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R in many areas of physics, applied mathematics and other of real numbers: ∀a1 ∀a2 ... ∀a4n pΦ(a1,...,a4n) ≥ 0. fields which model their questions within mathematics. Note that the formula ϕ has a special simple form - it is Indeed, assume that we are dealing with a scientific prob- a negation of an existential formula. Therefore it is pos- lem which has the following general form: sible to apply the results of J. Renegar from [24] which deals with the existential theory of real numbers, see also Determine whether some mathematical object ω [25, 26] for of quantifier elimination for possesses some property π. the field R. In this way we obtain a procedure which In many cases such assertions can be stated as first-order determines whether the formula ϕ holds or not. In par- formulas over R or C. If this is indeed the case, we ticular, we are not interested in the explicit quantifier- ′ can compute a quantifier-free formula which is equiva- free form ϕ of ϕ, but our way is completely sufficient for ′ lent with the original statement. As argued above, this applications (and rather close to determining ϕ ). formula can be easily verified, so we get a complete solu- Although similar approaches to the problem we con- tion to the problem we started with. sider are known (see especially [12, 13], [29] and [7]), this The above idea is realized in [22] where we apply quan- Letter is the first paper presenting a concrete procedure tifier elimination theory for algebraically closed fields which determines the validity of ϕ. in order to find computable conditions for irreducibil- We stress that, to the best of our knowledge, the results ity of completely positive maps [6, 10]. Assume that of J. Renegar [24–26] provide the most straightforward Φ: Mn(C) → Mn(C) is completely positive, that is, approach to the generally difficult quantifier elimination there are matrices K1,...,Ks ∈ Mn(C) (called Kraus for the field of real numbers. Importantly, they are also coefficients of Φ) such that quite effective from the point of view of computational complexity. However, it is the very nature of quantifier s ∗ elimination theory for real closed fields that yields its Φ(X)= KiXKi , algorithms are rather laborious. In some sense, this is a Xi=1 cost of the fact that these procedures can be applied to for any X ∈ Mn(C). The well-known result of D. any given formula. Thanks to this generality we are able Farenick [8] states that Φ is irreducible if and only if to determine whether an arbitrary hermiticity-preserving its Kraus coefficients do not have a non-trivial common superoperator is positive or not. It is our opinion that invariant subspace. This means that if V is a subspace such a goal cannot be achieved by any other methods. n of C such that KiV ⊆ V , for any i = 1,...,s, then Details on computational complexity of quantifier elim- V =0. We show in [22] that this condition can be stated ination algorithms can be found in papers [24–26] and as some first-order sentence ϕ over the field C of com- monographs [5, 20]. We recommend the huge monograph plex numbers. Moreover, we give a simple and straight- [5] for a comprehensive treatment of various aspects of forward proof of Tarski’s theorem which is based on the quantifier elimination theory. effective version of Hilbert’s Nullstellensatz [16]. Then, Assume now that Φ ∈ L(Mn(C)) is a superoperator applying this result, we compute the quantifier-free for- that preserves hermiticity. Our first goal is to introduce mula ϕ′ such that ϕ is equivalent with ϕ′, and hence ϕ′ some real homogeneous polynomial pΦ of degree 4 in 4n becomes the computable condition we are looking for. In variables such that Φ is positive if and only if pΦ ≥ 0. this way we give a complete solution of the problem stud- Recall that there exists an isomorphism of Hilbert spaces ied in [15, 23] and many other papers, see for example J : L(Mn(C)) → Mn(C) ⊗ Mn(C), known as the Choi- [1, 2, 9, 14, 28, 31]. Jamiołkowski isomorphism [12, 13], defined by the for- We stress that quantifier elimination for algebraically mula closed fields is generally much easier than the one for n real closed fields, especially in terms of its proof and J(Φ) = Eij ⊗ Φ(Eij ), computational complexity [11, 20]. However, this Let- i,jX=1 ter is devoted to show an application of quantifier elim- M C E i, j ,...,n ination for real closed fields in quantum information for any Φ ∈ L( n( )). Here ij , for =1 , de- n n e e , theory. Indeed, we study the problem of determining notes the × complex matrix [ kl] such that kl ∈{0 1} e k i l j whether a hermiticity-preserving superoperator is a pos- and kl =1 if and only if = and = . This isomor- itive map. Our strategy is the following. Assume that phism has the crucial property that a superoperatorΦ is positive if and only if J(Φ) is block positive. Recall that Φ: Mn(C) → Mn(C) is a superoperator which preserves hermiticity. We associate with Φ some real multivariate an operator polynomial p in 4n variables such that Φ is positive if n n Φ T ∈ L(C ⊗ C ) =∼ Mn(C) ⊗ Mn(C) and only if pΦ(a1,...,a4n) ≥ 0, for any a1,...,a4n ∈ R (this is denoted by pΦ ≥ 0). In terms of first-order logic, is block positive if and only if hx ⊗ y | T (x ⊗ y)i is a non- the latter condition means that we consider the validity negative real number, for any x, y ∈ Cn. It is well-known of the following first-order formula, say ϕ, over the field that if Φ ∈ L(Mn(C)) preserves hermiticity, then J(Φ) 3 is selfadjoint. Therefore we aim to solve a more general only if the positivity polynomial pΦ := pJ(Φ) satisfies problem. Indeed, we shall express the block positivity of pΦ ≥ 0. Observe that a selfadjoint operator T as the condition pT ≥ 0 for some s real multivariate polynomial pT . J(Φ) = etrΦ(E )e = α ar ar . Cn Cn (ij)(kl) l ki j r lk ji Assume that T ∈ L( ⊗ ) is a selfadjoint operator. Xr=1 We denote by T(ij)(kl) the complex numbers such that Then some straightforward calculations yield n n n T (ǫij )= T(ij)(kl)ǫkl pΦ = pJ(Φ) = J(Φ)(ij)(kl)xkylxiyj = k,lX=1 i,jX=1 k,lX=1 where ǫij = ei ⊗ ej , for i, j = 1,...,n, and e1,...,en are the elements of the standard C-basis of Cn. Since n n s α ar ar x y x y T is selfadjoint, we get T(ij)(kl) = T(kl)(ij) and thus = r lk ji k l i j = i,jX=1 k,lX=1 Xr=1 T(ij)(ij) ∈ R, for any i,j,k,l = 1,...,n. Assume that n x = (x1,...,xn),y = (y1,...,yn) ∈ C . Then we have n s x ⊗ y = i,j=1 xiyj ǫij which yields ∗ tr 2 P = αr [x1,...,xn] · Ar · [y1,...,yn] . n n Xr=1

hx ⊗ y | T (x ⊗ y)i = T(ij)(kl)xkylxiyj . This gives an interesting description of the positivity i,jX=1 k,lX=1 polynomial pΦ as the sum of some non-positive or non- It is clear that, for any i, j =1,...,n, the number negative real multivariate polynomials, depending on signs of the numbers α1,...,αs. As a consequence, we get σ(ij) := T(ij)(ij)xiyjxiyj an alternative proof of the fact that completely positive maps are positive. Indeed, if Φ is completely positive, is a real number. Moreover, for any i,j,k,l = 1,...,n, then α1,...,αs > 0, so in this case it is obvious that we have pΦ ≥ 0. This shows that the above description of pΦ is useful. T x y x y T x y x y , (ij)(kl) k l i j = (kl)(ij) i j k l Now we show that the condition pΦ ≥ 0 can be verified in an effective way. For this purpose, we apply some gen- so if (ij) < (kl) in the lexicographical order, then the eral techniques from the study of the existential theory of number real numbers [24]. These techniques get slightly simpler if we consider homogeneous polynomials of even degrees. τ(ij)(kl) := T(ij)(kl)xkylxiyj + T(kl)(ij)xiyj xkyl = Assume that g ∈ R[x1,...,xn] is a homogeneous poly- nomial of an even degree d. In Section 3 of [24] the author constructs some finite set of real multivariate polynomi- =2Re(T(ij)(kl)xkylxiyj ) als in variables u1,...,un,un+1, depending on g. This set is crucial in our considerations, but we omit its con- is also real. In fact, we may view both σ(ij) and τ(ij)(kl) as real homogeneous polynomials of degree 4 such that struction, because it is too long and technical. We denote the set by Rg in the Letter. R 1 2 1 2 1 2 1 2 R σ(ij), τ(ij)(kl) ∈ [xi , xi , xj , xj ,yk,yk,yl ,yl ] Assuming the set Rg ⊆ [u1,...,un,un+1] is given, define J(n, d) := {0,...,nd2n} and B(n, d) to be the set where x = x1 + x2ι, y = y1 + y2ι, for any i =1,...,n, i i i i i i n−1 n−2 Nn+1 2n and ι is the imaginary unit. Therefore the polynomial {(i ,i , . . . , i, 1, 0) ∈ | i =0,...,nd }.

n Let us fix j ∈ J(n, d), β ∈ B(n, d) and r ∈ Rg. We R pT := σ(ij) + τ(ij)(kl) define some univariate polynomials r1,...,rn+1 ∈ [t] un un R i,jX=1 (ijX)<(kl) and gj,β,r,+,gj,β,r,− ∈ [t] as follows: is a homogeneous polynomial of degree 4 in 4n variables. ∂r ri(t) = ( )(β + ten+1), We call pT the positivity polynomial for T . The above ∂ui arguments imply that a selfadjoint operator T is block for any i =1,...,n +1 and positive if and only if its positivity polynomial pT satisfies the condition pT ≥ 0. gun (t)= g(r(j)(t), r(j)(t),...,r(j)(t)), s ∗ r j,β,r,+ 1 2 n Assume that Φ(X)= r=1 αrArXAr and Ar = [aij ], for any r = 1,...,s. ThenP Φ preserves hermiticity, so un (j) (j) (j) J(Φ) is selfadjoint and it follows that Φ is positive if and gj,β,r,−(t)= g(−r1 (t), −r2 (t),..., −rn (t)) 4

(j) R where ri denotes the j-th derivative of ri. It follows For two non-zero f,g ∈ [x], denote by N(f,g) the from Section 4 of [24] that g ≥ 0 holds if and only if for value of the number any j ∈ J(n, d), β ∈B(n, d) and r ∈ Rg both conditions |{x ∈ R | f(x)=0 ∧ g(x) > 0}| un (∃t − gj,β,r,+(t) > 0 ∧ rn+1(t) > 0) diminished by the number and |{x ∈ R | f(x)=0 ∧ g(x) < 0}|

un (∃t − gj,β,r,−(t) > 0 ∧−rn+1(t) > 0) where |X| is the number of elements of a finite set X. Observe that if g is a polynomial such that g > 0 (e.g. do not hold. We show that these conditions can be any positive constant polynomial), then N(f,g) is the checked by applying the generalized Sturm’s theorem, number of all distinct real roots of the polynomial f. also known as the Sturm-Tarski theorem. In general, this The Sturm-Tarski theorem states that we have the theorem allows to calculate the number of distinct real equality ν(f,f ′g) = N(f,g), and hence the number roots of real univariate polynomials satisfying some ad- N(f,g) can be computed. Observe that if g =1 is a con- ditional conditions. stant polynomial, then the above theorem implies that In the Letter we apply the Sturm-Tarski theorem in the number N(f, 1) of all distinct real roots of f can be determining the validity of the sentence computed as ν(f,f ′). This is the assertion of the original Sturm’s theorem. ∃x (p(x) > 0 ∧ q(x) > 0) Now, assume that f,p,q ∈ R[x] are non-zero polyno- mials and define where p, q ∈ R[x]. In order to state the theorem, we recall the construction of the canonical Sturm sequence S(f,p,q) := {x ∈ R | f(x)=0 ∧ p(x) > 0 ∧ q(x) > 0}. [5, 17, 34] (preceded by the construction of the general- Then [17] yields |S(f,p,q)| is equal to the number ized Sturm sequence) associated with two non-zero poly- nomials p, q ∈ R[x]. Up to the sign, its elements are 1 (N(f,p2q2)+ N(f,p2q)+ N(f,pq2)+ N(f,pq)), polynomials that occur as remainders in the Euclid’s al- 4 gorithm for determining the greatest common divisor of so in particular |S(f,p,q)| can be computed. This gives p and q. us a way to determine whether the condition Assume that p, q ∈ R[x] are non-zero polynomials. The generalized Sturm sequence is defined recursively in ∃x (p(x) > 0 ∧ q(x) > 0), the following way. First we set h0 = p and h1 = q. say ϕ, holds. Indeed, first verify whether Assume that n ≥ 1 and h0,h1,...,hn are defined. If hn | hn−1, then the generalized Sturm sequence is the lim p(x) = lim q(x)= ∞ x→a x→a sequence (h0,h1,...,hn). Otherwise, we set hn+1 = −remhn (hn−1) where remhn (hn−1) is the remainder of where a = −∞ or a = ∞. If this is the case, then ϕ division of polynomial hn−1 by hn. If (h0,h1,...,hn) is holds. Otherwise, [17] or [26] yields ϕ is equivalent with the generalized Sturm sequence for p and q, then hn | hi, the condition for any i =0,...,n and the sequence ′ ∃x (p(x) > 0 ∧ q(x) > 0 ∧ (pq) (x)=0). h0 h1 hn , ,..., =1 The latter holds if and only if |S((pq)′,p,q)| 6=0. h h h  n n n Therefore our procedure for determining the positivity is the canonical Sturm sequence. of a hermiticity-preserving superoperator Φ is complete. Assume that (h0,h1,...,hn) is the canonical Sturm Indeed, for that purpose it suffices to calculate the posi- sequence for polynomials p and q. We define tivity polynomial pΦ and check whether pΦ ≥ 0 by apply- ing the methods described above. Although the proce- s σa(h1,...,hs) := (σa(h1),...,σa(hs)) ∈{+, −} dure is rather time-consuming from the point of view of computational complexity, it seems to be the best pos- where a ∈ {−∞, ∞} and σa(h)=+, if limx→a h(x)= ∞ sible, if the goal is to obtain a general solution to the and σa(h) = − otherwise, for any polynomial h ∈ R[x]. problem we consider. Define ν(p, q) as the number The authors are indebted to J. Renegar for his assis- tance in understanding the contents of [24]. λ(σ−∞(h0,...,hn)) − λ(σ∞(h0,...,hn)) where λ(α1,...,αs) denotes the number of all sign s changes in (α1,...,αs) ∈ {+, −} , that is, sequences of the form (−, +) or (+, −). ∗ [email protected], corresponding author 5

† [email protected] unitary subspaces and decoherence in quantum systems, ‡ jam@fizyka.umk.pl Electron. J. Linear Algebra 30, 253–270 (2015). [1] Yu. Alpin, A. George and Kh. Ikramov, Solving the two [24] J. Renegar, On the computational complexity and geom- dimensional CIS problem by a rational algorithm, Linear etry of the first-order theory of the reals I. Introduction. Algebra and its Applications 312, 115–123 (2000). Preliminaries. The geometry of semi-algebraic sets. The [2] Yu. Alpin and Kh. Ikramov, Rational procedures in the decision problem for the existential theory of the reals, J. problem of common invariant subspaces of two matri- Symbolic Comput. 13 no. 3, 255–299 (1992). ces, Journal of Mathematical Sciences 114(6), 1757–1764 [25] J. Renegar, On the computational complexity and geome- (2003). try of the first-order theory of the reals II. The general de- [3] D. Arnon, Geometric Reasoning with Logic and Algebra, cision problem. Preliminaries for quantifier elimination, Artificial Intelligence 31, 37–60 (1988). J. Symbolic Comput. 13 no. 3, 301–327 (1992). [4] D. S. Arnon and M. Mignotte, On Mechanical Quanti- [26] J. Renegar, On the computational complexity and geom- fier Elimination for Elementary Algebra and Geometry, etry of the first-order theory of the reals III. Quanti- J. Symbolic Computation 5, 237–259 (1988). fier elimination, J. Symbolic Comput. 13 no. 3, 329–352 [5] S. Basu, R. Pollack and M.-F. Roy, Algorithms in Real (1992). , Springer (2016). [27] P. Rothmaler, Introduction to , Algebra, [6] I. Bengtsson I. and K. Życzkowski, Geometry of Quan- Logic and Applications Series Volume 15, Gordon and tum States: An Introduction to Quantum Entanglement, Breach Science Publishers, Amsterdam (2000). Cambridge Univ. Press (2006). [28] D. Shemesh, Common eigenvectors of two matrices, Lin. [7] M.-D. Choi, Positive Semidefinite Biquadratic Forms, Alg. and Appl. 62, 11–18 (1984). Linear Algebra Appl. 12 no. 2, 95–100 (1975). [29] Ł. Skowronek and K. Życzkowski, Positive maps, positive [8] D. R. Farenick, Irreducible positive linear maps on oper- polynomials and entanglement witnesses, J. Phys. A 42 ator algebras, Proc. Amer. Math. Soc., 124 (11), 3381– no. 32, 325302, 14 pp (2009). 3390 (1996). [30] A. Tarski, A decision method for elementary algebra and [9] A. George and Kh. Ikramov, Common invariant sub- geometry, RAND Corporation, Santa Monica (1948). spaces of two matrices, Linear Algebra and its Applica- [31] M. Tsatsomeros, A criterion for the existence of common tions (287), 171–179 (1999). invariant subspaces of matrices, Linear Algebra and its [10] T. Heinosaari and M. Ziman, The Mathematical Lan- Applications 322, 51–59 (2001). guage of Quantum Theory, Cambridge Univ. Press [32] L. van den Dries, ’s elimination theory for (2012). real closed fields, J. Symbolic Logic 53 no. 1, 7–19 (1988). [11] J. Heintz, Definability and fast quantifier elimination in [33] B. L. Van Der Waerden, Algebra, Springer-Verlag New algebraically closed fields, Theoret. Comput. Sci. 24 no. York, Vol. 2 (1991). 3, 239–277, (1983). [34] B. Xia and L. Yang, Automated inequality proving and [12] A. Jamiołkowski, An effective method of investigation of discovering, World Scientific Publishing Co. Pte. Ltd., positive maps on the set of positive definite operators, Hackensack, NJ, xii+332 pp (2016). Rep. Math. Phys. 5, 415–424 (1974). [35] B. Xia and L. Yang, Quantifier Elimination for Quartics, [13] A. Jamiołkowski, On semipositive definiteness of 2n- In: J. Calmet, T. Ida, D. Wang (eds) Artificial Intelli- degree forms, Rep. Math. Phys. 10, 259–266 (1976). gence and Symbolic Computation. AISC 2006. Lecture [14] A. Jamiołkowski, T. Kamizawa and G. Pastuszak, On In- Notes in Computer Science, vol 4120. Springer, Berlin, variant Subspace In Quantum Control Systems and Some Heidelberg. Concepts of Integrable Quantum Systems, Int. J. Theor. [36] L. Yang, Recent Advances on Determining the Number Phys. Volume 54, Issue 8, 2662–2674 (2015). of Real Roots of Parametric Polynomials, J. Symbolic [15] A. Jamiołkowski and G. Pastuszak, Generalized Shemesh Computation 28, 225–242 (1999). Criterion, Common Invariant Subspaces and Irreducible Completely Positive Superoperators, Linear Multilinear A. 63(2), 314–325 (2015). [16] Z. Jelonek, On the effective Nullstellensatz, Invent. Math. 162 no. 1, 1–17 (2005). [17] A. Khovanskii and Y. Burda, Degree of rational map- pings, and the theorems of Sturm and Tarski, J. fixed point theory appl. 3, 79–93 (2008). [18] D. A. Lidar and T. A. Brun, Quantum Error Correction, Cambridge University Press, New York (2013). [19] D. Marker, Model Theory: An Introduction, Springer, Berkeley (2002). [20] B. Mishra, Algorithmic Algebra, Texts and Monographs in Computer Science, Springer-Verlag, New York (1993). [21] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, New York (2000). [22] G. Pastuszak, The common invariant subspace problem and Tarski’s theorem, Electron. J. Linear Algebra 32, 343–356 (2017). [23] G. Pastuszak and A. Jamiołkowski, Common reducing