr k [ | Φ ( n + k ) ⟩ ⟨ Φ ( n + k ) | ] − Z | θ ⟩ ⟨ θ | ⊗ n µ (d θ )     1 ≤ 2 d 2 n n + k =: ε d ( n, k ) , (2) where ∥·∥ 1 is the trace norm, tr k [ · ] denotes the partial trace of k subsystems, and d = dim( H ). It is w orth men tioning that the results stated in Equations (1) and (2) are essentially tight [ 20 , 13 ]. The main obstacle with these de Finetti results (classical and quan tum) is that they require k to b e large. Sp ecifically , if we w an t the error term ε d ( n, k ) to v anish in the limit n → ∞ , w e need k to b e sup erlinear in n , i.e. k = ω ( n ). 2 Therefore, w e need to ignore a large fraction of our data. This ma y be justified in certain scenarios where the num b er of data is by default h uge 3 , how ever, it can b e prohibitiv e in other settings. 4 A second, but often less drastic, disadv antage is that, by c hoosing k = p oly( n ), the error term ε d ( n, k ) is v anishing at a slow rate of order 1 / p oly ( n ). The insigh t of [ 33 , 34 ] w as that w e can o v ercome these limitations when relaxing the iid assumption. More precisely , the exp onential de Finetti the or em [ 33 , 34 ] shows that there exist a probabilit y measure ν on the unit sphere B ( H ) and a family {| Ψ ( n ) r,θ ⟩} θ of almost-iid states in | θ ⟩ with a defect of size r such that     tr k [ | Φ ( n + k ) ⟩ ⟨ Φ ( n + k ) | ] − Z | Ψ ( n ) r,θ ⟩ ⟨ Ψ ( n ) r,θ | ν (d θ )     1 ≤ 3 k d exp  − k ( r + 1) n + k  =: ε ′ d ( n, k ) . (3) The precise statement is given in Theorem 3.1 . Almost-iid states in | θ ⟩ are sup erp ositions of states that are equal to | θ ⟩ ⊗ n − r on n − r subsystems and arbitrary on the remaining r subsystems. Note that in each elemen t of the sup erp osition the p ositions of the defects ma y b e different. Usually , the n um b er of defects is sublinear in n , i.e. r = o ( n ). The precise definition of almost-iid states is giv en in Section 2 . Equation (3) has a drastically b etter parameter scaling compared to Equation (2) . It allo ws us to c hoose k sublinear in n , i.e. k = o ( n ), and still ha v e an error term that v anishes in the limit n → ∞ — even at an exp onen tial rate. T o see this, note that, for example, by choosing k = n 3 4 and r = √ n , w e obtain ε ′ d ( n, k ) = 3 n 3 d 4 exp( − n 5 / 4 + n 3 / 4 n + n 3 / 4 ) = O (exp( − n 1 4 )). While the exp onential de Finetti theorem justifies the imp ortance of almost-iid states, it is natural to ask if these states are also relev an t in a purely classical scenario. One may define almost-iid distributions as a conv ex combination of an iid distribution on n − r subsystems and an arbitrary distribution on the remaining r subsystems, where the p osition of the defects can b e differen t in eac h element of the con v ex combination. F or such distributions, w e show that no classical equiv alen t to Equation (3) exists. In other w ords, it is not p ossible to appro ximate a classical permutation- in v ariant distribution by a conv ex combination of almost-iid distributions for k sublinear in n . This is made precise in Section 3 . The lack of a classical exponential de Finetti theorem shows that almost-iid states are considerably more p ow erful than classical almost-iid distributions. This is due to the allow ed sup erp ositions which can contain long-range entanglemen t. F or that reason, almost-iid states are more complicated to ana- lyze. In particular, it is unclear if certain (robust) applications could b ehav e differently for almost-iid and iid states. How ever, it is exp ected that they should not b ehav e differently for practical purp oses. Quan tum tomography , which is used to infer the characteristics of a system, cannot distinguish b e- t w een almost-iid and iid states for practically feasible scenarios. W e refer to Prop osition 2.7 for a mathematically rigorous statemen t. Giv en the operational relev ance of almost-iid states (ensured by the exp onential de Finetti theorem), it is crucial to understand which applications b ehav e asymptotically equally under almost-iid and iid states. In this work, we introduce the definition of mixed almost-iid states and pro v e that for 2 Note that, by definition, f ( n ) = ω ( g ( n )) ⇐ ⇒ g ( n ) = o ( f ( n )). Hence, k = ω ( n ) ⇐ ⇒ n = o ( k ), or in other words, lim n →∞ n k = 0. 3 F or example, random samples of a coin toss, which in principle can be repeated arbitrarily many times. 4 F or example, in the setting of quan tum key distribution. 2 conditional en tropies, there is no asymptotic difference betw een almost-iid and iid states. F or n ∈ N and ρ A n 1 B n 1 a mixed almost-iid state in σ AB with a defect of size r = o ( n ), w e sho w that 1 n H ( A n 1 | B n 1 ) ρ = H ( A | B ) σ + o ( n ) n , (4) where H ( A | B ) σ := H ( AB ) σ − H ( B ) σ is the conditional en trop y and H ( B ) σ := − tr[ σ B log σ B ] for σ B = tr A [ σ AB ] is the von Neumann entrop y . The exact result is giv en in Theorem 4.1 . T o prov e our results, we use information-theoretic tools that ha ve been developed to go b eyond the traditionally assumed iid structure [ 33 , 17 , 36 ]. W e conclude in Section 5 with a discussion of whether p opular entanglemen t measures asymptot- ically coincide for almost-iid and iid states. W e prov e that this holds for the squashed entanglemen t (see Corollary 5.1 ); ho w ev er, it remains an op en question for other entanglemen t measures such as the distillable en tanglemen t, the en tanglemen t cost, and the relative en trop y of entanglemen t. 2 Almost-iid states In this section, we formally define almost-iid states and discuss some of their prop erties. Histori- cally [ 33 ], almost-iid states w ere introduced as pure and symmetric states with additional structure, as explained in the following. This family of states app ears, for example, in exp onential de Finetti theorems suc h as those presented in Equation (3) . Ho w ever, we will relax the definition from [ 33 ] to capture more general families of states, including mixed ones, that hav e a similar almost-iid structure. Let S n b e the set of p ermutations on { 1 , ..., n } and let S( H ) denote the set of density matrices on a Hilb ert space H whose dimension is denoted by d . The symmetric subspace is given by Sym n ( H ) := span {| ϕ ⟩ ⊗ n : | ϕ ⟩ ∈ H} . F or a fixed | θ ⟩ , let V ( H ⊗ n , | θ ⟩ ⊗ m ) := { π ( | θ ⟩ ⊗ m ⊗| Ω ( n − m ) ⟩ ) : π ∈ S n , | Ω ( n − m ) ⟩ ∈ H ⊗ n − m } and Sym n ( H , | θ ⟩ ⊗ m ) := Sym n ( H ) ∩ span V ( H ⊗ n , | θ ⟩ ⊗ m ) . (5) Let n, r ∈ N such that r ≤ n and | θ ⟩ ∈ H . In [ 33 ], an  n r  - almost-iid state in | θ ⟩ was defined as a pure state | Ψ ( n ) ⟩ ∈ Sym n ( H , | θ ⟩ ⊗ n − r ). Ho w ev er, for pure states outside the symmetric subspace or mixed states, this definition needs to b e relaxed in order to capture states that intuitiv ely ha v e an almost-iid structure suc h as those men tioned in Example 2.2 b elo w. W e therefore next in tro duce a no v el definition for mixed almost-iid states. Definition 2.1 (Almost-iid states) . Let H A b e a Hilbert space, σ A ∈ S( H A ), and n, r ∈ N suc h that r ≤ n . Then, ρ A n 1 ∈ S( H ⊗ n A ) is called a  n r  - almost-iid state in σ A if there exists a purification | θ ⟩ AE of σ A and an extension ρ A n 1 E n 1 of ρ A n 1 suc h that (i) ρ A n 1 E n 1 is p erm utation-in v ariant with resp ect to ( A i , E i ) ↔ ( A j , E j ); (ii) supp( ρ A n 1 E n 1 ) ⊆ span V ( H ⊗ n AE , | θ ⟩ ⊗ n − r AE ). W e then write ρ A n 1 ∈ S n ( H A , σ ⊗ n − r A ). Here, supp( X ) denotes the supp ort of a linear operator X and is defined as supp( X ) = k er( X ) ⊥ . W e next discuss a few p edagogical examples of mixed almost-iid states. 3 Example 2.2. A few simple instances of almost-iid states include: (a) T ensor p ow er states ρ A n 1 = σ ⊗ n A . These are almost-iid states with defect size r = 0, i.e. ρ A n 1 ∈ S n ( H A , σ ⊗ n A ). (b) Conv ex combinations of tensor p ow er states with a small n um ber of defects, i.e., states of the form ρ A n 1 = 1 n ! P π ∈S n π ( σ ⊗ n − r A ⊗ ω A r 1 ) π † , where ω A r 1 denotes an arbitrary densit y op erator of size r representing the defects. W e hav e ρ A n 1 ∈ S n ( H A , σ ⊗ n − r A ). T o see this, let | θ ⟩ AE and | ω ⟩ A r 1 E r 1 b e purifications of θ A and ω A r 1 , resp ectiv ely . Then the extension ρ A n 1 E n 1 = 1 n ! X π ∈S n π  | θ ⟩ ⟨ θ | ⊗ n − r AE ⊗ | ω ⟩ ⟨ ω | A r 1 E r 1  π † (6) of ρ A n 1 clearly satisfies prop erty ( i ) since it is permutation-in v ariant. It also satisfies the prop ert y ( ii ) as it can b e written as ρ A n 1 E n 1 = P i,j ∈T β i,j | Ψ i ⟩⟨ Ψ j | for β i,j = 1 n ! δ i,j and | Ψ i ⟩ ∈ V ( H ⊗ n AE , | θ ⟩ ⊗ n − r AE ) for all i ∈ T . (c) The an tisymmetric Bell state ρ A 2 1 = | Ψ − ⟩ ⟨ Ψ − | A 2 1 with | Ψ − ⟩ A 2 1 := 1 √ 2 ( | 0 ⟩| 1 ⟩ A 2 1 − | 1 ⟩| 0 ⟩ A 2 1 ) is a  2 1  -almost-iid state in | 0 ⟩ A . W e wan t to emphasize that the class of almost-iid states contains man y more families than the ones presented in Example 2.2 . In particular, it allows for sup erp ositions of pro duct states with a small n um ber of defects. These sup erp ositions are crucial for an exp onential de Finetti theorem as in Equation (3) to hold. On the other hand, classical almost-iid distributions ha v e a simpler structure as they do not hav e superp ositions and hence are just con v ex combinations of almost-pro duct distributions. This is discussed in Section 3 . Remark 2.3. A more restrictive definition of mixed almost-iid states was in tro duced in [ 9 ] by calling ρ A n 1 ∈ S( H ⊗ n A ) an  n r  - almost-iid state in σ A if there exists a purification | Ψ ( n ) ⟩ A n 1 E n 1 suc h that | Ψ ( n ) ⟩ ∈ Sym n ( H A ⊗ H E , | θ ⟩ ⊗ n − r AE ) for some purification | θ ⟩ AE of σ A . Such states clearly satisfy the assumptions of Definition 2.1 . Ho w ev er, Definition 2.1 is strictly broader as it includes states that do not meet the definition giv en in [ 9 ]. One such example is giv en b y Example 2.2 ( c ) as explained in App endix A . W e call ρ A n 1 a  n r  - gener alize d-almost-iid-state in σ A if it satisfies ( ii ) but not necessarily ( i ). W e then write ρ A n 1 ∈ ¯ S n ( H A , σ ⊗ n − r A ). T rivially , we hav e S n ( H A , σ ⊗ n − r A ) ⊆ ¯ S n ( H A , σ ⊗ n − r A ). With the follo wing trace-preserving completely p ositiv e map PERM : X 7→ 1 n ! X π ∈S n π X π † , (7) whic h symmetrizes the input, it is p ossible to conv ert a generalized-almost-iid-state into an almost-iid state in the sense that ρ A n 1 ∈ ¯ S n ( H A , σ ⊗ n − r A ) = ⇒ PERM( ρ A n 1 ) ∈ S n ( H A , σ ⊗ n − r A ) . (8) 4 Remark 2.4 (Properties of almost-iid states) . Let ρ A n 1 ∈ S n ( H A , σ ⊗ n − r A ) b e a mixed almost- iid state according to Definition 2.1 . Then it satisfies the following prop erties: (a) F or any purification | θ ⟩ AE of σ A there exists an extension ρ A n 1 E n 1 of ρ A n 1 that satisfies ( i ) and ( ii ). See Lemma B.1 . (b) ρ A n 1 is p erm utation-in v ariant. (c) There exists an orthonormal basis {| Ψ t ⟩} t ∈T of span V ( H ⊗ n AE , | θ ⟩ ⊗ n − r ) with vectors | Ψ t ⟩ ∈ V ( H ⊗ n AE , | θ ⟩ ⊗ n − r ) and with |T | ≤  n r  d r AE ≤ 2 nh ( r/n ) d r AE , (9) where h ( x ) := − x log x − (1 − x ) log (1 − x ) for x ∈ [0 , 1] is the binary entrop y function. With resp ect to this basis, condition ( ii ) can b e rewritten as ρ A n 1 E n 1 = X i,j ∈T β i,j | Ψ i ⟩⟨ Ψ j | , (10) for co efficien ts β i,j ∈ C with β i,i ∈ [0 , 1] and P i ∈T β i,i = 1. See Lemma B.2 . (d) ρ A n + m 1 ∈ S n + m ( H A , σ ⊗ n + m − r A ) implies ρ A n 1 ∈ S n ( H A , σ ⊗ n − r A ) for any m, n ∈ N . See Lemma B.3 . (e) ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) implies ρ A n 1 ∈ S n ( H A , σ ⊗ n − r A ). This follows directly from the definition of mixed almost-iid states. (f ) F or any ρ A n 1 ∈ ¯ S n ( H A , σ ⊗ n − r A ) we hav e for any s ∈ N that ( ρ A n 1 ) ⊗ s ∈ ¯ S ns ( H A , σ ⊗ ns − rs A ). See Lemma B.4 . (g) The set of almost-iid states is con v ex. This follo ws directly from the definition of mixed almost-iid states. In the analysis of almost-iid states, the represen tation in Equation (10) is particularly useful b ecause it exploits the pro duct structure of the space V ( H ⊗ n C E , | θ ⟩ ⊗ n − r C E ). How ev er, it is sometimes preferable to go to a purified description, i.e., to w ork with a purification of Equation (10) . The following lemma establishes that the underlying structure of V ( H ⊗ n C E , | θ ⟩ ⊗ n − r C E ) p ersists even in this purified form ulation. Lemma 2.5 (Purification in a preferred basis) . L et | Θ ⟩ AH ∈ H AH for some c omp osite Hilb ert sp ac e H AH , and let V A ⊆ H A b e any subset of ve ctors such that span V A admits an orthonormal b asis  | Ψ j ⟩ A  j ∈T with ve ctors | Ψ j ⟩ A ∈ V A ∀ j ∈ T . Then, the fol lowing ar e e quivalent: (i) supp  tr H  | Θ ⟩ ⟨ Θ | AH  ⊆ span V A (ii) | Θ ⟩ AH = P j ∈T α j | Ψ j ⟩ A | ˜ h j ⟩ H for c o efficients α j ∈ C and normalize d ve ctors | ˜ h j ⟩ H ∈ H H for al l j ∈ T . In addition, if | Θ ⟩ AH is normalize d and satisfies ( ii ) , we have 1 = ⟨ Θ | Θ ⟩ AH = P j ∈T | α j | 2 . Pr o of. ( i ) = ⇒ ( ii ): By the Schmidt decomposition, there exist orthonormal sets of v ectors  | χ k ⟩ A  k ⊂ H A and  | ϕ k ⟩ H  k ⊂ H H , and scalars γ k ≥ 0 suc h that | Θ ⟩ AH = P k γ k | χ k ⟩ A | ϕ k ⟩ H . T aking the partial 5 trace, w e obtain tr H  | Θ ⟩ ⟨ Θ | AH  = X m X k,l γ k γ l | χ k ⟩⟨ χ l | A ⟨ ϕ m | ϕ k ⟩ H ⟨ ϕ l | ϕ m ⟩ H = X k γ 2 k | χ k ⟩ ⟨ χ k | A . (11) By assumption ( i ), it follo ws that | χ k ⟩ ∈ span V A for all k such that γ k  = 0. (If, b y contradiction, there w ere | χ l ⟩ / ∈ span V A for some l with γ l  = 0, then w e would hav e tr H  | Θ ⟩ ⟨ Θ | AH  | χ l ⟩ = γ 2 l | χ l ⟩  = 0 = ⇒ supp( . . . ) ⊆ span V A . ) Therefore, w e may write | χ k ⟩ A = P j ∈T c ( k ) j | Ψ j ⟩ A for some co efficients c ( k ) j ∈ C ∀ j dep ending on k . With that, we find | Θ ⟩ AH = X k γ k | χ k ⟩ A | ϕ k ⟩ H = X k γ k X j ∈T c ( k ) j | Ψ j ⟩ A | ϕ k ⟩ H = X j ∈T | Ψ j ⟩ A  X k c ( k ) j γ k | ϕ k ⟩ H  (12) = X j ∈T | Ψ j ⟩ A | h j ⟩ H = X j ∈T α j | Ψ j ⟩ A | ˜ h j ⟩ H , (13) where we hav e introduced co efficien ts α j ≥ 0 to normalize the vector | h j ⟩ H := P k c ( k ) j γ k | ϕ k ⟩ H (with the con v en tion that α j = 0 and | ˜ h j ⟩ H are normalized but arbitrary if | h j ⟩ H = 0). ( ii ) = ⇒ ( i ): F or any orthonormal basis  | ϕ m ⟩ H  k of H H , w e simply ev aluate tr H  | Θ ⟩ ⟨ Θ | AH  = X i,j ∈T α i α ∗ j X m ⟨ ϕ m | ˜ h i ⟩ H ⟨ ˜ h j | ϕ m ⟩ H | Ψ i ⟩⟨ Ψ j | A , (14) whic h implies supp  tr H  | Θ ⟩ ⟨ Θ | AH  ⊆ span V A since | Ψ j ⟩ A ∈ V A ∀ j ∈ T . In addition, if | Θ ⟩ AH is normalized, w e find 1 = ⟨ Θ | Θ ⟩ AH = P j ∈T | α j | 2 since the v ectors  | Ψ j ⟩ A  j ∈T are orthonormal and the v ectors | ˜ h j ⟩ H are normalized for eac h j ∈ T . ■ An imp ortant prop erty of almost-iid states is that when tracing out man y subsystems, we obtain a state that is close to an iid state. Prop osition 2.6. L et n, r , s ∈ N such that r, s ≤ n , σ A ∈ S( H A ) , and ρ A n 1 ∈ S n ( H A , σ ⊗ n − r A ) . Then   tr n − s [ ρ A n 1 ] − σ ⊗ s A   1 ≤ 4 r r s n . (15) Pr o of. Let ˜ n := ⌊ n s ⌋ s ≤ n and define ρ A ˜ n 1 := tr n − ˜ n [ ρ A n 1 ]. Let | θ ⟩ AE b e a purification of σ A . By assumption ρ A n 1 ∈ S n ( H A , σ ⊗ n − r A ) which according to Remark 2.4 implies ρ A ˜ n 1 ∈ S ˜ n ( H A , σ ⊗ ˜ n − r A ). Hence, due to the definition of this vector space, there exists a family {| Ψ t ⟩} t ∈T of orthonormal v ectors from V ( H ⊗ ˜ n AE , | θ ⟩ ⊗ ˜ n − r AE ) suc h that ρ A ˜ n 1 E ˜ n 1 = X t,t ′ ∈T β t,t ′ | Ψ t ⟩⟨ Ψ t ′ | , (16) with P t ∈T β t,t = 1, where ρ A ˜ n 1 E ˜ n 1 is an extension of ρ A ˜ n 1 . Now, we may in terpret | Ψ t ⟩ as a state on the ⌊ n s ⌋ blo cks. Since | Ψ t ⟩ ∈ V ( H ⊗ ˜ n AE , | θ ⟩ ⊗ ˜ n − r AE ), we know that | Ψ t ⟩ is at least in ⌊ n s ⌋ − r blo cks of the form | θ ⟩ ⊗ s and in the remaining r blocks arbitrary . F or any t ∈ T , let F t denote the set of block indices i ∈ { 1 , . . . , ⌊ n s ⌋} on whic h | Ψ t ⟩ is not of the form | θ ⟩ ⊗ s and note that |F t | ≤ r . (17) 6 Then, for an y blo ck sp ecified by i , the total weigh t of the vectors | Ψ t ⟩ that deviate from | θ ⟩ ⊗ s for that blo c k is giv en b y w i := X t ∈T β t,t δ i ∈F t . (18) Summing o v er all blo c ks yields ⌊ n s ⌋ X i =1 w i Eq. (18) = ⌊ n s ⌋ X i =1 X t ∈T β t,t δ i ∈F t = X t ∈T β t,t ⌊ n s ⌋ X i =1 δ i ∈F t = X t ∈T β t,t |F t | Eq. (17) ≤ X t ∈T β t,t r = r , (19) where the final step uses P t ∈T β t,t = 1. Equation (19) implies that w j ≤ r ⌊ n s ⌋ for some j ∈ n 1 , . . . , j n s ko . (20) F or a fixed j ∈  1 , . . . ,  n s  , let Π b e the pro jector onto the subspace spanned by {| Ψ t ⟩ : j ∈ F t } , i.e. Π = X t ∈T s.t. j ∈F t | Ψ t ⟩ ⟨ Ψ t | and Π ⊥ = X t ∈T s.t. j ∈F t | Ψ t ⟩ ⟨ Ψ t | . (21) Note that Π Π ⊥ = 0 and Π + Π ⊥ = id span V ( H ⊗ ˜ n AE , | θ ⟩ ⊗ ˜ n − r AE ) . The op erator ρ ′ A ˜ n 1 E ˜ n 1 := Π ρ A ˜ n 1 E ˜ n 1 Π † is subnormalized, i.e., tr[ ρ ′ ] = tr[Π ρ Π † ] ≤ 1 , (22) whic h can b e seen, for example, via H¨ older’s inequality [ 35 , Prop osition 2.5]. The fidelity b etw een tw o densit y op erators is defined as F ( ρ, σ ) :=   √ ρ √ σ   2 1 . Hence, for any ω ≥ 0 we hav e F ( ω , ρ ′ ) =    √ ω p ρ ′    2 1 = tr[ ρ ′ ]      √ ω s ρ ′ tr[ ρ ′ ]      2 1 Eq. (22) ≤      √ ω s ρ ′ tr[ ρ ′ ]      2 1 = F  ω , ρ ′ tr[ ρ ′ ]  . (23) Using the p erm utation in v ariance of ρ A ˜ n 1 E ˜ n 1 w e obtain F ( ρ A s 1 E s 1 , | θ ⟩ ⟨ θ | ⊗ s ) Eq. (23) ≥ F ( ρ A j s ( j − 1) s +1 E j s ( j − 1) s +1 , ρ ′ A j s ( j − 1) s +1 E j s ( j − 1) s +1 ) DPI ≥ F ( ρ A ˜ n 1 E ˜ n 1 , ρ ′ A ˜ n 1 E ˜ n 1 ) , (24) where the final step uses the data-pro cessing inequality for the fidelity [ 21 , Lemma B.4]. By definition of the fidelit y w e ha v e q F ( ρ A ˜ n 1 E ˜ n 1 , ρ ′ A ˜ n 1 E ˜ n 1 ) =    √ ρ p ρ ′    1 = tr h q √ ρρ ′ √ ρ i = tr h q √ ρ Π ρ Π † √ ρ i = tr[Π ρ ] . (25) By definition of Π w e ha v e tr[Π ρ ] = 1 − tr[Π ⊥ ρ ] Eq. (21) = 1 − X t ∈T ,j ∈F t ⟨ Ψ t | ρ | Ψ t ⟩ Eq. (16) = 1 − X t ∈T ,j ∈F t β t,t Eq. (18) = 1 − w j . (26) The F uchs-v an der Graaf inequality [ 22 ] then gives   ρ A s 1 E s 1 − | θ ⟩ ⟨ θ | ⊗ s   1 ≤ 2 q 1 − F ( ρ A s 1 E s 1 , | θ ⟩ ⟨ θ | ⊗ s ) (27) Eqs. (24) to (26) ≤ 2 q 1 − (1 − w j ) 2 (28) 7 Eq. (20) ≤ 2 √ 2 r r ⌊ n s ⌋ . (29) Recalling that the trace distance is con tractive under the partial trace [ 43 , Theorem 8.16] implies   ρ A s 1 − σ ⊗ s A   1 ≤   ρ A s 1 E s 1 − | θ ⟩ ⟨ θ | ⊗ s   1 Eq. (29) ≤ 2 √ 2 r r ⌊ n s ⌋ . (30) T o conclude the pro of of the assertion, note that for q = ⌊ n s ⌋ w e ha v e sq = s j n s k ≤ n ≤ s ( q + 1) . (31) Hence p r /q p r s/n = r n q s Eq. (31) ≤ s s ( q + 1) q s = r 1 + 1 q ≤ √ 2 , (32) where the final step uses q ≥ 1. Putting everything together yields   ρ A s 1 − σ ⊗ s A   1 Eq. (30) ≤ 2 √ 2 r r ⌊ n s ⌋ Eq. (32) ≤ 4 r r s n . (33) ■ Another prop erty concerns the statistics of almost-iid and iid states when performing an iid mea- suremen t (as is used in a tomography pro cedure, for example) [ 33 , Theorem 4.5.2]. Let σ ∈ S( H ) and assume that n indep endent measuremen ts with resp ect to a POVM M = { M x } x ∈X are p erformed on n iid copies of σ , giving the outcomes x = ( x 1 , . . . , x n ) with x i ∈ X ∀ i . The outcomes x can b e c haracterized b y a fr e quency distribution (or typ e ) λ x , which is defined as the follo wing probabilit y distribution on X , λ x ( y ) := 1 n   { i : x i = y }   , (34) for all y ∈ X . In tuitiv ely , λ x ( y ) corresp onds to the relativ e num ber of o ccurrences of y in the sequence of outcomes x = ( x 1 , . . . , x n ). F or large v alues of n , it follows by the la w of large num bers that the frequency distribution λ x is close to the probability distribution P X defined b y P X ( y ) := tr( M y σ ). Prop osition 2.7 sho ws that a similar statemen t holds when performing n indep endent measuremen ts on a  n r  -almost-iid state in σ for small enough v alues of r . This prov es that almost-iid and iid states cannot be distinguished b y any practically feasible measurement. This is discussed in more detail in [ 29 ]. Prop osition 2.7 (Statistics of almost-iid states) . L et n, r ∈ N such that r ≤ 1 2 n , σ ∈ S( H ) , and ρ ( n ) ∈ S n ( H , σ ⊗ n − r ) . L et M = { M x } x ∈X b e a POVM on H , and P X ( x ) = tr[ M x σ ] for al l x ∈ X . Then, for any ε > 0 , we have P x  ∥ λ x − P X ∥ 1 > f ( ε, r, n )  ≤ ε , (35) wher e the pr ob ability is taken over the outc omes x = ( x 1 , . . . , x n ) of the pr o duct me asur ement M ⊗ n applie d to ρ ( n ) , and f ( ε, r, n ) = 2 s log  1 ε  n + |X | n log  n 2 + 1  + h  r n  + 2 r n log( d ) + 2 r n , (36) wher e h ( · ) is the binary entr opy and d = dim H . F urthermor e, if r = o ( n ) , then lim n →∞ f ( ε, r, n ) = 0 . 8 The pro of is giv en in App endix C . 3 Quan tum vs. classical exp onen tial de Finetti theorem In this section, we formally state the quantum exp onential de Finetti theorem [ 34 ] whic h justifies the definition of almost-iid states. W e then sho w that for classical almost-iid distributions, defined as con v ex combinations of pro duct distributions with a small num ber of defects, no classical exp onen tial de Finetti theorem can hold. Theorem 3.1 (Exp onen tial de Finetti [ 34 , Theorem 1]) . L et n, k , r ∈ N and let H b e a d - dimensional Hilb ert sp ac e. F or any | Φ ( n + k ) ⟩ ∈ Sym n + k ( H ) ther e exists a pr ob ability me asur e ν on the unit spher e B ( H ) and a family {| Ψ ( n ) r,θ ⟩} θ of states such that | Ψ ( n ) r,θ ⟩ ∈ Sym n ( H , | θ ⟩ ⊗ n − r ) and     tr k [ | Φ ( n + k ) ⟩ ⟨ Φ ( n + k ) | ] − Z | Ψ ( n ) r,θ ⟩ ⟨ Ψ ( n ) r,θ | ν (d θ )     1 ≤ 3 k d e − k ( r +1) n + k . (37) W e next define almost-iid distributions which is the classical counterpart to almost-iid states. F or a finite alphab et X , let Sym( X n ) denote the set of p ermutation-in v ariant distributions on X n . F urthermore, for a distribution q on X let V ( X n , q m ) := { π ( q m × R X n − m 1 ) : π ∈ S n , R X n − m 1 distribution on X n − m } (38) and Sym( X n , q m ) := Sym( X n ) ∩ conv V ( X n , q m ) . (39) Note that the set Sym( X n , q m ) is considerably simpler compared to Sym n ( H , | θ ⟩ ⊗ m ) since the former set consists of conv ex com binations of pro duct distributions, whereas the latter set con tains sup erp o- sitions of pro duct states. The follo wing prop osition states that no classical v ersion of the quan tum de Finetti theorem can exist where almost-iid states are replaced with almost-iid distributions. Prop osition 3.1 (No classical exponential de Finetti result) . L et n ∈ N , k = o ( n ) , r = o ( n ) , X b e a finite alphab et of size d . Ther e exists P X n + k 1 ∈ Sym( X n + k ) such that it is not p ossible to ap- pr oximate P X n 1 with a pr ob abilistic mixtur e of distributions { Q ( q ) X n 1 } q with Q ( q ) X n 1 ∈ Sym( X n , q n − r ) in the ℓ 1 -norm up to ε d ( n ) such that lim n →∞ ε d ( n ) = 0 . The proof of Prop osition 3.1 is given in App endix D . As men tioned in Equation (1) , if w e are willing to c ho ose k large, more precisely k = ω ( n ), then a classical de Finetti theorem holds. Ho w- ev er, Proposition 3.1 states that this is no longer the case for k = o ( n ), ev en if the error term would decrease at a non-exp onential rate. F urther note that Prop osition 3.1 do es not prohibit the existence of a classical exp onen tial de Finetti theorem if the definition of almost-iid distributions, Equation (39) , is relaxed. F or example, one could define the set V ( X n , q m ) in Equation (38) by only requiring the marginals of the distributions to yield an iid state instead of imp osing a pro duct structure b etw een the iid part and the defects. How ev er, suc h a structure w ould no longer be cov ered by the quan tum v ersion in Definition 2.1 and therefore, ma y p oten tially b e difficult to w ork with. One may still wonder what Theorem 3.1 pro duces if it is applied to a classical symmetric distribu- tion. Can the resulting family of almost-iid states that appro ximates the classical distribution b ecome classical as w ell? T o analyze this, let P X n + k 1 ∈ Sym( X n + k ) b e a symmetric classical distribution. W e 9 can apply Theorem 3.1 to P X n + k 1 b y em b edding it in to a p erm utation-in v ariant density matrix, ρ A n + k 1 = X x n + k 1 P X n + k 1 ( x n + k 1 ) | x 1 ⟩ ⟨ x 1 | · · · | x n + k ⟩ ⟨ x n + k | . (40) Let | Φ ( n + k ) ⟩ A n + k 1 E n + k 1 b e a symmetric purification of ρ A n + k 1 . Consider the measurement channel on the A -system M : Y AE 7→ X x | x ⟩ ⟨ x | A Y AE | x ⟩ ⟨ x | A . (41) Because ρ A n 1 is classical, w e ha v e ρ A n 1 = M ⊗ n ( ρ A n 1 ) = M ⊗ n  tr E n 1 [ ρ A n 1 E n 1 ]  = tr E n 1  M ⊗ n ( ρ A n 1 E n 1 )  (42) = tr E n 1  M ⊗ n (tr k [ | Φ ( n + k ) ⟩ ⟨ Φ ( n + k ) | ])  , (43) where ρ A n 1 E n 1 := tr k [ | Φ ( n + k ) ⟩ ⟨ Φ ( n + k ) | ], and the penultimate step uses that M only acts on the A -system and hence commutes with the partial trace ov er the E -system. Theorem 3.1 sho ws that there exist a probabilit y measure ν on the unit sphere B ( H ) and, for each | θ ⟩ ∈ B ( H ), a family {| Ψ ( n ) r,θ ⟩} θ of states such that | Ψ ( n ) r,θ ⟩ ∈ Sym n ( H , | θ ⟩ ⊗ n − r ), and such that for σ ( n ) θ := tr E n 1 [ M ⊗ n ( | Ψ ( n ) r,θ ⟩ ⟨ Ψ ( n ) r,θ | )] and ¯ ν denoting the induced measure obtained b y taking the partial trace, we ha v e     ρ A n 1 − Z σ ( n ) θ ¯ ν (d θ )     1 Eq. (43) =     tr E n 1  M ⊗ n (tr k [ | Φ ( n + k ) ⟩ ⟨ Φ ( n + k ) | ])  − tr E n 1 h M ⊗ n  Z | Ψ ( n ) r,θ ⟩ ⟨ Ψ ( n ) r,θ | ν (d θ ) i     1 ≤     tr k [ | Φ ( n + k ) ⟩ ⟨ Φ ( n + k ) | ]) − Z | Ψ ( n ) r,θ ⟩ ⟨ Ψ ( n ) r,θ | ν (d θ )     1 (44) Theorem 3.1 ≤ 3 k d e − k ( r +1) n + k , (45) where the second step uses the fact that the trace distance is con tractiv e under trace-preserving completely p ositive maps [ 43 , Theorem 8.16]. Prop osition 3.1 no w implies that the density op erators σ ( n ) θ cannot be given almost-iid distributions in the sense of Equation (39) . Intuitiv ely , this happ ens since the states | θ ⟩ appearing in Equation (44) are not necessarily classical and therefore, the almost-iid structure is not preserv ed after the measurement. 4 Conditional en trop y of almost-iid states In this section, we prov e that the conditional en trop y of almost-iid states asymptotically coincides with the conditional en tropy of iid states. In the pro of, w e develop tec hnical to ols that may be of indep enden t in terest. F or ρ AB ∈ S( H AB ) the conditional en trop y is defined as H ( A | B ) ρ = H ( AB ) ρ − H ( B ) ρ , where H ( B ) ρ := − tr[ ρ B log ρ B ] is the von Neumann entrop y . It is straightforw ard to see that the conditional en trop y is additiv e for iid states, i.e., 1 n H ( A n 1 | B n 1 ) ρ ⊗ n = H ( A | B ) ρ . (46) W e next show that this prop erty is preserved for almost-iid states in the limit n → ∞ . Theorem 4.1. L et σ AB ∈ S( H AB ) and ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) for r = o ( n ) . Then 1 n H ( A n 1 | B n 1 ) ρ = H ( A | B ) σ + o ( n ) n . (47) 10 The main difficulty in proving the assertion of Theorem 4.1 is the fact that almost-iid states are more general than just conv ex mixtures of pro duct states where eac h element in the conv ex sum has a certain num ber of defects (see Example 2.2 ). A look at Definition 2.1 reveals that almost-iid states may con tain sup erp ositions whic h store long range correlations and entanglemen t. Dealing with these sup erp ositions is the main technical challenge in the pro of. W e do this utilizing tools from one- shot information theory such as R´ en yi and smo oth entropies. W e also wan t to emphasize that the sup erp ositions in the definition of almost-iid states are crucial for making the exp onential de Finetti theorem ( Theorem 3.1 ) p ossible. As shown in Prop osition 3.1 , no exp onential de Finetti theorem can exist without sup erp ositions. Before presenting the proof of Theorem 4.1 , whic h is given in Section 4.4 , w e need to define some en tropic quantities. W e note that an alternative proof using differen t techniques that ma y therefore b e of indep enden t in terest is giv en in App endix E . 4.1 Entropic quantities F or α ∈ [1 / 2 , 1) ∪ (1 , ∞ ) the sandwiche d R ´ enyi diver genc e [ 31 , 41 ] is given by D α ( ρ ∥ σ ) := 1 α − 1 log tr  ( σ 1 − α 2 α ρ σ 1 − α 2 α ) α  . (48) F or α = 1 2 w e hav e D 1 2 ( ρ ∥ σ ) = − log F ( ρ, σ ). In the limits α → 1 and α → ∞ the sandwiched R´ enyi div ergence con v erges to the relativ e entrop y D ( ρ ∥ σ ) and the max-relative entrop y [ 33 , 16 ] D max ( ρ ∥ σ ) := inf { λ ∈ R : ρ ≤ 2 λ σ } , (49) resp ectiv ely . F or ρ AB ∈ S( H AB ), the R´ en yi div ergence can b e used to define a conditional R ´ enyi en trop y [ 36 ] H α ( A | B ) ρ := − min σ B ∈ S( H B ) D α ( ρ AB ∥ id A ⊗ σ B ) , (50) whic h conv erges to the conditional entrop y H ( A | B ) ρ for α → 1 and to the conditional min-en trop y H min ( A | B ) ρ for α → ∞ . F or α = 1 / 2 w e obtain the conditional max-en trop y H max ( A | B ) ρ . The trace distance betw een t w o states ρ, σ ∈ S( H ) is giv en b y ∆( ρ, σ ) := 1 2 ∥ ρ − σ ∥ 1 and the purified distance [ 36 ] is defined as P ( ρ, σ ) := p 1 − F ( ρ, σ ). The F uchs-v an der Graaf inequality [ 22 ] implies P ( ρ, σ ) ≥ ∆( ρ, σ ). F or ρ ∈ S( H ) and ε ∈ (0 , 1) define the ε -ball around ρ b y B ε ( ρ ) := { ρ ′ ∈ S( H ) : P ( ρ, ρ ′ ) ≤ ε } . W e then define a smo oth v arian t of the min- and max-entrop y by H ε min ( A | B ) ρ := max ρ ′ ∈B ε ( ρ ) H min ( A | B ) ρ ′ and H ε max ( A | B ) ρ := min ρ ′ ∈B ε ( ρ ) H max ( A | B ) ρ ′ . (51) 4.2 Asymptotic equipartition prop erty for almost-iid states Another statement which can b e pro v en using similar techniques and ma y b e of indep endent interest is a strong asymptotic equipartition prop ert y (AEP) for almost-iid states. T o understand this, let us recall the AEP for iid states [ 37 , 36 ]. This fundamental result ensures that for any densit y operator ρ AB ∈ S( H AB ) and an y ε ∈ (0 , 1) we hav e 1 n H ε min ( A n 1 | B n 1 ) ρ ⊗ n = H ( A | B ) ρ + o ( n ) n and 1 n H ε max ( A n 1 | B n 1 ) ρ ⊗ n = H ( A | B ) ρ + o ( n ) n . (52) Note that Equation (52) is called a str ong AEP as the error term o ( n ) n v anishes in the limit n → ∞ for an y fixed ε ∈ (0 , 1). F urthermore, it is understo o d how fast the error term v anishes for finite v alues of n [ 38 ]. W e sho w that Equation (52) remains v alid when replacing iid states with almost-iid states. 11 Prop osition 4.2 (Strong AEP for almost-iid states) . L et ε ∈ (0 , 1) , σ AB ∈ S( H AB ) , and ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) for r = o ( n ) . Then 1 n H ε min ( A n 1 | B n 1 ) ρ = H ( A | B ) σ + o ( n ) n and 1 n H ε max ( A n 1 | B n 1 ) ρ = H ( A | B ) σ + o ( n ) n . (53) In [ 33 , Theorem 4.4.1] it was shown that the conditional smo oth min-entrop y for almost-iid states, whic h are classical on one subsystem, asymptotically coincides with the conditional entrop y of iid states. This w as crucial to pro v e securit y of quantum key distribution via a de Finetti argumen t. Using the duality of conditional entrop y , the result can b e lifted to the smo oth max-entrop y of almost- iid states [ 44 ]. In addition, for the case of pure almost-iid states, a similar result has b een pro v en based on [ 33 ] in [ 44 , Lemma 11]. Here, the presented pro of of Prop osition 4.2 is more general as it applies for mixed almost-iid states and is also more modular allowing one to distill other results such as Theorem 4.1 . Beyond these results, to the b est of our knowledge, little is kno wn about how en tropic functions b eha v e for almost-iid states. 4.3 Pro of of Proposition 4.2 Let ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ). By definition, there exist a purification | θ ⟩ AB E of σ AB and an extension ρ A n 1 B n 1 E n 1 of ρ A n 1 B n 1 that can b e written as ρ A n 1 B n 1 E n 1 = X t,t ′ ∈T β t,t ′ | Ψ t ⟩⟨ Ψ t ′ | A n 1 B n 1 E n 1 , (54) for a family {| Ψ t ⟩ A n 1 B n 1 E n 1 } t ∈T of orthonormal vectors from V ( H ⊗ n AB E , | θ ⟩ ⊗ n − r AB E ) with β t,t ′ ∈ C satisfying P t ∈T β t,t = 1 and log |T | ≤ n h  r n  + r log d AB E . (55) Let ˜ ρ A n 1 B n 1 E n 1 T := X t ∈T β t,t | Ψ t ⟩ ⟨ Ψ t | A n 1 B n 1 E n 1 | {z } =: ˜ ρ ( t ) ⊗| t ⟩ ⟨ t | T . (56) Lemma 4.3. F or the setting define d ab ove, we have ρ A n 1 B n 1 E n 1 ≤ |T | ˜ ρ A n 1 B n 1 E n 1 , and hence ρ A n 1 B n 1 ≤ |T | ˜ ρ A n 1 B n 1 . (57) Pr o of. The pro of idea is similar to [ 33 , Proof of Lemma 3.1.13] but is based on pinc hing maps and therefore w orks for a more general setup. Consider the pinching map P : Z 7→ X t ∈T | Ψ t ⟩ ⟨ Ψ t | Z | Ψ t ⟩ ⟨ Ψ t | . (58) Using the fact that {| Ψ t ⟩} t ∈T is orthonormal, w e ha v e P ( ρ A n 1 B n 1 E n 1 ) Eq. (54) = X t,k,ℓ ∈T β k,ℓ | Ψ t ⟩ ⟨ Ψ t || Ψ k ⟩⟨ Ψ ℓ || Ψ t ⟩ ⟨ Ψ t | = X t ∈T β t,t | Ψ t ⟩ ⟨ Ψ t | Eq. (56) = ˜ ρ A n 1 B n 1 E n 1 . (59) Hence, w e find ˜ ρ A n 1 B n 1 E n 1 Eq. (59) = P ( ρ A n 1 B n 1 E n 1 ) pinching inequality ≥ 1 |T | ρ A n 1 B n 1 E n 1 . (60) 12 The interested reader can find more information on pinching maps, including a proof of the pinching inequalit y in [ 35 , Lemma 3.5]. Since the partial trace is a completely positive map Equation (60) implies ˜ ρ A n 1 B n 1 ≥ 1 |T | ρ A n 1 B n 1 . (61) ■ Lemma 4.4. L et n, r ∈ N such that r ≤ n , σ AB ∈ S( H AB ) , ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) , d AB = dim H AB , and d A = dim H A . Then 1 n H α ( A n 1 | B n 1 ) ρ ≥ H α ( A | B ) σ − 2 r n log d A − α α − 1  h  r n  + 2 r n log d AB  ∀ α > 1 (62) and 1 n H α ( A n 1 | B n 1 ) ρ ≤ H α ( A | B ) σ + 2 r n log d A + 1 1 − α  h  r n  + 2 r n log d AB  ∀ α ∈ [ 1 2 , 1) . (63) Pr o of. W e start by proving Equation (62) . F or α > 1 and ρ , ˜ ρ defined ab ov e, we hav e α 1 − α log |T | + H α ( A n 1 | B n 1 ) ˜ ρ Eqs. (48) and (50) = max σ ∈ S( H ⊗ n B ) 1 1 − α log tr  ( σ 1 − α 2 α ˜ ρ A n 1 B n 1 |T | σ 1 − α 2 α ) α  (64) Lem. 4.3 ≤ max σ ∈ S( H ⊗ n B ) 1 1 − α log tr  ( σ 1 − α 2 α ρ A n 1 B n 1 σ 1 − α 2 α ) α  (65) Eqs. (48) and (50) = H α ( A n 1 | B n 1 ) ρ , (66) where the inequality step used that the function X 7→ tr[ X α ] is monotone [ 11 , Theorem 2.10]. F ur- thermore, w e ha v e H α ( A n 1 | B n 1 ) ˜ ρ [ 36 , Eq. 5.41] ≥ H α ( A n 1 | B n 1 T ) ˜ ρ (67) [ 36 , Prop. 5.4] = α 1 − α log X t ∈T β t,t exp  1 − α α H α ( A n 1 | B n 1 ) ˜ ρ ( t )  ! (68) ≥ min t ∈T H α ( A n 1 | B n 1 ) ˜ ρ ( t ) , (69) where the final step uses that the logarithm is a quasi-linear function and that β t,t ∈ [0 , 1] with P t ∈T β t,t = 1. Recalling that ˜ ρ ( t ) = | Ψ t ⟩ ⟨ Ψ t | for | Ψ t ⟩ ∈ V ( H ⊗ n AB E , | θ ⟩ ⊗ n − r AB E ) and using the additivity of the R ´ enyi entropies under tensor pro ducts allows us to write for any t ∈ T H α ( A n 1 | B n 1 ) ˜ ρ ( t ) = ( n − r ) H α ( A | B ) σ + H α ( A r 1 | B r 1 ) Ω [ 36 , Lem. 5.11] ≥ ( n − r ) H α ( A | B ) σ − r log d A . (70) Putting ev erything together yields 1 n H α ( A n 1 | B n 1 ) ρ ≥ n − r n H α ( A | B ) σ − r n log d A − 1 n α α − 1 log |T | (71) [ 36 , Lem. 5.11]& Eq. (55) ≥ H α ( A | B ) σ − 2 r n log d A − α α − 1  h  r n  + 2 r n log d AB  , (72) where in the final step, w e used d AB E = d 2 AB . This prov es Equation (62) . 13 The statemen t from Equation (63) follo ws similarly . F or α ∈ [1 / 2 , 1) we hav e α 1 − α log |T | + H α ( A n 1 | B n 1 ) ˜ ρ Eqs. (48) and (50) = max σ ∈ S( H ⊗ n B ) 1 1 − α log tr  ( σ 1 − α 2 α ˜ ρ A n 1 B n 1 |T | σ 1 − α 2 α ) α  (73) Lem. 4.3 ≥ max σ ∈ S( H ⊗ n B ) 1 1 − α log tr  ( σ 1 − α 2 α ρ A n 1 B n 1 σ 1 − α 2 α ) α  (74) Eqs. (48) and (50) = H α ( A n 1 | B n 1 ) ρ , (75) where the inequalit y step used that the function X 7→ tr[ X α ] is monotone [ 11 , Theorem 2.10]. In addition, w e ha v e H α ( A n 1 | B n 1 ) ˜ ρ [ 36 , Eq. 5.96] ≤ H α ( A n 1 | B n 1 T ) ˜ ρ + log |T | (76) [ 36 , Prop. 5.4] = α 1 − α log X t ∈T β t,t exp  1 − α α H α ( A n 1 | B n 1 ) ˜ ρ ( t )  ! + log |T | (77) ≤ max t ∈T H α ( A n 1 | B n 1 ) ˜ ρ ( t ) + log |T | , (78) where the final step uses that the logarithm is a quasi-linear function and that β t,t ∈ [0 , 1] with P t ∈T β t,t = 1. Since ˜ ρ ( t ) = | Ψ t ⟩ ⟨ Ψ t | for | Ψ t ⟩ ∈ V ( H ⊗ n AB E , | θ ⟩ ⊗ n − r AB E ), the additivit y of the R ´ enyi en tropies under tensor pro ducts implies for an y t ∈ T H α ( A n 1 | B n 1 ) ˜ ρ ( t ) = ( n − r ) H α ( A | B ) σ + H α ( A r 1 | B r 1 ) Ω [ 36 , Lem. 5.11] ≤ ( n − r ) H α ( A | B ) σ + r log d A . (79) Com bining Equations (75) , (78) and (79) yields 1 n H α ( A n 1 | B n 1 ) ρ ≤ n − r n H α ( A | B ) σ + r n log d A + 1 1 − α 1 n log |T | (80) [ 36 , Lem. 5.11]& Eq. (55) ≤ H α ( A | B ) σ + 2 r n log d A + 1 1 − α  h  r n  + 2 r n log d AB  , (81) where in the final step w e used that d E = d AB . ■ W e are now equipp ed with all the tools we need to prov e the assertion of Prop osition 4.2 . This will b e done in four steps, by provin g tw o inequalities (direct and conv erse part) for b oth the smo oth min- and max-en trop y . (i) Direct part for smo oth min-entrop y: F or α > 1 consider the error term δ ε ( n, r , α ) := 2 r n log d A + α α − 1  h  r n  + 2 r n log d AB  + 1 n 1 α − 1 log 1 ε 2 + 1 n log 1 1 − ε 2 . (82) W e can write 1 n H ε min ( A n 1 | B n 1 ) ρ [ 10 , Prop. 2.2] ≥ 1 n H α ( A n 1 | B n 1 ) ρ − 1 n 1 α − 1 log 1 ε 2 − 1 n log 1 1 − ε 2 (83) Lem. 4.4 ≥ H α ( A | B ) σ − δ ε ( n, r , α ) (84) [ 37 , Lem. 8] ≥ H ( A | B ) σ − δ ε ( n, r , α ) − 4( α − 1)(log η ) 2 , (85) for a constant η = √ 2 − H min ( A | B ) σ + √ 2 H max ( A | B ) σ + 1. F or a choice α = 1 + 1 / log ( r /n ) and recalling that r = o ( n ) w e see that δ ε ( n, r , α ) = o ( n ) n , (86) 14 where we used that lim x → 0 (log( x ) + 1) h ( x ) = 0 and lim x → 0 (log( x ) + 1) x = 0 . Hence, we obtain 1 n H ε min ( A n 1 | B n 1 ) ρ ≥ H ( A | B ) σ − o ( n ) n . (87) (ii) Direct part for smo oth max-entrop y: F or α ∈ [1 / 2 , 1) consider the error term δ ′ ε ( n, r , α ) := 2 r n log d A + 1 1 − α  h  r n  + 2 r n log d AB  + 1 n α 1 − α log 1 ε . (88) W e can write 1 n H ε max ( A n 1 | B n 1 ) ρ [ 10 , Prop. 2.2] ≤ 1 n H α ( A n 1 | B n 1 ) ρ + 1 n α 1 − α log 1 ε (89) Lem. 4.4 ≤ H α ( A | B ) σ + δ ′ ε ( n, r , α ) (90) [ 37 , Lem. 8] & [ 36 , Prop. 5.7] ≤ H ( A | B ) σ + δ ′ ε ( n, r , α ) + 4(1 − α )(log η ) 2 . (91) Similarly as ab o v e, c ho osing α = 1 − 1 / log ( r /n ) yields δ ′ ε ( n, r , α ) = o ( n ) n and hence 1 n H ε max ( A n 1 | B n 1 ) ρ ≤ H ( A | B ) σ + o ( n ) n . (92) (iii) Conv erse part for smo oth min-entrop y: F or a fixed ε ∈ (0 , 1) consider an arbitrary ε ′ ∈ (0 , 1 − ε ). Then 1 n H ε min ( A n 1 | B n 1 ) ρ [ 36 , Eq. 6.107] ≤ 1 n H ε ′ max ( A n 1 | B n 1 ) ρ + 1 n log 1 1 − ( ε + ε ′ ) 2 Eq. (92) ≤ H ( A | B ) σ + o ( n ) n . (93) (iv) Conv erse part for smo oth max-entrop y: F or a fixed ε ∈ (0 , 1) consider an arbitrary ε ′ ∈ (0 , 1 − ε ). Then 1 n H ε ′ max ( A n 1 | B n 1 ) ρ [ 36 , Eq. 6.107] ≥ 1 n H ε min ( A n 1 | B n 1 ) ρ − 1 n log 1 1 − ( ε + ε ′ ) 2 Eq. (87) ≥ H ( A | B ) σ − o ( n ) n . (94) Com bining Equations (87) and (92) to (94) completes the pro of. ■ 4.4 Pro of of Theorem 4.1 The monotonicit y of the R ´ enyi divergence in α [ 31 ] implies that for α > 1 we hav e 1 n H ( A n 1 | B n 1 ) ρ ≥ 1 n H α ( A n 1 | B n 1 ) ρ (95) Lem. 4.4 ≥ H α ( A | B ) σ − 2 r n log d A − α α − 1  h  r n  + 2 r n log d AB  (96) [ 37 , Lem. 8] ≥ H ( A | B ) σ − 2 r n log d A − α α − 1  h  r n  + 2 r n log d AB  − 4( α − 1)(log η ) 2 , (97) for a constant η = √ 2 − H min ( A | B ) σ + √ 2 H max ( A | B ) σ + 1. Cho osing α = 1 + 1 / log ( r /n ) and recalling that r = o ( n ) yields 5 1 n H ( A n 1 | B n 1 ) ρ ≥ H ( A | B ) σ − o ( n ) n . (98) 5 Note that lim x → 0 (log( x ) + 1) h ( x ) = 0 and lim x → 0 (log( x ) + 1) x = 0. 15 T o see the other direction, note that for ε n = 2 p r n with r = o ( n ) 1 n H ( A n 1 | B n 1 ) ρ chain rule = 1 n n X i =1 H ( A i | A i − 1 1 B n 1 ) ρ (99) SSA [ 27 ] ≤ 1 n n X i =1 H ( A i | B i ) ρ (100) perm. inv. = H ( A 1 | B 1 ) ρ (101) Prop. 2.6 ≤ H ( A | B ) σ + 2 ε n log d A + (1 + ε n ) h  ε n 1 + ε n  (102) = H ( A | B ) σ + o ( n ) n , (103) where the p en ultimate step uses the con tin uit y of en trop y [ 42 ]. Combining Equations (98) and (103) completes the pro of. 5 Robustness of information measures for almost-iid states In this work, we justified the imp ortance of almost-iid states. This prompts the question if almost-iid states are as effectiv e as p erfect iid states for information-pro cessing tasks. T o answer this, it is crucial to understand if certain functionals (that c haracterize specific information-pro cessing tasks) behav e equally or differen tly for almost-iid and p erfect iid states. In Section 4 we hav e seen that the conditional entrop y is robust for almost-iid states in the sense that it asymptotically coincides with the en trop y of iid states. This implies that also the mutual information is robust for almost-iid states. T o make this precise, recall that for a bipartite densit y matrix σ AB the m utual information is defined as I ( A : B ) σ := H ( A ) σ − H ( A | B ) σ . (104) The m utual information is a popular correlation measure in the sense that it satisfies (i) I ( A : B ) σ ≥ 0, (ii) I ( A : B ) σ = 0 iff σ AB = σ A ⊗ σ B and (iii) I ( A : B C ) σ ≥ I ( A : B ). 6 Let σ AB ∈ S( H AB ) and ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) for r = o ( n ). Then 1 n I ( A n 1 : B n 1 ) ρ Eq. (104) = 1 n H ( A n 1 ) ρ − 1 n H ( A n 1 | B n 1 ) ρ (105) Thm. 4.1 = H ( A ) σ − H ( A | B ) σ + o ( n ) n (106) Eq. (104) = I ( A : B ) σ + o ( n ) n . (107) It is natural to ask if p opular entanglemen t measures are also robust for almost-iid states. In abstract terms, let E ( · ) b e an arbitrary en tanglemen t measure. Let σ AB ∈ S( H AB ) and ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) for r = o ( n ). Is it true that 1 n E ( A n 1 : B n 1 ) ρ ? = 1 n E ( A n 1 : B n 1 ) σ ⊗ n + o ( n ) n (108) In the follo wing, w e discuss the robustness of (a) squashed en tanglement, (b) en tanglemen t distillation, (c) en tanglemen t cost, and (d) relativ e entrop y of entanglemen t. 6 Properties (i) and (ii) follow b y noting that I ( A : B ) ρ = D ( ρ AB ∥ ρ A ⊗ ρ B ). The third prop erty follo ws from strong subadditivity together with the chain rule as I ( A : B C ) σ = I ( A : B ) σ + I ( A : C | B ) σ ≥ I ( A : B ) σ . 16 5.1 Robustness of squashed en tanglemen t Ab o v e w e ha ve seen that the m utual information is robust under almost-iid states. The same argumen t can b e extended to see that the conditional mutual information also coincides for almost-iid and iid states. The squashe d entanglement [ 14 ] is an entanglemen t measure that is based on the conditional m utual information. Given a biparitite density matrix ρ AB ∈ S( H AB ), the squashed en tanglemen t is defined as E sq ( A : B ) ρ := 1 2 inf ρ ABE ∈ S( H ABE )  I ( A : B | E ) ρ : tr E [ ρ AB E ] = ρ AB  , (109) where there is no b ound on the dimension of E . It features many desirable prop erties such as b eing ad- ditiv e on tensor pro ducts and sup eradditive in general. W e next sho w that the squashed en tanglemen t for almost-iid and iid states coincide. Corollary 5.1. L et σ AB ∈ S( H AB ) and ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) for r = o ( n ) . Then 1 n E sq ( A n 1 : B n 1 ) ρ = E sq ( A : B ) σ + o ( n ) n . (110) Pr o of. Let d A := dim( H A ) and d B := dim( H B ). W e can employ the p ermutation inv ariance of ρ and the sup eradditivit y of the squashed en tanglemen t [ 14 , Prop osition 4] to write for ε n := 4 p r n 1 n E sq ( A n 1 : B n 1 ) ρ superadditivity ≥ 1 n n X i =1 E sq ( A i : B i ) ρ (111) perm. inv. = E sq ( A : B ) ρ (112) contin uit y & Prop. 2.6 ≥ E sq ( A : B ) σ − 12 ε n log( d A d B ) − 6 h ( ε n ) (113) = E sq ( A : B ) σ + o ( n ) n , (114) where the contin uit y of squashed en tanglement follo ws from the contin uit y of the conditional entrop y [ 1 ] as explained in [ 14 , Section IV]. It thus remains to prov e the other direction. F or any ξ > 0 there exists an extension σ AB E or σ AB with d E := dim( H E ) < ∞ such that    E sq ( A : B ) σ − 1 2 I ( A : B | E ) σ    ≤ ξ . (115) T o see this, note that by the definition of the squashed en tanglemen t there exists an extension σ ′ AB E of σ AB (with p ossibly un b ounded E -system) such that    E sq ( A : B ) σ − 1 2 I ( A : B | E ) σ ′    ≤ ξ 2 . (116) Cho ose a finite-dimensional pro jector Π E on the E -system such that tr[Π E σ ′ AB E ] = 1 − ε for some ε > 0. Let σ AB E := Π E σ ′ AB E Π † E +  1 − tr[Π E σ ′ AB E ]  | e ⟩ ⟨ e | AB E , (117) where | e ⟩ is a state orthogonal to the supp ort of Π E . By the contin uit y of the conditional entrop y [ 1 ] w e can c ho ose ε > 0 such that | I ( A : B | E ) σ ′ − I ( A : B | E ) σ | ≤ ξ , (118) 17 where w e used that the con tin uit y of the conditional entrop y do es not dep end on the dimension of the conditioning system. The triangle inequality implies    E sq ( A : B ) σ − 1 2 I ( A : B | E ) σ    ≤    E sq ( A : B ) σ − 1 2 I ( A : B | E ) σ ′    +    1 2 I ( A : B | E ) σ ′ − 1 2 I ( A : B | E ) σ    Eqs. (116) and (118) ≤ ξ , (119) whic h th us justifies Equation (115) . Due to Lemma B.5 , there exists an extension ρ A n 1 B n 1 E n 1 of ρ A n 1 B n 1 whic h is an  n r  -almost-iid state in σ AB E . Hence, 1 n E sq ( A n 1 : B n 1 ) ρ ≤ 1 2 n I ( A n 1 : B n 1 | E n 1 ) ρ (120) = 1 2 n H ( A n 1 | E n 1 ) ρ − 1 2 n H ( A n 1 | B n 1 E n 1 ) ρ (121) Thm. 4.1 = 1 2 H ( A | E ) σ − 1 2 H ( A | B E ) σ + o ( n ) n (122) = 1 2 I ( A : B | E ) σ + o ( n ) n (123) Eq. (115) ≤ E sq ( A : B ) σ + ξ + o ( n ) n . (124) Since this holds for an y ξ > 0 we can consider ξ → 0, which concludes the pro of. ■ 5.2 Robustness of en tanglemen t distillation and en tanglemen t cost Let | Φ ⟩ AB denote an en tangled Bell state. Giv en a bipartite density matrix ρ AB ∈ S( H AB ), recall the definitions of entanglement distil lation [ 3 , 4 , 5 ] E D ( A : B ) ρ := lim ε → 0 lim n →∞ sup n m n : inf P n ∈ LOCC 1 2   P n ( ρ ⊗ n ) − | Φ ⟩ ⟨ Φ | ⊗ m   1 ≤ ε o (125) and entanglement c ost [ 24 ] E C ( A : B ) ρ := lim ε → 0 lim n →∞ inf n m n : inf P n ∈ LOCC 1 2   P n ( | Φ ⟩ ⟨ Φ | ⊗ m ) − ρ ⊗ n   1 ≤ ε o . (126) Question 5.2. Let σ AB ∈ S( H AB ) and ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) for r = o ( n ). Is it true that 1 n E D ( A n 1 : B n 1 ) ρ ? = E D ( A : B ) σ + o ( n ) n (127) As discussed in [ 29 ], there are strong indications that one direction of Equation (127) holds, namely that 1 n E D ( A n 1 : B n 1 ) ρ ≥ E D ( A : B ) σ + o ( n ) n . (128) Whether the other direction holds also remains an op en question. The equiv alent question for the entanglemen t cost asks: Question 5.3. Let σ AB ∈ S( H AB ) and ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) for r = o ( n ). Is it true that 1 n E C ( A n 1 : B n 1 ) ρ ? = E C ( A : B ) σ + o ( n ) n (129) 18 Ho w ev er, none of the t w o directions of Equation (129) are known to hold. A t this p oint, we emphasize that, unlike squashed entanglemen t, already the definitions of both en tanglemen t cost and en tanglemen t distillation rely on a tensor pow er (iid) structure. This may suggest that, rather than asking about the robustness of Equations (125) and (126) , one should in- corp orate robustness directly in to the definition itself. One may therefore wonder ho w these notions w ould change if an almost-iid structure were built into the definition from the outset. In [ 29 ], the authors inv estigate this question b y introducing new asymptotic state transformation rates that av oid the standard iid assumption. W e refer the in terested reader to that pap er for further details. 5.3 Robustness of relativ e en tropy of entanglemen t A popular measure to quan tify the amoun t of entanglemen t is the r elative entr opy of entanglement [ 39 ] defined as E R ( A : B ) ρ := min σ AB ∈ SEP( A : B ) D ( ρ AB ∥ σ AB ) , (130) where SEP( A : B ) := con v {| ϕ ⟩ ⟨ ϕ | A ⊗ | φ ⟩ ⟨ φ | B : | ϕ ⟩ A ∈ H A , | φ ⟩ B ∈ H B , ⟨ ϕ | ϕ ⟩ = ⟨ φ | φ ⟩ = 1 } . It is kno wn [ 40 ] that the relative entrop y of en tanglemen t is not additiv e under the tensor product, whic h justifies the definition of a regularized version E ∞ R ( A : B ) ρ := lim k →∞ 1 k E R ( A k 1 : B k 1 ) ρ ⊗ k . The limit in the regularization exists due to F ekete’s subadditivity lemma. Question 5.4. Let σ AB ∈ S( H AB ) and ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) for r = o ( n ). Is it true that 1 n E R ( A n 1 : B n 1 ) ρ ? = 1 n E R ( A n 1 : B n 1 ) σ ⊗ n + o ( n ) n (131) If Equation (131) were true, this would sav e the original proof of the generalized quantum Stein’s lemma [ 23 , 26 ] b y Brand˜ ao and Plenio [ 9 ] (see also [ 6 , 7 ]). One direction of Equation (131) follows from the results developed in this pap er. T o see this, c ho ose s n a monotonically increasing sequence of in tegers such that s n r = o ( n ) and lim n →∞ s n = ∞ . Let k n = ⌈ n/s n ⌉ . Note that n ≤ s n k n ≤ n + s n . Hence, using the monotonicit y of the en tanglement of formation under partial trace, 1 n E R ( A n 1 : B n 1 ) ρ ≤ 1 n E R  A s n k n 1 : B s n k n 1  ρ (132) ≤ 1 s n k n E R  A s n k n 1 : B s n k n 1  ρ + s n n log d A d B , (133) where the final step uses E R ( A m : B m ) ρ ≤ m log ( d A d B ) and s n = o ( n ). In the following steps, we omit the subscripts n for b etter readability . Let ω s ∈ arg min τ A s 1 B s 1 ∈ SEP D ( ρ A s 1 B s 1 ∥ τ A s 1 B s 1 ). Then for ρ s = tr n − s [ ρ n ] w e ha v e 1 n E R ( A n 1 : B n 1 ) ρ Eq. (133) ≤ 1 sk E R  A sk 1 : B sk 1  ρ + o ( n ) n (134) = 1 sk min τ A sk 1 B sk 1 ∈ SEP D ( ρ A sk 1 B sk 1 ∥ τ A sk 1 B sk 1 ) + o ( n ) n (135) ≤ 1 sk D  ρ sk ∥ ( ω s ) ⊗ k  + o ( n ) n (136) = − 1 sk H ( ρ sk ) − 1 s tr[ ρ s log ω s ] + o ( n ) n (137) Thm. 4.1 = − 1 s H ( σ ⊗ s ) − 1 s tr[ ρ s log ω s ] + o ( n ) n (138) 19 contin uit y & Prop. 2.6 ≤ 1 s H ( ρ s ) − 1 s tr[ ρ s log ω s ] + o ( n ) n (139) = 1 s E R ( A s 1 : B s 1 ) ρ + o ( n ) n (140) contin uit y & Prop. 2.6 ≤ 1 s E R ( A s 1 : B s 1 ) σ ⊗ s + o ( n ) n (141) = 1 n E R ( A n 1 : B n 1 ) σ ⊗ n + o ( n ) n , (142) where the contin uit y of the von Neumann entrop y and the relative entrop y of entanglemen t can b e found in [ 2 , 32 , 42 ]. In the last step, we use that lim n →∞ s n = ∞ and that the limit in the RHS of Equation (131) exists due to F ekete’s subadditivity lemma. The other direction of Equation (131) app ears more complicated and remains an op en question. Ac kno wledgemen ts W e thank F ernando Brand˜ ao for his talk on almost-iid states at the SwissMAP Researc h Station (SRS) conference in Les Diablerets 2024, which motiv ated us to write this pap er. W e further thank F r´ ed´ eric Dupuis and Ludo vico Lami for insightful discussions on this topic at the same conference. GM and RR ackno wledge supp ort from the NCCR SwissMAP , the ETH Zurich Quantum Cen ter, the SNSF pro ject No. 20QU-1 225171, and the CHIST-ERA pro ject MoDIC. App endix A Justification of Remark 2.3 T o justify the assertion of Remark 2.3 , we need to show that for all purifications | ψ ⟩ A 2 1 E 2 1 of ρ A 2 1 and for all purifications | θ ⟩ AE of | 0 ⟩ A it follo ws that | ψ ⟩ A 2 1 E 2 1 ∈ Sym 2 ( H AE ) ∩ span V ( H ⊗ 2 AE , | θ ⟩ AE ) T o see this note that, b ecause | 0 ⟩ A is pure, | θ ⟩ AE = | 0 ⟩ A ⊗ | ϑ ⟩ E . Similarly , b ecause ρ A 2 1 = | Ψ − ⟩ ⟨ Ψ − | is pure, any purification on E m ust b e of the form | Ψ − ⟩ A 1 A 2 ⊗ | φ ⟩ E 1 E 2 =: | ψ ⟩ A 2 1 E 2 1 . T o ensure that ρ A 2 1 is an almost-iid state according to the definition from [ 9 ], we need | ψ ⟩ A 2 1 E 2 1 ∈ Sym 2 ( H AE ) ∩ span V ( H ⊗ 2 AE , | θ ⟩ AE ). Th us, the swap op eration π must satisfy π | ψ ⟩ A 2 1 E 2 1 = ( π | Ψ − ⟩ A 1 A 2 ) ⊗ ( π | φ ⟩ E 1 E 2 ) = −| Ψ − ⟩ A 1 A 2 ⊗ ( π | φ ⟩ E 1 E 2 ) ! = | Ψ − ⟩ A 1 A 2 ⊗ | φ ⟩ E 1 E 2 . (143) This yields π | φ ⟩ E 1 E 2 = −| φ ⟩ E 1 E 2 , i.e., | φ ⟩ is anti-symmetric. Expressing this in an orthonormal basis {| i ⟩ E } i of E with | 0 ⟩ E = | ϑ ⟩ E giv es | φ ⟩ E 1 E 2 = P d E − 1 i,j =0 α i,j | i ⟩ E 1 | j ⟩ E 2 with α i,j = − α j,i for all i, j ∈ { 0 , . . . , d E − 1 } . Hence, | ψ ⟩ A 2 1 E 2 1 = 1 √ 2 X i 0 we hav e P h ∥ λ x ′ − P X ∥ 1 > r 2(ln 2)  δ + |X | log( n − r + 1) n − r i [ 33 , Cor. B.3.3] ≤ 2 − ( n − r ) δ . (169) Using r ≤ n 2 this can b e simplified to P h ∥ λ x ′ − P X ∥ 1 > r 2(ln 2)  δ + 2 |X | log( n 2 + 1) n i ≤ 2 − nδ 2 . (170) Using λ x = n − r n λ x ′ + r n λ x ′′ yields ∥ λ x − P X ∥ 1 triangle ≤ n − r n ∥ λ x ′ − P X ∥ 1 + r n ∥ λ x ′′ − P X ∥ 1 ≤ ∥ λ x ′ − P X ∥ 1 + 2 r n . (171) Hence, P h ∥ λ x − P X ∥ 1 > r 2(ln 2)  δ + 2 |X | log( n 2 + 1) n  + 2 r n i Eq. (171) ≤ P h ∥ λ x ′ − P X ∥ 1 > r 2(ln 2)  δ + 2 |X | log( n 2 + 1) n i (172) Eq. (170) ≤ 2 − nδ 2 . (173) This can b e rewritten as P x ←| Ψ t ⟩ [ x ∈ W δ ] ≤ 2 − nδ 2 , (174) for W δ = n x ∈ X n : ∥ λ x − P X ∥ 1 > r 2(ln 2)  δ + 2 |X | log( n 2 + 1) n  + 2 r n o . (175) The notation x ← | Ψ t ⟩ indicates that x is distributed according to the outcomes of the measurement applied to | Ψ t ⟩ . F or M x = M x 1 ⊗ . . . ⊗ M x n w e find P x ← ρ A n 1 E n 1 [ x ∈ W δ ] = X x ∈W δ tr[ ρ A n 1 E n 1 M x ] (176) Eq. (57) ≤ X x ∈W δ |T | X t ∈T β t,t ⟨ Ψ t | M x | Ψ t ⟩ (177) = |T | X t ∈T β t,t P x ←| Ψ t ⟩ [ x ∈ W δ ] (178) Eq. (174) ≤ |T | 2 − nδ 2 (179) Remark 2.4 ≤ 2 − n ( δ 2 − h ( r n )) d 2 r . (180) Cho osing δ = 2 log ( 1 ε ) n + 2 h ( r n ) + 4 r n log( d ) yields P h ∥ λ x − P X ∥ 1 > s 4(ln 2)  log( 1 ε ) n + h  r n  + 2 r n log( d ) + |X | log( n 2 + 1) n  i Eq. (180) ≤ ε , (181) whic h completes the pro of. ■ 24 D Pro of of Prop osition 3.1 Consider a ( n + k )-bit string with m ones and n + k − m zeros. If we throw a w ay k bits, then the probabilit y of ha ving j ones in the remaining n -bit string is p j =  n j  k m − j   n + k m  . (182) Note that P m j =0 p j = 1 follows from a kno wn identit y due to V andermonde which states that for all r , s, t ∈ N we hav e  r + s t  = t X ℓ =0  r ℓ  s t − ℓ  . (183) T o see Equation (182) , let v ∈ { 0 , 1 , . . . , k } denote the num ber of ones that hav e b een thro wn a w a y . Hence, p j = 1 c  n j  k v  = 1 c  n j  k m − j  . (184) for some normalization constan t c = m X j =0  n j  k m − j  Eq. (183) =  n + k m  . (185) F act D.1. F or the setting ab ove, we have V ar p [ X ] = k mn ( n + k − m ) ( n + k − 1)( n + k ) 2 . (186) Pr o of. Recall that j  n j  = n  n − 1 j − 1  . (187) With this w e can write E p [ X ] = m X j =0 j p j (188) Eq. (182) = m X j =0 j  n j  k m − j   n + k m  (189) Eq. (187) = n  n + k m  m X j =1  n − 1 j − 1  k m − j  (190) = n  n + k m  m − 1 X j =0  n − 1 j  k m − 1 − j  (191) Eq. (183) = n  n + k m   n + k − 1 m − 1  (192) Eq. (187) = nm n + k . (193) 25 Similarly , we find E p [ X 2 ] = m X j =0 j 2 p j (194) Eq. (182) = 1  n + k m  m X j =0 j 2  n j  k m − j  (195) = n  n + k m  m − 1 X j =0 ( j + 1)  n − 1 j  k m − 1 − j  (196) = n  n + k m    m − 1 X j =0 j  n − 1 j  k m − 1 − j  + m − 1 X j =0  n − 1 j  k m − 1 − j    (197) Eqs. (183) and (187) = n  n + k m    ( n − 1) m − 2 X j =0  n − 2 j  k m − 2 − j  +  n + k − 1 m − 1    (198) Eq. (183) = n  n + k m   ( n − 1)  n + k − 2 m − 2  +  n + k − 1 m − 1  (199) = n ( n − 1)  n + k − 1 m − 1  n + k − 2 m − 2   n + k m  n + k − 1 m − 1  + n  n + k − 1 m − 1   n + k m  (200) = n ( n − 1) m ( m − 1) ( n + k )( n + k − 1) + n m n + k . (201) Com bining ev erything yields V ar p [ X ] = E p [ X 2 ] − ( E p [ X ]) 2 Eqs. (193) and (201) = k mn ( n + k − m ) ( n + k − 1)( n + k ) 2 . (202) ■ Let Q ( q ) X n 1 ∈ V ( X n , q n − r ) and consider a binary random string X n 1 ∼ Q ( q ) X n 1 . Then without loss of generalit y assume that the r defects are at the end of the random string, and hence V ar h n X i =1 X i i = V ar h n − r X i =1 X i i + V ar h n X i = n − r +1 X i i | {z } ≥ 0 ≥ ( n − r )V ar q [ X ] , (203) where the first equalit y uses that the defects are indep enden t of iid parts. Example D.2. Let α ∈ (0 , 1), m = n + k 2 , k = n α and r = o ( n ). Then Fact D.1 giv es V ar p [ X ] = n 1+ α 4( n + n α − 1) = Θ( n α ). F urthermore, Equation (203) yields V ar h P n i =1 X i i ≥ Θ( n ). F act D.3. L et t ∈ [0 , 1] and p, q b e two pr ob ability distributions. Then, V ar tp +(1 − t ) q [ X ] ≥ t V ar p [ X ] + (1 − t )V ar q [ X ] . (204) Pr o of. By definition of the v ariance, we hav e V ar tp +(1 − t ) q [ X ] = E tp +(1 − t ) q [ X 2 ] − ( E tp +(1 − t ) q [ X ]) 2 (205) 26 = t E p [ X 2 ] + (1 − t ) E q [ X 2 ] − ( t E p [ X ] + (1 − t ) E q [ X ]) 2 (206) ≥ t E p [ X 2 ] + (1 − t ) E q [ X 2 ] − ( t E p [ X ] 2 + (1 − t ) E q [ X ] 2 ) (207) = t V ar p [ X ] + (1 − t )V ar q [ X ] . (208) ■ Putting ev erything together, w e obtain for α ∈ (0 , 1) V ar p [ X ] Ex. D.2 = Θ( n α ) and V ar R Q ( q ) X ν (d q ) [ X ] Fact D.3 ≥ Z ν (d q )V ar Q ( q ) X [ X ] Ex. D.2 = Θ( n ) . (209) This prov es the assertion of Prop osition 3.1 . Note that for b etter readabilit y , the ab ov e proof has b een done for the sp ecific c hoice k = n α = o ( n ) for α ∈ (0 , 1), but the same argumen t remains v alid for an arbitrary k = o ( n ). ■ E Alternativ e pro of of Theorem 4.1 In this section, w e presen t an alternative pro of for Theorem 4.1 which uses an en tirely different pro of tec hnique which may b e of indep endent interest. How ev er, we note that the scaling of the defects r is sligh tly w orse than in Theorem 4.1 . Note that it suffices to prov e the following result. Theorem E.1. L et σ A ∈ S( H A ) and ρ A n 1 ∈ S n ( H A , σ ⊗ n − r A ) for r = o ( √ n ) . Then 1 n H ( A n 1 ) ρ = H ( A ) σ + o ( n ) n . (210) Theorem E.1 implies that the conditional en tropy of almost-iid states coincides asymptotically with the conditional en trop y of iid states. Corollary E.2. L et σ AB ∈ S( H AB ) and ρ A n 1 B n 1 ∈ S n ( H AB , σ ⊗ n − r AB ) for r = o ( √ n ) . Then 1 n H ( A n 1 | B n 1 ) ρ = H ( A | B ) σ + o ( n ) n . (211) Pr o of. By definition of the conditional entrop y , we hav e 1 n H ( A n 1 | B n 1 ) ρ = 1 n H ( A n 1 B n 1 ) ρ − 1 n H ( B n 1 ) ρ Thm. E.1 = H ( AB ) σ − H ( B ) σ + o ( n ) n = H ( A | B ) σ + o ( n ) n . ■ E.1 Pro of of Theorem E.1 One direction of Equation (210) is simple. T o see this, let ε n = 2 p rs n for s = ⌊ √ n ⌋ and consider 1 n H ( A n 1 ) ρ subadditivity ≤ H ( A 1 ) ρ Prop. 2.6 ≤ H ( A ) σ + ε n log d + h ( ε n ) = H ( A ) σ + o ( n ) n , (212) where the p en ultimate step uses the con tin uit y of entrop y [ 2 , 32 , 42 ]. The other direction is more complicated. Recall that strong subadditivit y of quantum en tropy (SSA) [ 27 , 28 ] ensures I ( A : C | B ) ρ ≥ 0. F urthermore, the conditional m utual information satisfies a c hain rule I ( A : B C ) ρ = I ( A : B ) ρ + I ( A : C | B ) ρ . Let | θ ⟩ AE b e a purification of σ A and let ρ A n 1 E n 1 27 b e an extension of ρ A n 1 that satisfies the tw o conditions of Definition 2.1 . Since ρ A n 1 E n 1 is p erm utation- in v ariant, for en trop y and m utual information terms the indices of the considered subsystems can b e c hanged. Hence, w e find for any k ≤ ⌊ n 2 ⌋ =: n 0 , k ′ ≤ n 0 + 1 and for any ℓ = k , . . . , n 0 , ℓ ′ = k ′ , . . . , n 0 + 1 I ( A n 0 1 E n 0 1 : A n n 0 +1 E n n 0 +1 ) ρ SSA ≥ I ( A n 0 1 : A n n 0 +1 ) ρ (213) chain rule = I ( A ℓ 1 : A n n 0 +1 | A n 0 ℓ +1 ) ρ + I ( A n 0 ℓ +1 : A n n 0 +1 ) ρ (214) SSA ≥ I ( A k 1 : A n n 0 +1 | A n 0 ℓ +1 ) ρ (215) chain rule = I ( A k 1 : A n 0 + ℓ ′ n 0 +1 | A n 0 ℓ +1 A n n 0 + ℓ ′ +1 ) ρ + I ( A k 1 : A n n 0 + ℓ ′ +1 | A n 0 ℓ +1 ) ρ (216) SSA ≥ I ( A k 1 : A n 0 + k ′ n 0 +1 | A n 0 ℓ +1 A n n 0 + ℓ ′ +1 ) ρ (217) perm . inv . = I ( A k 1 : A k + k ′ k +1 | A m k + k ′ +1 ) ρ , (218) where in the last step, we p ermute all the remaining systems app earing in the conditioning (there are n − ℓ − ℓ ′ man y) in to neighboring systems labeled b y indices from ( k + k ′ + 1) to m := ( k + k ′ + n − ℓ − ℓ ′ ). Th us, for an y m ≥ k + k ′ w e find I ( A k 1 : A k + k ′ k +1 | A m k + k ′ +1 ) ρ Eq. (218) ≤ I ( A n 0 1 B n 0 1 : A n n 0 +1 E n n 0 +1 ) ρ (219) = H ( A n 0 1 E n 0 1 ) ρ + H ( A n n 0 +1 E n n 0 +1 ) ρ − H ( A n 1 E n 1 ) ρ (220) ≤ H ( A n 0 1 E n 0 1 ) ρ + H ( A n n 0 +1 E n n 0 +1 ) ρ . (221) In the pro of of Lemma B.3 it is sho wn that ρ A n 0 1 E n 0 1 ∈ span V ( H ⊗ n 0 AE , | θ ⟩ ⊗ n 0 − r AE ). This implies H ( A n 0 1 E n 0 1 ) ρ Eq. (9) ≤ log  2 n 0 h ( r/n 0 ) d r AE  = n 0 h  r n 0  + r log d AE . (222) Similarly , we obtain H ( A n n 0 +1 E n n 0 +1 ) ρ ≤ ( n − n 0 ) h  r n − n 0  + r log d AE . (223) Hence w e find I ( A k 1 : A k + k ′ k +1 | A m k + k ′ +1 ) ρ Eq. (221) ≤ H ( A n 0 1 B n 0 1 ) ρ + H ( A n n 0 +1 B n n 0 +1 ) ρ (224) Eqs. (222) and (223) ≤ nh  2 r n  + 2 r log d AE . (225) F or any ℓ < s = ⌊ (log n ) 3 2 ⌋ , w e th us ha v e    ρ A ℓ +1 1 − σ ⊗ ( ℓ +1) A    1 ≤   ρ A s 1 − σ ⊗ s A   1 Prop. 2.6 ≤ ε n , (226) where the first step follows since the trace distance is contractiv e under trace-preserving completely p ositiv e maps [ 43 , Theorem 8.16]. The contin uit y of conditional entrop y [ 1 ] then implies that | H ( A 1 | A ℓ +1 2 ) ρ − H ( A ) σ | = | H ( A 1 | A ℓ +1 2 ) ρ − H ( A 1 | A ℓ +1 2 ) σ ⊗ ( ℓ +1) | ≤ 4 ε n log d A + 2 h ( ε n ) =: δ n , (227) where d A = dim( A ). The chain rule together with p ermutation inv ariance allows us to write H ( A n 1 ) ρ chain rule = H ( A n − ℓ 1 | A n n − ℓ +1 ) ρ + H ( A n n − ℓ +1 ) ρ (228) ≥ H ( A n − ℓ 1 | A n n − ℓ +1 ) ρ (229) 28 chain rule = n − ℓ X i =1 H ( A i | A n n − ℓ +1 ) ρ − n − ℓ X i =1 I ( A i − 1 1 : A i | A n n − ℓ +1 ) ρ (230) perm. inv. = ( n − ℓ ) H ( A 1 | A ℓ +1 2 ) ρ − n − ℓ X i =1 I ( A i − 1 1 : A i | A n n − ℓ +1 ) ρ (231) Eq. (227) ≥ ( n − ℓ ) H ( A ) σ − n − ℓ X i =1 I ( A i − 1 1 : A i | A n n − ℓ +1 ) ρ − nδ n (232) = ( n − ℓ ) H ( A ) σ − n − s X i =1 I ( A i − 1 1 : A i | A n n − ℓ +1 ) ρ − n − ℓ X i = n − s +1 I ( A i − 1 1 : A i | A n n − ℓ +1 ) ρ − nδ n (233) ≥ ( n − ℓ ) H ( A ) σ − n − s X i =1 I ( A i − 1 1 : A i | A n n − ℓ +1 ) ρ − 2( s − ℓ ) log d A − nδ n (234) perm. inv. = ( n − ℓ ) H ( A ) σ − n − s X i =1 I ( A i − 1 1 : A i | A n − s + ℓ n − s +1 ) ρ − 2( s − ℓ ) log d A − nδ n . (235) Claim E.3. L et n ∈ N , exists ℓ ≤ s = ⌊ √ n ⌋ such that for P n − s i =1 I ( A i − 1 1 : A i | A n − s + ℓ n − s +1 ) ρ =: ξ n we have lim n →∞ ξ n n = 0 . Pr o of. The proof follows the idea from [ 8 , Equation (5)], whic h prov es a v ariant of Claim E.3 for a purely classical scenario. By the p ermutation-in v ariance w e ha v e for any 1 ≤ i < k ≤ m ≤ n I ( A i − 1 1 : A i | A m k +1 ) ρ = I ( A i − 1 1 : A m | A m − 1 k ) ρ . (236) This allo ws us to write n X m = k I ( A i − 1 1 : A i | A m k +1 ) ρ ( 236 ) = n X m = k I ( A i − 1 1 : A m | A m − 1 k ) ρ chain rule = I ( A i − 1 1 : A n k ) ρ . (237) Summing Equation (237) o v er all 1 ≤ i < k and dividing by n − k + 1 gives 1 n − k + 1 n X m = k k − 1 X i =1 I ( A i − 1 1 : A i | A m k +1 ) ρ = 1 n − k + 1 k − 1 X i =1 I ( A i − 1 1 : A n k ) ρ . (238) Hence, there exists m ∗ ∈ { k , k + 1 , . . . , n } such that k − 1 X i =1 I ( A i − 1 1 : A i | A m ∗ k +1 ) ρ ≤ 1 n − k + 1 k − 1 X i =1 I ( A i − 1 1 : A n k ) ρ . (239) Cho osing k = n − s , Equation (239) c an b e rewritten (as there exists 0 ≤ ℓ ≤ s such that m ∗ = k + ℓ ) as n − s − 1 X i =1 I ( A i − 1 1 : A i | A n − s + ℓ n − s +1 ) ρ ≤ 1 s + 1 n − s X i =1 I ( A i − 1 1 : A n n − s ) ρ (240) W e next show that Equation (225) implies I ( A i − 1 1 : A n n − s ) ρ ≤ 2 nh  2 r n  + 4 r log d AE ∀ i = 1 , . . . , n − s . (241) 29 T o see this, we can assume w.l.o.g. 8 that n is large enough suc h that k ′ := s + 1 = j (log n ) 3 2 k + 1 ≤ j n 2 k + 1 = n 0 + 1 . (242) Let us no w consider tw o cases: In case i − 1 ≤ n 0 , the assertion follo ws from Equation (225) b y choosing m = k + k ′ , since I ( A i − 1 1 : A n n − s ) ρ perm. inv. = I ( A i − 1 1 : A i + s i ) ρ ≤ nh  2 r n  + 2 r log d AE . (243) In case i − 1 > n 0 , w e can use the c hain rule to write I ( A i − 1 1 : A n n − s ) ρ = I ( A n 0 1 : A n n − s ) ρ + I ( A i − 1 n 0 +1 : A n n − s | A n 0 1 ) ρ (244) perm. inv. = I ( A n 0 1 : A n 0 +1+ s n 0 +1 ) ρ + I ( A i − 1 − n 0 1 : A i − n 0 + s i − n 0 | A i + s i − n 0 + s +1 ) ρ , (245) where b oth terms on the right-hand side are b ounded from ab o v e by nh ( 2 r n ) + 2 r log d AE via Equa- tion (225) . Putting things together yields n − s X i =1 I ( A i − 1 1 : A i | A n − s + ℓ n − s +1 ) ρ = n − s − 1 X i =1 I ( A i − 1 1 : A i | A n − s + ℓ n − s +1 ) ρ + I ( A n − s − 1 1 : A n − s | A n − s + ℓ n − s +1 ) ρ (246) Eq. (240) ≤ 1 s + 1 n − s X i =1 I ( A i − 1 1 : A n n − s ) ρ + 2 log d A (247) Eq. (241) ≤ n − s s + 1  2 nh  2 r n  + 4 r log d AE  + 2 log d A (248) = o ( n ) , (249) where the final step uses that for s = ⌊ √ n ⌋ and r = o ( n ) w e ha v e lim n →∞ n s h  2 r n  = 0 . (250) ■ Since lim n →∞ δ n = 0, com bining Claim E.3 with Equation (235) yields for some ℓ ≤ s = ⌊ √ n ⌋ H ( A n 1 ) ρ ≥ ( n − ℓ ) H ( A ) σ − o ( n ) = nH ( A ) σ − o ( n ) . (251) ■ References [1] R. Alicki and M. F annes. Con tin uit y of quantum conditional information. Journal of Physics A: Mathematic al and Gener al , 37(5):55–57, 2004. DOI: doi:10.1088/0305-4470/37/5/L01 . [2] K. M. R. Audenaert. A sharp con tin uit y estimate for the v on Neumann en tropy . 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