Tip of the Quantum Entropy Cone

Relations among von Neumann entropies of different parts of an N -partite quantum system have direct impact on our understanding of diverse situations ranging from spin systems to quantum coding theory and black holes. Best formulated in terms of the set Σ (cid:1) N of possible vectors comprising the entropies of the whole and its parts, the famous strong subaddivity inequality constrains its closure ¯ Σ (cid:1) N , which is a convex cone. Further homogeneous constrained inequalities are also known. In this Letter we provide (nonhomogeneous) inequalities that constrain Σ (cid:1) N near the apex (the vector of zero entropies) of ¯ Σ (cid:1) N , in particular showing that Σ (cid:1) N is not a cone for N ≥ 3 . Our inequalities apply to vectors with certain entropy constraints saturated and, in particular, they show that while it is always possible to upscale an entropy vector to arbitrary integer multiples it is not always possible to downscale it to arbitrarily small size, thus answering a question posed by Winter.


I. INTRODUCTION
Entropy is a very important concept in physics, whose role and status have vastly expanded past its original boundaries within thermodynamics.It is a main object of study in many areas of research, including quantum cryptography, information theory, black holes and more.
In models of the world, it is often very advantageous and natural to consider large systems as composed of smaller distinct subsystems.This calls for a good understanding of the relations among entropies of different subsystems of a joint system.The most important such relation is without a doubt the strong subaddivity inequality [1], which entails all other known entropy inequalities for multi-partite quantum systems and has long been appreciated in quantum information theory.There has naturally been a great interest in finding new such inequalities.The problem of finding new entropy inequalities is an aspect of a more general research endeavour to adequately describe the set of possible values that the different allocations of entropy in a multi-partite system can take, i.e. to determine whether or not any given ordered set of numbers corresponds to an achievable entropy-vector, by which we mean the entropy-values of the marginals of some quantum state.
In this Letter, we prove a new relationship between the entropies of a multi-partite system, which rules out the possibility of constructing certain small entropy-vectors that otherwise satisfy strong subadditivity and related inequalities.This result, interestingly, entails that the set of achievable entropy vectors is neither a cone nor a closed set -thus answering a question left open in an influential paper by Pippenger [2].We additionally discuss applications of the new results to a diverse set of areasnamely topological materials, entanglement theory, and quantum cryptography.
In the remainder of this introduction, we shall introduce some necessary notation and relevant background concerning the quantum entropy cone.The main results are presented in the following section, after which we discuss some applications and provide a conclusion and outlook for this work.Given a quantum system X in a state described by a density operator ρ, i.e. a non-negative operator of trace 1 on a (finite dimensional) Hilbert space H X , its von Neumann entropy is given by where λ i are the eigenvalues of ρ, and log denotes the binary logarithm.We shall be concerned with multipartite systems N consisting of N constituent systems X 1 , ..., X N with associated Hilbert spaces H X1 , ..., H XN , such that the state of N is given by a density operator ρ on H X1 ⊗ ... ⊗ H XN .The reduced state of a subsystem X ⊆ N is then given by where Tr N \X [•] denotes the partial trace over ⊗ Xi / ∈X H Xi (and in particular ρ = ρ N ).The entropy H ρX of the reduced state will also be denoted by H(X ) ρ or by called the entropy vector of ρ, whose coordinates are labelled by the nonempty subsystems of N .E.g., for N = 2 and N = {A, B} we have H ρ = (H(A), H(B), H(AB)) ρ ∈ R 3 , while for N = 3 and N = {A, B, C} we write The main object of study in this context is the set Σ * N of all possible entropy vectors associated to N -partite systems, It is a fundamental result of Pippenger [2] that the topological closure Σ * N of Σ * N in R 2 N −1 is a convex cone, called the quantum entropy cone of N -partite systems, i.e.Σ * N is closed under addition and under multiplication by positive scalars.It is also known, and easy to demonstrate, that Σ * N has full dimension, i.e. it spans all of R 2 N −1 as a vector space, and that Σ * N and Σ * N have identical interiors and hence also identical boundaries.For N = 2 it is even true that Σ * 2 = Σ * 2 as will be commented on further below.But for general N ≥ 3 an appropriate characterisation of the boundary entropy vectors is missing [3].
A related but different long standing problem is to determine whether or not Σ * N is a polyhedral cone, i.e. if it can be specified in terms of a finite number of linear inequalities.The known general inequalities of this sort are of two types: called strong subadditivity and weak monotonocity, respectively.Here, X and Y are arbitrary subsystems, and by convention we have H(∅) = 0. We emphasize that not all inequalities of the forms above are independent.
Strong subadditivity was first established in [5], but a variety of proofs exist in the literature, see e.g.[1,[6][7][8][9][10].To obtain weak monotonicity one makes use of the fact, referred to as purification [8], that given a state ρ of N it is always possible to extend N by a system Y and to define a pure state The polyhedral cone defined by (3) and ( 4) is a closed convex cone, and will here be denoted Σ N .The question of whether Σ N = Σ * N , or if there exist further independent linear inequalities beyond (3) and ( 4), remains open for N ≥ 4. For N ≤ 3 the two closed cones coincide as shown in [2].While it is quite easy to see that Σ N = Σ * N = Σ * N hold for N ≤ 2, the case N ≥ 3 is different.It has been shown that for N ≥ 4 there exist further constrained homogeneous linear inequalities [11][12][13].
We shall now delve a bit deeper into the details of the case N = 3 where the relevant inequalities are This makes a total of twelve inequalities, three of each type.A key observation is that each of which has seven facets, corresponding to their seven defining inequalities.By a slight elaboration of Pippenger's approach [2] it can be shown that Σ + 3 ⊂ Σ * 3 , while Σ − 3 behaves differently.For any H ∈ Σ − 3 one finds that there exists a quantum state ρ and a vector l belonging to the 1-dimensional face (half-line) ℓ of Σ − 3 defined by the six equations ℓ : such that If it so happened that ℓ ⊂ Σ * 3 , it would follow by the additivity of entropy vectors in suitably constructed product states that Σ − 3 ⊂ Σ * 3 and hence that Σ 3 = Σ * 3 .However, as a consequence of Theorem 1 below there is an open line segment of ℓ ending at the apex which is not contained in Σ * 3 , and so Σ 3 = Σ * 3 .On the other hand, Pippenger identifies a state ρ l such that H ρ l ∈ ℓ, which by the cone property implies that ℓ ⊂ Σ * 3 .Using (9) one then obtains that Σ − 3 ⊂ Σ * 3 and consequently Σ 3 = Σ * 3 , which is the already mentioned main result of [2].
In order to satisfy (8), the entropy vector H ρ l must satisfy (10) for any pair {X, Y } in N = {A, B, C} with Z = X, Y , where the more standard notation I(X : Y ) has been used instead of I XY for the quantum mutual information.By purification one can alternatively consider a state η = |V V | on a 4-partite system {A, B, C, D} such that ρ l = η N .Such a pure state makes the equations (10) take on the more symmetric form I(X i : X j ) η = 0 (11) for all pairs X i , X j in {A, B, C, D}.Indeed, the state ρ l is obtained in [2] by first constructing such a pure state η.Our main theorem below concerns pure states of arbitrary N -partite systems that fulfill the conditions (11) for fixed i, showing that sufficiently small scalar multiples of their entropy vectors lie outside Σ * N i.e. cannot be realized by quantum states.For the sake of completeness we exhibit in Appendix C, for arbitrary N ≥ 4, states which fulfill the stated conditions, and thus generalising the pure state η mentioned above.

II. MAIN RESULTS
The goal of this section is to establish the following entropy bound. .The conditions of Theorem 1 are here represented with each circle denoting a constituent system Xi.The double lines indicate that the mutual information between the two systems is 0, and it is assumed that the total state is pure.
Then the following bound holds: The conditions in the theorem are illustrated in Fig. 1.Note that they can only be satisfied if N ≥ 4.
To establish Theorem 1, we first list three lemmas below which are the main ingredients in the subsequent proof.Their demonstrations are provided in Appendix A. We will use the following notation.Given a state ρ of N , we denote by λ i 1 ≥ λ i 2 ≥ . . . the eigenvalues of ρ Xi in decreasing order and by |e i 1 , |e i 2 , . . .a corresponding orthonormal eigenstate basis such that Moreover, we define Clearly, xi>1 λ i xi = ǫ i and one easily verifies that where h denotes the binary entropy function, Assuming ρ to be pure, i.e. ρ = |V V | where V |V = 1, we represent |V with respect to the basis for H N consisting of tensor products of eigenstates |e i a for the single-party density matrices, that is A sum over dummy-indices x i ∈ N will here always run up to dim(H Xi ).The matrix elements of ρ N and the reduced states are quadratic expressions of the components of |V ; e.g., Extensive use will be made of the fact that I(X i : X j ) = 0 holds if and only if ρ XiXj is a product state, which in our notation and choice of basis means that The announced lemmas relate the ǫ i 's to the components of V as follows.
Lemma 1.For any pure state ρ it holds that Lemma 2. For any pure state ρ such that I(X 1 : X j ) = 0 for all j = 1 we have Lemma 3.For any pure state ρ it holds that We remark that Lemma 1 is used for the proof of Lemma 3, while only Lemma 2 and Lemma 3 are used in the proof of Theorem 1.
Proof of Theorem 1. Combining Lemma 2 and Lemma 3 we get Since ǫ 1 > 0 as a consequence of the assumption H(X 1 ) = 0, this is equivalent to Since the left-hand side of this inequality is larger than 1, it follows that ε > 1 2 which in turn implies (12) by use of (15) and the definition of ε.This completes the proof of Theorem 1.
In case the given state ρ is not pure, we can apply Theorem 1 to its purification and obtain (see Appendix B for more details and further elaboration of the main theorem) Corollary 1.Let H be a realizable entropy-vector for a system N = {X 1 , ..., X N } which fulfills for all i ∈ {1, ..., N }.Then the following bound holds: We note that the conditions in the corollary can be satisfied if N ≥ 3.This result excludes a range of vectors in Σ N from Σ * N that satisfy N linear constraints and hence can be labeled by 2 N − N − 1 parameters.See Figure 2 for a visualization in case N = 3.In Appendix C we provide a 4-parameter family of realizable entropy vectors on the boundary of Σ N satisfying the conditions of the corollary.

III. APPLICATIONS
The entropy concept itself originally arose from thermodynamical considerations of macroscopic systems consisting of many particles, such as gases.Quantum lations of such systems can be quantified in terms of the scaling of the entanglement entropy, that is the entropy of a subregion A. It has been found for many systems that this entropy is roughly proportional to the size of the boundary ∂A and not to the volume, a statement known as the area law [14].For topologically ordered systems it is expected that up to terms vanishing as the "area" |∂A| gets large.Moreover, the constant additive term −γ is expected to be universal and is dubbed the topological entanglement entropy.Actually, −γ equals an alternating sum of entropies, called M , encountered above in (5).As shown in [15,16] the value of γ in a class of systems is always positive, and M is thus negative.This is precisely the regime in which we identified restrictions on entropy vectors and they may therefore have implications for the attainable values of the topological entropy.We point out, however, that the entropy vectors of the particular finite systems calculated in [15,16] in terms of their total quantum dimension do not satisfy the conditions of our theorem.Also, as the constraints we obtained are not balanced [12], our results have no direct bearing on the usual situation when a large system size is considered.
Many functions in quantum information theory are defined in terms of optimizations of von Neumann entropies [17] or even optimization with entropic constraints [18].An example from entanglement theory is the squashed entanglement [19] where the minimization is over extensions ρ ABE of ρ AB .
The results of the present work constrain such optimization and it remains to be explored whether they could lead to simplified computations in specific cases.Finally, let us consider a cryptographic situation, known as quantum secret sharing [20][21][22]: Alice (A) wishes to distribute information to N −1 parties (N ≥ 4) • purely, in the sense that the overall state of her and the constituent systems is pure • secretly, in the sense that every share is in product with hers • non-trivially, in the sense that H(A) > 0 .These are precisely the conditions of Theorem 1 and thus it follows from our work that she cannot do so unless the average share carries a minimum entropy, equal to 1/N , putting a lower bound on the communication required.

IV. CONCLUSION
We conclude this letter by summarizing the new results.Theorem 1 concerns pure states and establishes, for general values of N ≥ 4, that inside certain faces of Σ N , defined by requiring one constituent system, say X 1 , to have vanishing mutual information with all others, there is a strictly positive lower bound on the distance from the apex to any entropy vector corresponding to a pure state with H(X 1 ) = 0.
Corollary 1 concerns arbitrary states for N -partite system with N ≥ 3.In particular, for the case N = 3, it entails a positive lower bound on the distance from the apex to any realizable entropy vector on any given ray within the 4-dimensional face of Σ 3 defined by III XY = 0 for all X, Y, and H(ABC) = 0.This answers, in particular, a question posed by A. Winter [23] concerning the possibility of down-scaling certain realizable entropy vectors.For general values of N ≥ 3, Corollary 1 provides non-homogeneous bounds (24), which rule out down-scaled versions of realisable entropy vectors -such as those presented in Appendix C. It follows that Σ * N is not a cone for N ≥ 3. On the other hand, the closure of Σ * N is a cone [2], so it likewise follows that Σ * N is not closed for N ≥ 3.This confirms a previous statement from [11] and solves an open problem from [2].
We emphasise that our results apply to the case of finite dimensional as well as infinite dimensional state spaces, provided the states in question have well-defined entropies.
In section III, we highlighted the potential impact in macroscopic systems, quantum information theory, and quantum cryptography -pointing to the importance of a further investigation of the shape of Σ * N .
We now use the Cauchy-Schwarz inequality to bound the last expression from above by Using again relations (18) and (13) to rewrite these sums in terms of matrix elements of ρ X1 , we obtain Rewriting this inequality as and summing over a > 1 then yields Finally, applying the lower bound from Lemma 1 on the right-hand side gives the desired inequality (23), which proves the lemma.
Appendix B: Further elaboration and refinement of the main theorem In this section we discuss a slight refinement of Theorem 1, or rather Corollary 1, concerning the entropyvector H = H ρ for a system N = {X 1 , ..., X N } in a state ρ that is not necessarily pure.
Assume H fulfills either of the following conditions: i) There exists an i ∈ {1, ..., N } such that H(X i ) = 0 and I(X i : X j ) = 0 for j = i and H( We then claim the bound max({H(X 1 ), ..., H(X N ), holds, where N ′ is the number of non-zero elements among H(X 1 ), ..., H(X N ), H(N ).
Indeed, we first note that the constituent systems with vanishing entropy can be disregarded since they are in a pure state in product with the rest and do not contribute to any of the component entropies of H.By purification of the resulting system we obtain an N ′ -partite system N ′ in a pure state that satisfies the conditions of Theorem 1 by using (15) and either of the assumptions i) or ii).In case of i) X 1 is replaced by X i while in case ii) X 1 is replaced by the purifying system E whose entropy is H(N ).From the proof of Theorem 1 as applied to the system N ′ we find that ε ≡ i ǫ i + ǫ E > 1 2 , with obvious notation.Using the intermediate inequality from the elementary entropy bound (15) and the fact that max [h(x), − log(1 − x)] is an increasing function of x ∈ [0, 1], this implies that the left-hand side of (B1) is strictly lower bounded by as claimed.This result entails, for example, that vectors on the open line-segment of ℓ between the origin and α(1, 1, 1, 2, 2, 2, 1), where α = h 1 8 ≈ 0.54, cannot be realized as entropy vectors of a quantum system.Note also that condition ii) above coincides with the assumptions of Corollary 1.
Next, we discuss further restrictions on states satisfying the conditions in Theorem 1. Suppose N = 4 and let ρ be a pure state which satisfies H(X 1 ) > 0 and I(X 1 : X i ) = 0 for i ∈ {2, 3, 4}.
To establish these inequalities we use that any reduced state of a total pure state have equal entropy to its complement, sub-additivity, and the assumption I(X 1 : X 4 ) = 0, together with the fact that ρ is pure, imply that H(X 1 ) + H(X 4 ) ≤ H(X 2 ) + H(X 3 ) , (B3) and hence H(X 1 ) ≤ H(X 2 ) which proves (B2).The inequality (B3) also entails by (B1), thus proving the other two inequalities.By a similar argument we get for general N ≥ 4 that, if the conditions of Theorem 1 are satisfied and we assume H(X 2 ) ≤ • • • ≤ H(X N ), then with the claimed properties.For N ≥ 4 the method can be further refined and obviously leads to a larger number of parameters than 4, but we refrain from elaborating on this issue here.

Figure 1
Figure1.The conditions of Theorem 1 are here represented with each circle denoting a constituent system Xi.The double lines indicate that the mutual information between the two systems is 0, and it is assumed that the total state is pure.

Figure 2 .
Figure 2. The solid figure represents the set of permissible values for (H(A), H(C), H(ABC)) satisfying IIIXY = 0 for all X, Y , given the the inequalities (3) and (4) and Corollary 1.We have further made the projection H(A) = H(B) to get a 3-dimensional surface.The dashed lines span a part of Σ3 ruled out by Corollary 1, and O denotes the apex of Σ3.The ray ℓ is the top edge in the figure.