# Quantum Information Theory and Free Semialgebraic Geometry: One Wonderland Through Two Looking Glasses

QQUANTUM INFORMATION THEORY ANDFREE SEMIALGEBRAIC GEOMETRY:ONE WONDERLAND THROUGH TWO LOOKINGGLASSES

GEMMA DE LAS CUEVAS AND TIM NETZER

Abstract.

We illustrate how quantum information theory and free (i.e.noncommutative) semialgebraic geometry often study similar objectsfrom diﬀerent perspectives. We give examples in the context of positivityand separability, quantum magic squares, quantum correlations in non-local games, and positivity in tensor networks, and we show the beneﬁtsof combining the two perspectives. This paper is an invitation to considerthe intersection of the two ﬁelds, and should be accessible for researchersfrom either ﬁeld.

Live free or die.

Motto of New Hampshire Introduction

The ties between physics, computer science and mathematics are histori-cally strong and multidimensional. It has often happened that mathematicalinventions which were mere products of imagination (and thus thought tobe useless for applications) have later played a crucial role in physics orcomputer science. A superb example is that of imaginary numbers andtheir use in complex Hilbert spaces in quantum mechanics. Other examplesinclude number theory and its use in cryptography, or Riemannian geometryand its role in General Relativity. It is also true that physicists tend to beonly aware of the mathematical tools useful for them — so there are manybranches of mathematics which have not found an outlet in physics.The relevance of a statement depends on the glass through which welook at it. There are statements which are mathematically unimpressive butphysically very impressive. A good example is entanglement. Mathematically,the statement that the positivity cone of the tensor product space is larger

Institut f¨ur Theoretische Physik, Technikerstr. 21a, A6020 Innsbruck, Aus-triaInstitut f¨ur Mathematik, Technikerstr. 13, A6020 Innsbruck, Austria

Date : February 9, 2021. Who would have thought that the square root of − a r X i v : . [ qu a n t - ph ] F e b ONE WONDERLAND THROUGH TWO LOOKING GLASSES

Figure 1.

Free semialgebraic geometry and quantum in-formation often look at similar landscapes from diﬀerentperspectives, as in the fantastic world of this woodcut by M.C. Escher.than the tensor product of the local cones is interesting, but not particularlywild or surprising. Yet, the physical existence of entangled particles is, fromour perspective, truly remarkable. In other words, while the mathematics iseasy to understand, the physics is mind-blowing. This is particularly trueregarding Bell’s Theorem: while it is mathematically not specially deep, weregard the experimental violation of Bell inequalities [31, 35, 52] as very deepindeed. Another example is the no-cloning theorem — it is mathematicallytrivial, yet it has very far-reaching physical consequences. On the otherhand, there are many mathematically impressive statements which are — sofar — physically irrelevant. Finally, there are statements which can be bothmathematically deep and central for quantum information theory, such asStinespring’s Dilation Theorem.The goal of this paper is to illustrate how two relatively new disciplines inphysics and mathematics — quantum information theory and free semialge-braic geometry — have a lot in common (Fig. 1). ‘Free’ means noncommu-tative, because it is free of the commutation relation . So free semialgebraicgeometry studies noncommutative versions of semialgebraic sets. On theother hand, quantum information theory is (mathematically) a noncommu-tative generalisation of classical information theory. So, intuitively, ‘free’ isnaturally linked to ‘quantum’. Moreover, in both ﬁelds, positivity plays avery important role. Semialgebraic geometry examines questions arising fromnonnegativity, like polynomial inequalities. In quantum information theory,quantum states are represented by positive semideﬁnite matrices. Positivityalso gives rise to convexity, which is central in both ﬁelds, as we will see.

NE WONDERLAND THROUGH TWO LOOKING GLASSES 3

So the two disciplines often study the same mathematical objects fromdiﬀerent perspectives. As a consequence, they often ask diﬀerent questions.For example, in quantum information theory, given an element of a tensorproduct space, one wants to know whether it is positive semideﬁnite, and howthis can be eﬃciently represented and manipulated. In free semialgebraicgeometry, the attention is focused on the geometry of the set of all suchelements (see Table 1).Quantum information theory Free semialgebraic geometry

Emphasis on the element Emphasis on the set

Given ρ = (cid:80) α A α ⊗ B α , is it positivesemideﬁnite (psd)? Given { A α } , characterise the set of { B α } such that (cid:80) α A α ⊗ B α is psd.Block positive matrices / Separablepsd matrices. Largest / smallest operator systemover the psd cone.Every POVM can be dilated to aPVM. The free convex hull of the set ofPVMs is the set of POVMs.Can a correlation matrix p be re-alised by a quantum strategy? Is p in the free convex hull of thefree independence model? Table 1.

Examples of the diﬀerent approaches of quantuminformation theory and free semialgebraic geometry in study-ing essentially the same mathematical objects. The variousnotions will be explained throughout the paper.We believe that much is to be learnt by bridging the gap among the twocommunities — in knowledge, notation and perspective. In this paper wehope to illustrate this point. This paper is thus meant to be accessible forphysicists and mathematicians.Obviously we are not the ﬁrst or the only ones to notice these similari-ties. In this paper, however, we will mainly concentrate on what we havelearnt from collaborating in recent years, and we will review a few otherworks. The selection of other works is not comprehensive and reﬂects ourpartial knowledge. We also remark that quantum information theory andfree semialgebraic geometry are not the only ones studying positivity intensor product spaces.

Compositional distributional semantics , for example,represents the meaning of words by positive semideﬁnite matrices, and thecomposition of meanings is thus given by positivity preserving maps — seee.g. [13, 21].This article is organised as follows. We will ﬁrst explain basic conceptsin quantum information theory and free semialgebraic geometry (Section 2)— the reader familiar with them can skip the corresponding section. Thenwe will explain how they are related (Section 3), and we will end with someclosing words (Section 4).

ONE WONDERLAND THROUGH TWO LOOKING GLASSES Some basic concepts

Here we present some basic concepts in quantum information theory(Section 2.1) and free semialgebraic geometry (Section 2.2).Throughout the paper we denote the set of r × s complex matrices byMat r,s , the set of r × r complex matrices by Mat r , and we use the identiﬁcationof Mat r ⊗ Mat s with Mat rs . We will also often use the real subspace of Mat r containing the Hermitian elements, called Her r , and use that Her r ⊗ Her s isidentiﬁed with Her rs . The d -fold cartesian product of Her r is denoted Her dr .2.1. Basic concepts from quantum information theory.

Here we brieﬂyintroduce some concepts from quantum information theory. We focus onﬁnite-dimensional quantum systems of which we do not assume to haveperfect knowledge (in the language of quantum information, these are called mixed states ). See, e.g. [45, 56], for a more general overview.The state of a quantum system is modelled by a normalized positivesemideﬁnite matrix , i.e. a ρ ∈ Mat d with ρ (cid:60) ρ ) = 1 , where (cid:60) (cid:62) measurement on the system ismodelled by a positive operator valued measure (POVM), i.e. a set of psdmatrices τ i that sum to the identity: τ , . . . , τ n ∈ Mat d with all τ i (cid:60) (cid:88) i τ i = I d . The probability to obtain outcome i on state ρ is given bytr( ρτ i ) . (1)Note that these probabilities sum to 1 because of the normalisation conditionon the τ i ’s and ρ .When the system is composed of several subsystems, the global state spaceis modelled as a tensor product of the local spaces,Mat d = Mat d ⊗ · · · ⊗ Mat d n , (2)where d = d · · · d n .A state ρ is called separable (w.r.t. a given tensor product structure) if itcan be written as ρ = r (cid:88) i =1 ρ (1) i ⊗ · · · ⊗ ρ ( n ) i with all ρ ( j ) i (cid:60) . This is obviously a stronger requirement than ρ being psd — not every ρ is separable. Separable states are not too interesting from a quantuminformation perspective: not separable states are called entangled , andentanglement is necessary for many quantum information tasks. NE WONDERLAND THROUGH TWO LOOKING GLASSES 5 A quantum channel is the most general transformation on quantum states.Mathematically, it is modelled by a linear trace-preserving map T : Mat d → Mat s that is completely positive . Complete positivity means that the mapsid n ⊗ T : Mat nd → Mat ns are positive (i.e. map psd matrices to psd matrices) for all n , where id n isthe identity map on Mat n .Any linear map T : Mat d → Mat s is uniquely determined by its Choimatrix C T := d (cid:88) i,j =1 E ij ⊗ T ( E ij ) ∈ Mat ds , where E ij is the matrix with a 1 in the ( i, j )-position and 0 elsewhere. Itis a basic fact that T is completely positive if and only if C T is psd (see forexample [48, 55]). Moreover, a completely positive map T is entanglement-breaking [36] if and only if C T is a separable matrix, and T is a positive mapif and only if C T is block positive , i.e.tr(( σ ⊗ τ ) C T ) (cid:62) σ (cid:60) , τ (cid:60) . Note that this is weaker than C T being psd, in which case tr( χC T ) (cid:62) χ (cid:60) T : Mat d → Mat s ρ ∈ Mat d ⊗ Mat s Entanglement-breaking map Separable matrixCompletely positive map Positive semideﬁnite matrixPositive map Block positive matrix

Table 2.

Correspondence between notions of positivity forlinear maps and their Choi matrices. Entanglement-breakingmaps are a subset of completely positive maps, which area subset of positive maps. The same is true for the rightcolumn, of course. If this is expressed in the so-called computational basis (which is one speciﬁc orthonor-mal basis), this is written E ij = | i (cid:105)(cid:104) j | in quantum information. ONE WONDERLAND THROUGH TWO LOOKING GLASSES

Basic concepts from (free) semialgebraic geometry.

We nowintroduce some basic concepts from free (i.e. noncommutative) semialgebraicgeometry. For a slightly more detailed introduction, see [43] and referencestherein.Our setup starts by considering a C -vector space V with an involution ∗ .The two relevant examples are, ﬁrst, the case where V is the space of matricesand * is the transposition with complex conjugation — denoted † in quantuminformation —, and, second, C d with entrywise complex conjugation.The ﬁxed points of the involution are called self-adjoint, or Hermitian,elements. We denote the set of Hermitian elements of V by V her . This is an R -subspace of V , in which the real things happen. In the free setup, we do not only consider V but also higher levels thereof.Namely, for any s ∈ N , we consider the space of s × s -matrices with entriesover V , Mat s ( V ) = V ⊗ Mat s . Recall that Mat s refers to s × s -matrices with entries over C . Mat s ( V ) isa C -vector space with a ‘natural’ involution, consisting of transposing andapplying ∗ entrywise. This thus promotes V and ∗ to an entire hierarchy oflevels, namely Mat s ( V ) for all s ∈ N with the just described involution.We are now ready to deﬁne the most general notion of a free real set . Thisis nothing but a collection C = ( C s ) s ∈ N where each C s ⊆ Mat s ( V ) her =: Her s ( V ). We call C s the set at level s .To make things more interesting, one often imposes conditions that connectthe levels. One important example is free convexity , which is deﬁned asfollows. For any τ i ∈ C t i with i = 1 , . . . , n, and v i ∈ Mat t i ,s with n (cid:88) i =1 v ∗ i v i = I s , (3)it holds that n (cid:88) i =1 v ∗ i τ i v i ∈ C s . (4)Note that in (4) matrices over the complex numbers (namely v i ) are multipliedwith matrices over V (namely τ i ). This is deﬁned as matrix multiplicationin the usual way for v ∗ i τ i v i , and using that elements of V can be multipliedwith complex numbers and added. For example, for n = 1, t = 2, s = 1, τ = ( µ i,j ) with µ i,j ∈ V for i = 1 ,

2, and v = ( λ , λ ) t with λ i ∈ C , we have v ∗ τ v = (cid:88) i,j ¯ λ i λ j µ i,j . Because it is where positivity and other interesting phenomena happen.

NE WONDERLAND THROUGH TWO LOOKING GLASSES 7

Note that if free convexity holds, then every C s is a convex set in the realvector space Her s ( V ). But free convexity is generally a stronger conditionthan ‘classical’ convexity, as we will see.In addition, a conic version of free convexity is obtained when giving upthe normalization condition on the v i , i.e. the right hand side of Eq. (3). Inthis case, C is called an abstract operator system (usually with the additionalassumption that every C s is a proper convex cone).Now, free semialgebraic sets are free sets arising from polynomial inequal-ities. This will be particularly important for the connection we hope toillustrate in this paper. In order to deﬁne these, take V = C d with the invo-lution provided by entrywise conjugation, so that V her = R d . Let z , . . . , z d denote free variables, that is, noncommuting variables. We can imagine each z i to represent a matrix of arbitrary size — later we will substitute z i by amatrix of a given size, and this size will correspond to the level of the freesemialgebraic set.Now let ω be a ﬁnite word in the letters z , . . . , z d , that is, an ordered tupleof these letters. For example, ω could be z z z or z z z . In addition, let σ ω ∈ Mat m be a matrix (of some ﬁxed size m ) that speciﬁes the coeﬃcientsof word ω ; this is called the coeﬃcient matrix. A matrix polynomial in thefree variables z , . . . , z d is an expression p = (cid:88) ω σ ω ⊗ ω, where the sum is over all ﬁnite words ω , and where only ﬁnitely manycoeﬃcient matrices σ ω are nonzero.We denote the reverse of word ω by ω ∗ . For example, if ω = z z z then ω ∗ = z z z . In addition, ( σ ω ) ∗ is obtained by transposition and complexconjugation of σ ω . If the coeﬃcient matrices fulﬁll( σ ω ) ∗ = σ ω ∗ , (5)then for any tuple of Hermitian matrices ( τ , . . . , τ d ) ∈ Her ds we have that p ( τ , . . . , τ d ) = (cid:88) ω σ ω ⊗ ω ( τ , . . . , τ d ) ∈ Her ms . That is, p evaluated at the Hermitian matrices τ , . . . , τ d is a Hermitianmatrix itself.So, for a given matrix polynomial p satisfying condition (5), we deﬁne thefree semialgebraic set at level s as the set of Hermitian matrices of size s such that p evaluated at them is psd: C s ( p ) := (cid:110) ( τ , . . . , τ d ) ∈ Her ds | p ( τ , . . . , τ d ) (cid:60) (cid:111) . Finally we deﬁne the free semialgebraic set as the collection of all such levels: C ( p ) := ( C s ( p )) s ∈ N . ONE WONDERLAND THROUGH TWO LOOKING GLASSES

For example, let E ii denote the matrix with a 1 in entry ( i, i ) and 0elsewhere. Then the matrix polynomial p = d (cid:88) i =1 E ii ⊗ z i deﬁnes the following free semialgebraic set at level sC s ( p ) = (cid:110) ( τ , . . . , τ d ) ∈ Her ds | E ⊗ τ + . . . + E dd ⊗ τ d (cid:60) (cid:111) . The positivity condition is equivalent to τ i (cid:60) i , which gives this freeset the name free positive orthant . Note that for s = 1, the ‘free’ variablesbecome real numbers, C ( p ) = (cid:110) ( a , . . . , a d ) ∈ R d | a i (cid:62) ∀ i (cid:111) , which deﬁnes the positive orthant in d dimensions.It is easy to see that any free semialgebraic set is closed under direct sums,meaning that if ( τ , . . . , τ d ) ∈ C s ( p ), ( χ , . . . , χ d ) ∈ C r ( p ) then( τ ⊕ χ , . . . , τ d ⊕ χ d ) ∈ C r + s ( p ) , where τ i ⊕ χ i denotes the block diagonal sum of two Hermitian matrices.This is because p ( τ ⊕ χ , . . . τ d ⊕ χ d ) = p ( τ , . . . τ d ) ⊕ p ( χ , . . . χ d ) , which ispsd if and only if each of the terms is psd.Note also that a semialgebraic set is a Boolean combination of C ( p i ) fora ﬁnite set of polynomials p i . A ‘free semialgebraic set’ is thus a noncom-mutative generalisation thereof, with the diﬀerence that usually a singlepolynomial p is considered.A very special case of free semialgebraic sets are free spectrahedra , whicharise from linear matrix polynomials. A linear matrix polynomial is a matrixpolynomial where every word ω depends only on one variable, i.e. (cid:96) = σ ⊗ d (cid:88) i =1 σ i ⊗ z i with 1 being the empty word, and all σ i ∈ Her m . The corresponding free setat level s is given by C s ( (cid:96) ) = (cid:40) ( τ , . . . , τ d ) ∈ Her ds | σ ⊗ I s + d (cid:88) i =1 σ i ⊗ τ i (cid:60) (cid:41) and C ( (cid:96) ) is called a free spectrahedron . The ﬁrst level set, C ( (cid:96) ) = (cid:110) ( a , . . . , a d ) ∈ R d | σ + a σ + · · · + a d σ d (cid:60) (cid:111) , is known as a classical spectrahedron , or simply, a spectrahedron (see Fig. 2for some three-dimensional spectrahedra). If all σ i are diagonal in the samebasis, then the spectrahedron C ( (cid:96) ) becomes a polyhedron. (Intuitively, NE WONDERLAND THROUGH TWO LOOKING GLASSES 9

Figure 2.

Some three-dimensional spectrahedra taken from[44]. Spectrahedra are convex sets described by a linear matrixinequality, and polyhedra are particular cases of spectrahedra.polyhedra have ﬂat facets whereas the borders of spectrahedra can be round,as in Fig. 2.) Thus, every polyhedron is a spectrahedron, but not vice versa.While the linear image (i.e. the shadow ) of a polyhedron is a polyhedron,the shadow of a spectahedron need not be a spectahedron. The forthcomingbook [44] presents a comprehensive treatment of spectrahedra and theirshadows. One wonderland through two looking glasses

Let us now explain some recent results that illustrate how concepts andmethods from the two disciplines interact. We will focus on positivity andseparability (Section 3.1), quantum magic squares (Section 3.2), non-localgames (Section 3.3), and positivity in tensor networks (Section 3.4).3.1.

Positivity and separability.

For ﬁxed d, s ∈ N consider the set ofstates and separable states in Mat d ⊗ Mat s , namely State d,s and Sep d,s , re-spectively. Both sets are closed in the real vector space Her d ⊗ Her s . Moreover,both are semialgebraic, since State d,s is a classical spectrahedron, and Sep d,s can be proven to be semialgebraic using the projection theorem/quantiﬁerelimination in the theory of real closed ﬁelds (see, e.g., [49]).It has long been known that Sep d,s is a strict subset of State d,s whenever d, s >

1. A recent work by Fawzi [28], building on Scheiderer’s [51], strength-ens this result, by showing that the geometry of these two sets is signiﬁcantlydiﬀerent:

Theorem 1 ([28]) . If d + s > then Sep d,s is not a spectrahedral shadow.

Recall that a spectahedral shadow is the linear image of a spectrahedron.Together with the relations of Table 2, it follows from the previous resultthat the corresponding sets of linear maps T : Mat d → Mat s satisfy that:(i) Entanglement-breaking maps form a convex semialgebraic set whichis not a spectrahedral shadow, Shadows can be very diﬀerent from the actual thing, as this shadow art by KumiYamashita shows. (ii) Completely positive maps form a spectrahedron, and(iii) Positive maps form a convex semialgebraic set which is not a spectra-hedral shadow. This follows from (i), the duality of positive maps andentanglement-breaking maps, and the fact that duals of spectrahedralshadows are also spectrahedral shadows [44].Let us now consider the set of states and separable states as free sets.Namely, for ﬁxed d (cid:62) d := (State d,s ) s ∈ N and Sep d := (cid:0) Sep d,s (cid:1) s ∈ N . This is a particular case of the setup described above, where V = Mat d andthe involution is provided by † . Moreover, both sets satisfy the condition offree convexity (Eq. (4)). In addition, State d is a free spectrahedron, whereasSep d is not, since for ﬁxed s it is not even a classical spectrahedral shadowat level s due to Theorem 1.Viewing states as free sets also leads to an easy conceptual proof of thefollowing result [16], which was ﬁrst proven by Cariello [10]. Theorem 2 ([10, 16]) . For arbitrary d, s ∈ N , if ρ ∈ State d,s is of tensorrank , i.e. it can be written as ρ = σ ⊗ τ + σ ⊗ τ , (6) where σ i and τ i are Hermitian, then it is separable. Note that σ i and τ i need not be psd. Let us sketch the proof of [16] toillustrate the method. Proof.

Consider the linear matrix polynomial (cid:96) = σ ⊗ z + σ ⊗ z , where σ , σ are given in Eq. (6). The fact that ρ is a state means that thecorresponding free set of level s contains ( τ , τ ):( τ , τ ) ∈ C s ( (cid:96) ) . At level one, the spectrahedron C ( (cid:96) ) is a convex cone in R . A convexcone in the plane must be a simplex cone , i.e. a cone whose number of extremerays equals the dimension of the space. In R this means that the cone isspanned by two vectors, C ( (cid:96) ) = cone { v , v } , where v , v ∈ R . When the cone at level one is a simplex cone, the freeconvex cone is fully determined [27, 30].In addition, the sets T s := { v ⊗ η + v ⊗ η | (cid:52) η i ∈ Her s } also give rise to a free convex cone ( T s ) s ∈ N , and we have that T = C ( (cid:96) ).These two facts imply that T s = C s ( (cid:96) ) for all s ∈ N . Using a represen-tation for ( τ , τ ) in T s , and substituting into Eq. (6) results in a separabledecomposition of ρ . (cid:3) NE WONDERLAND THROUGH TWO LOOKING GLASSES 11

The crucial point in the proof is that when the cone at level one is asimplex cone, the free convex cone is fully determined. This is not a verydeep insight — it can easily be reduced to the case of the positive orthant,where it is obvious.Note that the separable decomposition of ρ obtained in the above proofcontains only two terms — in the language of [16, 24], ρ has separable rank 2.References [7, 8] also propose to use free spectrahedra to study someproblems in quantum information theory, but from a diﬀerent perspective.Given d Hermitian matrices σ , . . . , σ d ∈ Her m , one would like to knowwhether they fulﬁll 0 (cid:52) σ i (cid:52) I m , because this implies that each σ i gives rise to the binary POVM consistingof σ i , I m − σ i . In addition, one would like to know whether σ , . . . , σ d are jointly measurable , meaning that these POVMs are the marginals of onePOVM (see [7] for an exact deﬁnition).Now use σ , . . . , σ d to construct the linear matrix polynomial (cid:96) := I m ⊗ − d (cid:88) i =1 (2 σ i − I m ) ⊗ z i and consider its free spectrahedron C ( (cid:96) ) = ( C s ( (cid:96) )) s ∈ N . Deﬁne the matrixdiamond as the free spectrahedron D = ( D s ) s ∈ N with D s := (cid:40) ( τ , . . . , τ d ) ∈ Her ds | I s − d (cid:88) i =1 ± τ i (cid:60) (cid:41) , where all possible choices of signs ± are taken into account. Note that D isjust the unit ball of R d in 1-norm, which explains the name diamond . Notealso that D ⊆ S ( (cid:96) ) is equivalent to 0 (cid:52) σ i (cid:52) I m for all i = 1 , . . . , d . Sincethese ﬁnitely many conditions can be combined into a single linear matrixinequality (using diagonal blocks of matrix polynomials), D is indeed a freespectrahedron. The following result translates the joint measurability to thecontainment of free spectrahedra: Theorem 3 ([7]) . σ , . . . , σ d are jointly measurable if and only if D ⊆ C ( (cid:96) ) . That one free spectrahedron is contained in another,

D ⊆ C ( (cid:96) ), meansthat each of their corresponding levels satisfy the same containment, i.e. D s ⊆ C s ( (cid:96) ) for all s ∈ N .The containment of spectrahedra and free spectrahedra has received con-siderable attention recently [5, 30, 33, 34, 47]. One often studies inclusionconstants for containment, which determine how much the small spectrahe-dron needs to be shrunk in order to obtain inclusion. In [7, 8] this is used to Figure 3. (Left) The magic square on the fa¸cade of theSagrada Fam´ılia in Barcelona, where every row and columnadds to 33. (Right) The magic square in Albrecht D¨urer’slithograph

Melencolia I , where every row and column adds to34.quantify the degree of incompatibility, and to obtain lower bounds on thejoint measurability of quantum measurements.3.2.

Quantum magic squares.

Let us now look at magic squares and theirquantum cousins.A magic square is a d × d -matrix with positive entries such that everyrow and column sums to the same number (see Fig. 3 for two beautifulexamples.) A doubly stochastic matrix is a d × d -matrix with real nonnegativeentries, in which each row and each column sums to 1. So doubly stochasticmatrices contain a probability measure in each row and each column. Forexample, dividing every entry of D¨urer’s magic square by 34 results in adoubly stochastic matrix. Now, the set of doubly stochastic matrices formsa polytope, whose vertices consist of the permutation matrices , i.e. doublystochastic matrices with a single 1 in every row and column and 0 elsewhere(that is, permutations of the identity matrix). This is the content of thefamous Birkhoﬀ–von Neumann Theorem.A ‘quantum’ generalization of a doubly stochastic matrix is obtained byputting a POVM (deﬁned in Section 2.1) in each row and each column of a d × d -matrix. This deﬁnes a quantum magic square [17]. That is, in passingfrom doubly stochastic matrices to quantum magic squares, we promote thenonnegative numbers to psd matrices. The normalisation conditions on thenumbers (that they sum to 1) become the normalisations of the POVM (thatthey sum to the identity matrix).What is a quantum generalisation of a permutation matrix? Permuta-tion matrices only contain 0s and 1s, so in passing to the quantum version,we promote 0 and 1 to orthogonal projectors (given that 0 and 1 are theonly numbers that square to themselves). The relevant notion is thus that NE WONDERLAND THROUGH TWO LOOKING GLASSES 13 of a projection valued measure (PVM), in which each measurement opera-tor τ , . . . , τ d is an orthogonal projection, τ i = τ i . Quantum permutationmatrices are magic squares containing a PVM in each row and column [3]. While PVMs are a special case of POVMs, every POVM dilates to a PVM(see, e.g., [48]):

Theorem 4 (Naimark’s Dilation Theorem) . Let τ , . . . , τ d (of size m × m )form a POVM. Then there exists a PVM σ , . . . , σ d (of size n × n , for some n ) and a matrix v ∈ Mat n,m such that v ∗ σ i v = τ i for all i = 1 , . . . , d. In terms of free sets, this theorem states that the free convex hull of theset of PVMs is precisely the set of POVMs . Both sets are free semialgebraic,and the POVMs even form a free spectrahedron.Through the glass of free semialgebraic geometry, quantum magic squaresform a free spectrahedron over the space V = Mat d , equipped with entrywisecomplex conjugation as an involution. Level s corresponds to POVMs withmatrices of size s × s , and thus level 1 corresponds to doubly stochasticmatrices. We thus recover the magic in the classical world at level 1, and wehave an inﬁnite tower of levels on top of that expressing the quantum case.Furthermore, quantum permutation matrices form a free semialgebraicset whose ﬁrst level consists of permutation matrices. The ‘classical magic’is thus again found at level 1, and the quantum magic is expressed in aninﬁnite tower on top of it.Now, recall that the Birkoﬀ–von Neumann theorem says that the convexhull of the set of permutation matrices is the set of doubly stochastic matrices.So the permutation matrices are the vertices of the polytope of doublystochastic matrices. In the light of the towers of quantum magic squaresand quantum permutation matrices, this theorem fully characterises whathappens at level one. We ask whether a similar characterisation is possible forthe quantum levels: Is the free convex hull of quantum permutation matricesequal to the set of quantum magic squares?

This question can be phrased in terms of dilations as follows. By Naimark’sDilation Theorem we know that every POVM dilates to a PVM. The questionis whether this also holds for a two-dimensional array of POVMs, i.e. whetherevery square of POVMs can dilated to a square of PVMs. The non-trivial partis that the dilation must work simultaneously for all POVMs in the rows andcolumns. The two-dimensional version of Naimark’s Dilation Theorem canthus be phrased as:

Does every quantum magic square dilate to a quantumpermutation matrix?

The answer to these questions is ‘no’: these quantum generalisations failto be true in the simplest nontrivial case. This means that there must existvery strange (and thus very interesting) quantum magic squares: See the closely related notion of quantum Latin squares [37, 39], which in essence arequantum permutation matrices with rank 1 projectors.

Theorem 5 ([17]) . For each d (cid:62) , the free convex hull of the free semial-gebraic set of d × d quantum permutation matrices is strictly contained inthe free spectrahedron of quantum magic squares. This strict containmentalready appears at level s = 2 . The latter statement means that there is a d × d -matrix with POVMs ofsize 2 × Non-local games and quantum correlations.

Consider a gamewith two players, Alice and Bob, and a referee. The referee chooses a questionrandomly from ﬁnite sets Q A and Q B for Alice and Bob, respectively, andsends them to Alice and Bob. Upon receiving her question, Alice choosesfrom a ﬁnite set A A of answers, and similarly Bob chooses his answer fromthe ﬁnite set A B . They send their answers to the referee, who computes awinning function w : Q A × Q B × A A × A B → { , } to determine whether they win or lose the game (value of w being 1 or 0,respectively).During the game, Alice and Bob know both the winning function w and the probability measure on Q A × Q B used by the referee to choosethe questions. So before the game starts Alice and Bob agree on a jointstrategy. However, during the game Alice and Bob are ‘in separate rooms’(or in separate galaxies) so they cannot communicate. In particular, Alicewill not know Bob’s question and vice versa. In order to ﬁnd the strategythat maximises the winning probability, Alice and Bob have to solve anoptimisation problem. What kind of strategies may Alice and Bob choose? It depends on theresources they have. First, in a classical deterministic strategy , both Aliceand Bob reply deterministically to each of their questions, and they do soindependently of each other. This is described by two functions c A : Q A → A A and c B : Q B → A B , which specify which answer Alice and Bob give to each question.Slightly more generally, in a classical randomised strategy , Alice andBob’s answers are probabilistic, but still independent of each other. This isdescribed by r A : Q A → Pr( A A ) and r B : Q B → Pr( A B ) , Thus, strictly speaking, this is not a game in the game-theoretic sense, but (just) anoptimisation problem.

NE WONDERLAND THROUGH TWO LOOKING GLASSES 15 where Pr( S ) denotes the set of probability measures on the set S . Namely,if Alice receives question a , the probability that she answers x is given by r A ( a )( x ), where r A ( a ) is the probability measure on A A corresponding toquestion a . Similarly, Bob answers y to b with probability r B ( b )( y ). SinceAlice and Bob answer independently of each other, the joint probability ofanswering x, y upon questions a, b is the product of the two, p ( x, y | a, b ) = r A ( a )( x ) · r B ( b )( y ) . (7)Finally, a quantum strategy allows them to share a bipartite state ρ ∈ State d,s . The questions determine which measurement to apply to their partof the state, and the measurement outcomes determine the answers. This isdescribed by functions q A : Q A → POVM d ( A A ) and q B : Q B → POVM s ( A B )(8)whose image is the set of POVMs with matrices of size d × d and s × s ,respectively, on the respective sets of answers. The probability that Aliceanswers x upon receiving a is described by q A ( a )( x ), which is the psd matrixthat the POVM q A ( a ) assigns to answer x . Similarly, Bob’s behaviour ismodelled by q B ( b )( y ). Since they act independently of each other, this isdescribed by the tensor product of the two. Using rule (1), we obtain thattheir joint probability is given by p ( x, y | a, b ) = tr( ρ ( q A ( a )( x ) ⊗ q B ( b )( y ))) . (9)Now, the table of conditional probabilities( p ( x, y | a, b )) ( a,b,x,y ) ∈Q A ×Q B ×A A ×A B is called the correlation matrix of the respective strategy. For any givenkind of strategy, the set of correlation matrices is the feasible set of theoptimisation problem that Alice and Bob have to solve. The objectivefunction of this optimisation problem is given by the winning probability.Since this objective function is linear in the correlation matrix entries, onecan replace the feasible set by its convex hull.The important fact is that quantum strategies cannot be reproduced byclassical randomised strategies: Theorem 6 ([4, 12]) . If at least questions and answers exist for bothAlice and Bob, the convex hull of correlation matrices of classical randomisedstrategies is strictly contained in the set of correlation matrices of quantumstrategies. For classical randomised strategies, passing to the convex hull has thephysical interpretation of including a hidden variable . The latter is a variablewhose value is unknown to us, who are describing the system, and it isusually denoted λ . However, this mysterious variable λ is shared betweenAlice and Bob, and it will determine the choice of their POVMs together with their respective questions a, b . This is the physical interpretation of theconvex hull p ( x, y | a, b ) = (cid:88) λ q λ r A ( a, λ )( x ) · r B ( b, λ )( y ) , where q λ is the probability of the hidden variable taking the value λ . Forexample, we can imagine that Alice and Bob are listening to a radio stationwhich plays songs from a certain list, but this is a ‘private’ radio station towhich we have no access. The song at the moment of playing the game (i.e.receiving the questions) will determine the value of λ (i.e. λ is an index ofthat list).Theorem 6 thus states that quantum strategies cannot be emulated byclassical strategies, even if we take into account ‘mysterious’ hidden variables.Let us now approach these results from the perspective of free sets. Assumefor simplicity that all four sets Q A , Q B , A A , A B have two elements. A quan-tum strategy consists of a state ρ ∈ State d,s and the following psd matricesfor Alice and Bob, respectively, satisfying this normalisation condition: σ ( i ) j (cid:60) τ ( i ) j (cid:60) (cid:88) j σ ( i ) j = I d and (cid:88) j τ ( i ) j = I s , (10)where i, j = 1 ,

2. The superscript refers to the questions and the subscriptto the answers. The correlation matrix is given by (cid:16) tr (cid:16) ρ (cid:16) σ ( i ) k ⊗ τ ( j ) l (cid:17)(cid:17)(cid:17) i,j,k,l . Using the spectral decomposition of ρ = (cid:80) r v r v ∗ r , it can be written as (cid:32)(cid:88) r v ∗ r (cid:16) σ ( i ) k ⊗ τ ( j ) l (cid:17) v r (cid:33) i,j,k,l , (11)where v r ∈ C d ⊗ C s and (cid:80) r v ∗ r v r = 1. Through the looking glass of freesemialgebraic geometry, this is ﬁrst level of a free convex hull. To see this,deﬁne the free set I as I = (cid:91) d,s ≥ (cid:26)(cid:16) σ ( i ) k ⊗ τ ( j ) l (cid:17) i,j,k,l ∈ Mat (Mat d ⊗ Mat s ) | σ ( i ) k and τ ( j ) l satisfy (10) (cid:27) (12)(Note that the 4 is due the fact that we have 2 questions and 2 answers; moregenerally we would have a matrix of size |Q A ||Q B | × |A A ||A B | . Note alsothat the ordering of questions and answers of Alice and Bob is irrelevant forthe following discussion.) NE WONDERLAND THROUGH TWO LOOKING GLASSES 17

If we look at level 1 of this free set, we encounter that I is the subset ofMat ( R ) consisting precisely of the correlation matrices of classical random-ized strategies. In other words, when d = s = 1, the formula coincides withthat of (7). Furthermore, higher levels of this free set contain the tensorproducts of POVMs of Alice and Bob in the corresponding space Mat d andMat s . Since I is called the independence model in algebraic statistics [25], wecall I the free independence model , since this is the natural noncommutativegeneralisation of independent strategies.Let us now consider the free convex hull of I . First of all, computingthe conditional probabilities of a pair of POVMs with a given state ρ corre-sponds to compressing to level 1 with the vectors { v r } given by the spectraldecomposition of ρ , as in (11). So the set of quantum correlations is the ﬁrstlevel of the free convex hull of the free independence model. We thus encounter an interesting phenomenon: the free convex hull of afree set can be larger than the classical convex hull at a ﬁxed level. Speciﬁcally,the convex hull of I is the set of classical correlations, whereas the freeconvex hull of I at level 1 is the set of quantum correlations, which arediﬀerent by Theorem 6. In fact, wilder things can happen: fractal sets canarise in the free convex hull of free semialgebraic sets [1]. We wonder whatthese results imply for the corresponding quantum information setup.Now, in the free convex hull of I , what do higher levels correspond to?Compressing to lower levels (i.e. with smaller ds ) corresponds to taking thepartial trace with a psd matrix of size smaller than ds . This results in 4 psdmatrices (one for each i, j, k, l ), each of size < ds , and which not need be anelementary tensor product.What about ‘compressing’ to higher levels? Any compression to a higherlevel can be achieved by direct sums of the POVMs of Alice and Bob and acompression to a lower level as we just described. The number of elementsin this direct sum is precisely n in (4). Another way of seeing that thedirect sum is needed is by noting that, if n = 1, the matrices v i cannotfulﬁll the normalisation condition on the right hand side of (4). In quantuminformation terms, this says that a POVM in a given dimension cannot betransformed to a POVM in a larger dimension by means of an isometry,because the terms will sum to a projector instead of the identity.Let us make two ﬁnal remarks. The ﬁrst one is that I is not a freesemialgebraic set, for the simple reason that it is not closed under directsums (which is a property of these sets, as we saw in Section 2.2), as is easilychecked.The second remark is that the free convex hull of the free independencemodel is not closed. This follows from the fact that, at level 1, this freeconvex hull fails to be closed, as shown in [53] and for smaller sizes in [26]. Theorem 7 ([26, 53]) . For at least questions and answers, the set ofquantum correlation matrices is not closed. In our language, this implies that the level ds — which is to be compressedto level 1 in the construction of the free convex hull — cannot be upperbounded. That is, the higher ds the more things we will obtain in itscompression to level 1.In the recent preprint [29] the membership problem in the closure of theset of quantum correlations is shown to be undecidable, for a ﬁxed (and largeenough) size of the sets of questions and answers.A computational approach to quantum correlations, comparable to sums-of-squares and moment relaxation approaches in polynomial optimisation[6, 44], is the NPA-hierarchy [40–42]. We brieﬂy describe the approach here,omitting technical details. Assume one is given a table p = ( p ( x, y | a, b )) ( a,b,x,y ) ∈Q A ×Q B ×A A ×A B , and the task is to check whether it is the correlation matrix of a quantumstrategy. The NPA hierarchy provides a family of necessary conditions, eachmore stringent than the previous one, for p to be a quantum strategy.In order to understand the NPA hierarchy, we will ﬁrst assume that p is a correlation matrix, i.e. there is a state ρ and strategies such that (9)holds. We will use this state and strategies to deﬁne a positive functional ona certain algebra. Namely, we consider the game ∗ -algebra G := C (cid:104)Q A × A A , Q B × A B (cid:105) . This is an algebra of polynomials in certain noncommuting variables. Explic-itly, for each question and answer pair from Alice and Bob, ( a, x ) and ( b, y ),there is an associated self-adjoint variable, z ( a,x ) and z ( b,y ) , respectively. G consists of all polynomials with complex coeﬃcients in these variables; forexample, the monomial z ( a,x ) z ( b,y ) ∈ G . Now, if we had the strategy ρ, q A , q B we could construct a linear functional ϕ : G → C by evaluating the variables z ( a,x ) and z ( b,y ) at the psd matrices q A ( a )( x ) ⊗ I s and I d ⊗ q B ( b )( y ), respectively, and computing the trace inner product withthe state ρ . So, in particular, evaluating ϕ at the monomial z ( a,x ) z ( b,y ) wouldyield ϕ ( z ( a,x ) z ( b,y ) ) = tr( ρ (( q A ( a )( x ) ⊗ I s ) · ( I d ⊗ q B ( b )( y ))))(13) = p ( x, y | a, b ) . (14)The crucial point is that ϕ evaluated at this monomial needs to have thevalue p ( x, y | a, b ) for any strategy realising p . In other words, the linearconstraint on ϕ expressed in Equation (14) must hold even if we do not knowthe strategy. This functional must satisfy other nice properties independentlyof the strategy too, such as being positive. NE WONDERLAND THROUGH TWO LOOKING GLASSES 19

This perspective is precisely the one we now take. Namely, we assumethat the strategy ρ, q A , q B is not given (since our question is whether p is a quantum strategy at all), and we search for a functional on G thathas the stated properties (or other properties, depending on the kind ofstrategies one is looking for). When restricted to a ﬁnite-dimensional subspaceof G , this becomes a semideﬁnite optimisation problem, as can be easilychecked. The dimension of this subspace will be the parameter indicating thelevel of the hierarchy, which is gradually increased. Solvability of all thesesemideﬁnite problems is thus a necessary condition for p to be a quantumcorrelation matrix. In words, the levels of the NPA hierarchy form an outerapproximation to the set of correlations. Conversely, if all/many of theseproblems are feasible, one can apply a (truncated) Gelfand-Naimark-Segal(GNS) construction (see for example [48]) to the obtained functional, andthereby try to construct a quantum strategy that realises p . This is thecontent of the NPA hierarchy from the perspective of free semialgebraicgeometry.3.4. Positivity in tensor networks.

Let us ﬁnally explain some resultsabout positivity in tensor networks. The results are not as much related tofree semialgebraic geometry as to positivity and sums of squares, as we willsee.Since the state space of a composite quantum system is given by thetensor product of smaller state spaces (Eq. (2)), the global dimension d grows exponentially with the number of subsystems n . Very soon it becomesinfeasible to work with the entire space — to describe n = 270 qubits d i = 2,we would need to deal with a space dimension d ∼ ∼ , the estimatednumber of atoms in the Universe. To describe anything at the macro-scaleinvolving a mole of particles, ∼ , we would need a space dimensionof ∼ , which is much larger than a googol (10 ), but smaller thana googolplex (10 ). These absurd numbers illustrates how quickly theHilbert space description becomes impractical — in practice, it works wellfor a few tens of qubits. Fortunately, many physically relevant states admit an eﬃcient description.The ultimate reason is that physical interactions are local (w.r.t. a particulartensor product decomposition; this decomposition typically reﬂects spatiallocality). The resulting relevant states admit a description using only a fewterms for every local Hilbert space. The main idea of tensor networks isprecisely to use a few matrices for every local Hilbert space Mat d i (Eq. (2);see, e.g., [11, 46]).Now, this idea interacts with positivity in a very interesting way (Fig. 4).Positivity is a property in the global space Mat d which cannot be easilytranslated to positivity properties in the local spaces. As we will see, there The lack of scalability of this description is far from being a unique case in physics —most theories are not scalable. One needs to ﬁnd the new relevant degrees of freedom atthe new scale, which will deﬁne an emergent theory.

Figure 4.

The notion of positivity gives rise to convexity,which gives rise to many surprising eﬀects when interactingwith the multiplicity of systems, as in this lithograph by M.C. Escher.is a ‘tension’ between using a few matrices for each local Hilbert spaceand representing the positivity locally. This mathematical interplay hasimplications for the description of quantum many-body systems, amongothers.Let us see one example of a tensor network decomposition where this positivity problem appears. To describe a mixed state in one spatial dimensionwith periodic boundary conditions we use the matrix product density operatorform (MPDO) of ρ , ρ = r (cid:88) i ,...,i n =1 ρ (1) i ,i ⊗ ρ (2) i ,i ⊗ · · · ⊗ ρ ( n ) i n ,i . The smallest such r is called the operator Schmidt rank of ρ [54, 57]. Clearly,every state admits an MPDO form, and the ones with small r can be handledeﬃciently. But how is the positivity of ρ reﬂected in the local matrices?Clearly, if all local matrices are psd (i.e. ρ ( k ) ij (cid:60)

0) then ρ will be psd. Butsome sums of non-psd matrices will also give rise to a global psd matrix,since negative subspaces may cancel in the sum. Can one easily characterisethe set of local matrices whose sum is psd? The short answer is ‘no’.For further reference, if all local matrices are psd, so that ρ is separable,the corresponding r is called the separable rank of ρ [20, 24].To obtain a local certiﬁcate of positivity, we ﬁrst express ρ = ξξ ∗ (whichis possible only if ρ is psd) and then apply the tensor network ‘philosophy’to ξ , i.e. express ξ as an MPDO: ρ = ξξ ∗ with ξ = r (cid:88) i ,...,i n =1 ξ (1) i ,i ⊗ ξ (2) i ,i ⊗ · · · ⊗ ξ ( n ) i n ,i . NE WONDERLAND THROUGH TWO LOOKING GLASSES 21

This is the local puriﬁcation form of ρ . Note that there are many ξ thatsatisfy ρ = ξξ ∗ , as ξ need not be Hermitian or a square matrix (it could be acolumn vector). The smallest r among all such ξ is called the puriﬁcationrank of ρ .The interesting point for the purposes of this paper is that the puriﬁcationrank is a noncommutative generalisation of the positive semideﬁnite rank ofa nonnegative matrix. There are many more such connections: the separablerank, the translational invariant (t.i.) puriﬁcation rank, and the t.i. separablerank are noncommutative generalisations of the nonnegative rank, the cpsdrank and the cp rank of nonnegative matrices, respectively [24]. As a matterof fact, this connection holds in much greater generality, as we will explainbelow. In all of these cases, the ranks coincide for quantum states that arediagonal in the computational basis.From our perspective, this connection is beneﬁcial for both sides. Forexample, for quantum many-body systems, this insight together with theresults by [32] leads to the following result: Theorem 8 ([14, 24]) . The puriﬁcation rank cannot be upper bounded by afunction of the operator Schmidt rank only. The separable rank cannot beupper bounded by a function of the puriﬁcation rank only. (It is worth noting these separations are not robust, as they disappear inthe approximate case for certain norms [22].)Conversely, the quantum perspective provides a natural and well-motivatedpath for generalisation of the ‘commutative’ results about cpsd rank, cprank, etc. For example, in [32] it is shown that the extension complexityof a polytope w.r.t. a given cone is given by the rank of the slack matrixof that polytope w.r.t. that cone. We wonder whether this result could begeneralisation to the noncommutative world. This would give a geometric interpretation of the puriﬁcation rank, the separable rank and their symmetricversions, perhaps as extension complexities of some objects.

Symmetry is a central property in physics, both conceptually and practi-cally. Conceptually, symmetry is the other side of the coin of a conservedquantity (by Noether’s theorem). Practically, it allows for more eﬃcientmathematical descriptions, as symmetric objects have fewer degrees of free-dom. For example, in the above context, ρ is translational invariant if itremains unchanged under cyclic permutations of the local systems. Thisraises the question: is there an MPDO form that explicitly expresses thissymmetry? For example, the following form does, ρ = r (cid:88) i ,...,i n =1 ρ i ,i ⊗ ρ i ,i ⊗ · · · ⊗ ρ i n ,i , because it uses the same matrices on every site, and the arrangement ofindices is such that a cyclic permutation of the local systems does not change ρ . But does this hold for other symmetries too?The existence of such invariant decompositions and their correspondingranks has been studied in a very general framework [20]. Explicitly, everytensor decomposition is represented as a simplicial complex, where theindividual tensor product spaces are associated to the vertices, and thesummation indices to the facets. The symmetry is modelled by a groupaction on the simplicial complex. The central result is that an invariantdecomposition exists if the group action is free on the simplicial complex [20].Just to give one example, if ρ ∈ Mat d ⊗ Mat d is separable and symmetric, itwill in general not admit a decomposition of the type ρ = (cid:88) α ρ α ⊗ ρ α with all ρ α psd,but it will have one of the type ρ = (cid:88) α,β ρ α,β ⊗ ρ β,α with all ρ α,β psd . From the perspective of our framework, this is due the fact that the grouppermuting the two end points of an edge does not act freely on the edgeconnecting them. But this group action can be made free if the two pointsare connected by two edges, leading to the two indices α, β in the above sum.This is one example of a reﬁnement of a simplicial complex, which makesthe action of the group free [20].Finally, we remark that this framework of tensor decompositions withinvariance can not only be applied to quantum-many body systems, butto any object in a tensor product space. One example are multivariatesymmetric polynomials with positivity conditions [23].A related question is the existence of invariant decompositions uniform in the system size. Namely, given a tensor ρ = ( ρ α,β ) α,β =1 ,...,r with all ρ α,β ∈ Mat d , deﬁne τ n ( ρ ) := r (cid:88) i ,...,i n =1 ρ α ,α ⊗ ρ α ,α ⊗ · · · ⊗ ρ α n ,α ∈ Mat d n for all n ∈ N . The result, in this case, is very diﬀerent from the ﬁxed n case: Theorem 9 ([15]) . Let d, r (cid:62) . Then it is undecidable whether τ n ( ρ ) (cid:60) for all n ∈ N . Using this result it can be shown that a translationally invariant localpuriﬁcation of τ n ( ρ ) uniform in the system size need not exist [15].The proof of this theorem uses a reduction from the matrix mortalityproblem. In the latter, given a ﬁnite set of matrices M α ∈ Mat d ( Z ), one NE WONDERLAND THROUGH TWO LOOKING GLASSES 23 is asked whether there is a word w such that 0 = M w · · · M w n ∈ Mat d ( Z ).While this problem is noncommutative (because matrix multiplication is),the problem about τ n ( ρ ) is ‘more’ noncommutative. Intuitively, if all ρ α,β are diagonal, we recover a version of the matrix mortality problem. Notealso that the space where τ n ( ρ ) lives grows with n , in contrast to the matrixmortality problem.The decidability of a similar problem can be studied for more generalalgebras. In that case, ρ α,β is in a certain algebra, and τ n ( ρ ) is asked to bein a certain cone [19].Let us ﬁnally explain a computational approach for the ﬁnite case. Soconsider n ﬁxed and recall that after specifying some local matrices ρ ( j ) i , onewants to know whether ρ = (cid:88) i ρ (1) i ⊗ · · · ⊗ ρ ( n ) i is psd. Since the n is ﬁxed, this problem is decidable, but computing anddiagonalising ρ is impossible for large values of n (in fact it is NP -hard [38]).So one has to come up with a diﬀerent idea. What can be computed arecertain moments of ρ , i.e. the numbers tr( ρ k ) for small enough k . This followsfrom the observation that the moments only require local matrix products,tr( ρ k ) = r (cid:88) i ,...,i k =1 tr (cid:16) ρ (1) i · · · ρ (1) i k (cid:17) · · · tr (cid:16) ρ ( n ) i · · · ρ ( n ) i k (cid:17) . These few moments can then be used to compute optimal upper and lowerbounds on the distance of ρ to the cone of psd matrices [18]. Speciﬁcally, tocompute this distance it suﬃces to compare ρ with f ( ρ ), where f : R → R is the function that leaves the positive numbers unchanged, and sets thenegative numbers to zero. We then approximate f by polynomial functions q of low degree, so that tr( q ( ρ )) only uses a few moments of ρ . The bestresults were obtained with certain sums of squares approximations , whichcan be computed with a linear or semideﬁnite optimisation.4. Closing words

We have illustrated how quantum information theory and free semialge-braic geometry often study very similar mathematical objects from diﬀerentperspectives. We have given the examples of positivity and separability(Section 3.1), quantum magic squares (Section 3.2), non-local games (Sec-tion 3.3), and positivity in tensor networks (Section 3.4). In all of these cases,we have tried to illustrate how results can be transferred among the twoﬁelds, and how this can be beneﬁcial for the two perspectives. As mentionedin the introduction, there are many similar such connections which have notbeen covered here.

Going back to New Hampshire’s motto, we conclude that it is undecidableto determine whether to live free or die, because both the question of whethermatrices generate a free semigroup and the matrix mortality problem areundecidable.

Acknowledgements .— GDLC acknowledges support from the AustrianScience Fund (FWF) with projects START Prize Y1261-N and the StandAlone project P33122-N. TN acknowledges support from the Austrian ScienceFund (FWF) with Stand Alone project P29496-N35.

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