0305-4470_39_2_007

Algebraic invariants of five qubits This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2006 J. Phys. A: Math. Gen. 39 371 (http://iopscience.iop.org/0305-4470/39/2/007) Download details: IP Address: 202.113.13.10 The article was downloaded on 08/05/2012 at 02:38 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 39 (2006) 371–377 doi:10.1088/0305-4470/39/2/007 Algebraic invariants of five qubits Jean-Gabriel Luque and Jean-Yves Thibon Institut Gaspard Monge, Universite´ de Marne-la-Valle´e, 77454 Marne-la-Valle´e Cedex, France Received 6 September 2005, in final form 16 November 2005 Published 14 December 2005 Online at stacks.iop.org/JPhysA/39/371 Abstract We consider the action of the group SL(2,C )⊗5 on fifth rank tensors on a two-dimensional space, that is, the Hilbert space of a five-qubit system. The Hilbert series of the algebra of polynomial invariants of five qubits pure states is obtained, and the simplest invariants are computed. PACS numbers: 02.20.Qs, 03.65.Ud, 02.10.Xm 1. Introduction The invariant theory of hypermatrices, which aims to describe the action of the full product group G = SL(V1) ⊗ · · · ⊗ SL(Vr) on a tensor space V1 ⊗ · · · ⊗ Vr , has recently been connected to various problems in mathematical physics, including calculation of multiple integrals [1, 2], and the investigation of entanglement patterns in quantum information theory. Indeed, quantifying entanglement in multipartite systems is a fundamental issue. However, for systems with more than two parts, very little is known in this respect. A few useful entanglement measures for pure states of three or four qubits have been investigated [3–5], but one is still far from a complete understanding. Furthermore, for systems of up to four qubits, a complete classification of entanglement patterns and of corresponding invariants under the group G, called in this context the group of invertible local filtering operations, is known [6, 7]. Klyachko [8, 9] proposed to associate entanglement (of pure states) in a k-partite system (or perhaps, one should say ‘pure k-partite’ entanglement) with the mathematical notion of semi-stability, borrowed from geometric invariant theory, which means that at least one G invariant is nonzero. For such states, the absolute values of these invariants provide some kind of entanglement measure. However, even for system of k qubits, the complexity of these invariants grows very rapidly with k. For k = 2, they are given by simple linear algebra [10, 11]. The case k = 3 is already nontrivial but appears in the physics literature in [12] and boils down to a mathematical result which was known by 1880 [13]. The case k = 4 is quite recent [7], and to the best of our knowledge, nothing is known for five-qubit systems1. 1 Just after the first version of this note was posted (quant-ph/0506058), A Osterloh and J Siewert informed us of their independent work [14] on the five-qubit problem (see our section ‘Conclusion’ for a short discussion). Since then, an alternative interpretation of some of our invariants has been given by Le´vay [15]. 0305-4470/06/020371+07$30.00 © 2006 IOP Publishing Ltd Printed in the UK 371 372 J-G Luque and J-Y Thibon Our main result is a closed expression of the Hilbert series of the algebra of SLOCC invariants of pure five-qubit states. This result, which determines the number of linearly independent homogeneous invariants in any degree, was obtained through intensive symbolic computations relying on a very recent algorithm for multivariate residue calculations [16]. We point out a few properties which can be read off from the series, and determine the simplest invariants, which are of degrees 4 and 6 in the component of the states. 2. Hilbert series Denote by V = C 2 the local Hilbert space of a two-state particle. The state space of a five-particle system is H = V ⊗5, which will be regarded as the natural representation of the group of invertible local filtering operations, also known as reversible stochastic local quantum operations assisted by classical communication G = GSLOCC = SL(2,C)⊗5, that is, the group of 5-tuples of complex unimodular 2 × 2 matrices. We will denote by |�〉 = 1∑ i1,i2,i3,i4,i5=0 Ai1i2i3i4i5 |i1〉|i2〉|i3〉|i4〉|i5〉 a state of the system. An element g = (kgji ) of G maps |�〉 to the state |� ′〉 = g|�〉 whose components are given by A′i1i2i3i4i5 = ∑ j 1g j1 i1 2g j2 i2 3g j3 i3 4g j4 i4 5g j5 i5 Aj1j2j3j4j5 . (1) We are interested in the dimension of the space Id of all G-invariant homogeneous polynomials of degree d = 2m (Id = 0 for odd d) in the 32 variables Ai1i2i3i4i5 . It is known that it is equal to the multiplicity of the trivial character of the symmetric groupS2m in the fifth power of its irreducible character labelled by the partition [m,m], hence given by the following scalar product of characters (cf [17]): dim Id = 〈χ2m|(χmm)5〉 = 1 (2m)! ∑ σ∈S2m χmm(σ )5. (2) The generating function of these numbers h(t) = ∑ d�0 dim Id td (3) is called the Hilbert series of the algebra I =⊕d Id . Standard manipulations with symmetric functions allow us to express it as a multidimensional residue: h(t) = ∮ du1 2π iu1 · · · ∮ du5 2π iu5 A(u) B(u; t) (4) where the contours are small circles around the origin, A(u) = 5∏ i=1 ( 1 + 1/u2i ) (5) and B(u; t) = ∏ ai=±1 ( 1 − t ua11 ua22 ua33 ua44 ua55 ) . (6) Invariants of five qubits 373 Table 1. Coefficients of P(t). n an n an n an n an 0 1 30 24 659 54 225 699 78 9664 8 16 32 36 611 56 214 238 80 5604 10 9 34 52 409 58 195 358 82 3024 12 82 36 71 847 60 172 742 84 1659 14 145 38 95 014 62 146 849 86 770 16 383 40 119 947 64 119 947 88 383 18 770 42 14 849 66 95 014 90 145 20 1659 44 172 742 68 71 847 92 82 22 3024 46 195 358 70 52 409 94 9 24 5604 48 214 238 72 36 611 96 16 26 9664 50 225 699 74 24 659 104 1 28 15 594 52 229 752 76 15 594 Such multidimensional residues are notoriously difficult to evaluate. After trying various approaches, we eventually succeeded by means of a recent algorithm due to Guoce Xin [16], in a Maple implementation. The result can be cast in the form h(t) = P(t) Q(t) (7) where P(t) is an even polynomial of degree 104 with non-negative integer coefficients an, P(t) = 52∑ k=0 a2kt 2k given in table 1 2, and Q(t) = (1 − t4)5(1 − t6)(1 − t8)5(1 − t10)(1 − t12)5. On this expression, it is clear that a complete description of the algebra of G-invariant polynomials by generators and relations is out of reach of any computer system. Nevertheless, inspection of the Hilbert series suggests the following kind of structure for this algebra. We know, since dimH− dimG = 25 − 3 × 5 = 17, that there must exist a set of 17 algebraically independent invariants. The denominator of the series, which is precisely a product of 17 factors, makes it plausible that these invariants can be chosen as five polynomials of degree 4 (to be denoted by Dx,Dy,Dz,Dt ,Du), one polynomial of degree 6 (F), five polynomials of degree 8 (H1,H2, . . . , H5), one polynomial of degree 10 (J ) and five polynomials of degree 12 (L1, . . . , L5). Such a set of 17 polynomials is called a set of primary invariants. Their choice is of course not unique. The numerator should then describe the secondary invariants, that is, a set of 3014 400 homogeneous polynomials (1 of degree 0 , 16 of degree 8, 9 of degree 10, 82 of degree 12, etc) such that any invariant polynomial can be uniquely expressed as a linear combination of secondary invariants, the coefficients being themselves arbitrary polynomials in the primary invariants. Actually, it is known that the algebra of G-invariants admits such a structure (this is called a Cohen–Macaulay ring, see [18]), with a set of 17 primary invariants. But the representation of h(t) as P(t)/Q(t) is not unique, and the knowledge of h(t) is not sufficient to determine 2 Note that there is no known way to predict this degree without completing the calculation. 374 J-G Luque and J-Y Thibon the degrees of the primary invariants. However, evidence for our conjecture is provided by the subset of primary invariants computed in section 3. This picture, which is the simplest kind of description to be expected, is far too complex for physical applications. The best that can be done is to use the Hilbert series as a guide for finding explicitly a small set of reasonably simple invariants, in particular, the primary invariants of lowest degrees. We have computed the first primary invariants, those of degrees 4 and 6, and found good candidates in degree 8, using methods from classical invariant theory (cf [19]). 3. The simplest invariants 3.1. Transvectants and Cayley’s Omega process In order to apply the formalism of classical invariant theory, a state |�〉 will be interpreted as a quintilinear form on C 2 (called the ground form) f := 1∑ i1,i2,i3,i4i5=0 Ai1i2i3i4i5xi1yi2zi3 ti4ui5 . A covariant of f is a G-invariant polynomial in the coefficients Ai1i2i3i4i5 and the variables xi, yi, zi, ti and ui . A complete set of covariants can, in principle, be computed from the ground form by means of the so-called Omega process (see [19] for notations). Cayley’s Omega process consists of applying iteratively differential operators called transvections and defined by (P,Q)�1···�5 = tr��1x · · ·��5u P (x ′, . . . , u′)Q(x ′′, . . . , u′′) where �x = det ∣∣∣∣∣∣∣∣ ∂ ∂x ′0 ∂ ∂x ′1 ∂ ∂x ′′0 ∂ ∂x ′′1 ∣∣∣∣∣∣∣∣ and tr : x ′, x ′′ → x is the map which erases the primes and the double primes3. There is no systematic way to guess which transvectants will lead to interesting (in particular, nonzero) invariants or covariants, but in small degrees, a systematic computer exploration remains possible. 3.2. Degree 4 In degree 4, previous experience of the four-qubit system suggests the following starting point. Regarding x as a parameter, write f as a quadrilinear binary form in the variables yi, zi, ti and ui , f = ∑ Axi1i2i3i4yi1zi2 ti3ui4 . It is known that such a quadrilinear form admits an invariant of degree 2 (called Cayley’s hyperdeterminant [1, 21, 22]) which is a quadratic binary form bx := (f, f )01 111 = αx20 + 2βx0x1 + γ x21 (8) in the variables x = (x1, x2). Hence, taking the discriminant β − αγ of bx one obtains an invariant Dx of degree 4. We repeat this operation for the other binary variables and obtain four other invariants Dy,Dz,Dt and Du. 3 This multivariate version of the Omega process seems to have been first used by Peano in 1882 [20]. His results are reproduced in Olver’s book [19]. Invariants of five qubits 375 3.3. Degree 6 We obtain the primary invariant of degree 6 by a succession of transvections. First, we compute a triquadratic covariant of degree 2 B22 020 = (f, f )00 101. This covariant allows us to construct a cubico-quadrilinear covariant of degree 3 C31 111 = (B22 020, f )01 010 which gives a triquadratic polynomial of degree 4 D22 200 = (C31 111, f )10 011. Hence, one obtains a quintilinear covariant of degree 5 E11 111 = (D22 200, f )11 100. Finally, we find the invariant of degree 6 F = (E11 111, f )11 111. 3.4. Degree 8 We can compute a set of five linearly independent invariants of degree 8 in the following way. First, we compute some covariants of degree 2 which are triquadratic forms B22 200 = (f, f )00 011, B00 222 = (f, f )11 000. The invariants of degree 4 of these triquadratic forms are invariants of degree 8 of our quintilinear form. Hence, we compute D40 000 = (B22 200, B22 200)02 200, D04 000 = (B22 200, B22 200)20 200, D00 400 = (B22 200, B22 200)22 000, D00 040 = (B00 222, B00 222)00 202, D00 004 = (B00 222, B00 222)00 220, and Fx = (D40 000, B22 200)20 000, Fy = (D04 000, B22 200)02 000, Fz = (D00 400, B22 200)00 200, Ft = (D00 040, B00 222)00 020, Fu = (D00 004, B00 222)00 002. Finally, we find five invariants Hx = (Fx, B22 200)22 200, Hy = (Fy, B22 200)22 200, Hz = (Fz, B22 200)22 200, Ht = (Ft , B00 222)00 222, Hu = (Fu, B00 222)00 022. (9) 3.5. Algebraic independence To prove that the polynomials Dx, . . . ,Du, F,Hx, . . . , Hu are algebraically independent, we need to compute the Jacobian determinant of the set of polynomials {Dx, . . . ,Du, F, Hx, . . . , Hu,A01 011, A01 100, A01 101, . . . , A11 111} for the variables A00 000, A00 001, . . . , A11 111. The direct calculation of this determinant is certainly out of reach on a personal computer, but it is sufficient to compute it for the numerical values given in table 2. This gives the value −1147 501 176 422 400 �= 0 and implies that the 11 polynomials are algebraically independent. From the Hilbert series the polynomials Dx, . . . ,Du, F are primary invariants. From the previous computation, we cannot conclude that Hx, . . . , Hu are primary invariants. Nevertheless, their independence makes them good candidates. 376 J-G Luque and J-Y Thibon Table 2. Random values of Aijklm used in the Jacobian. A00 000 A00 001 A00 010 A00 011 1 1 1 3 A00 100 A00 101 A00 110 A00 111 3 2 1 1 A01 000 A01 001 A01 010 A01 011 3 3 3 3 A01 100 A01 101 A01 110 A01 111 1 1 2 1 A10 000 A10 001 A10 010 A10 011 2 2 3 3 A10 100 A10 101 A10 110 A10 111 2 1 3 3 A11 000 A11 001 A11 010 A11 011 2 3 1 1 A11 100 A11 101 A11 110 A11 111 2 3 1 2 Table 3. Evaluation of SLOCC covariants for Osterloh and Siewert states (× means that the evaluation is not 0). |�1〉 |�2〉 |�3〉 |�4〉 Dx × × 0 0 Dy × × 0 0 Dz × 0 0 0 Dt × 0 0 0 Du × 0 0 0 F 0 0 0 0 Bx × × × × C31 111 0 0 × × E11 111 0 × 0 × 4. Conclusion From the Hilbert series, it appears that the algebra of polynomial invariants of a five-qubit system has a very high complexity. Furthermore, as is already the case with smaller systems [6, 7, 22], the knowledge of the invariants is not sufficient to classify entanglement patterns. In the case of four qubits or three qutrits, this classification can be achieved due to hidden symmetries which have their roots in very subtle aspects of the theory of semi-simple Lie algebras (Vinberg’s theory [18]). However, such symmetries are absent in the case of five qubits. Then, the only known general approach for classifying orbits (entanglement patterns) requires the computation of the algebra of covariants, which is already almost intractable in the case of four qubits. It has 170 generators, which have been found [7], but the description of their algebraic relations (syzygies) is definitely out of reach. However, a closer look at the four-qubit system reveals that the classification of Verstraete et al [6, 23] can be reproduced by means of only a small set of covariants. We hope that our results will allow the identification and the calculation of such a small set of invariants and covariants, sufficient to separate the physically relevant entanglement patterns, which are probably not so numerous. To illustrate this principle, let us consider a result of Osterloh and Siewert [14]. Having introduced a notion Invariants of five qubits 377 of filter which can be used to separate SLOCC orbits in the same way as covariants, these authors show that the four states |�1〉 = 1√ 2 (|11 111〉 + |00 000〉) |�2〉 = 12 (|11 111〉 + |11 100〉 + |00 010〉 + |00 001〉) |�3〉 = 1√6 (√ 2|11 111〉 + |11 000〉 + |00 100〉 + |00 010〉 + |00 001〉) |�4〉 = 1 2 √ 2 ( √ 3|11 111〉 + |10 000〉 + |01 000〉 + |00 100〉 + |00 010〉 + |00 001〉) are in different orbits. As can be seen in table 3, the orbits of these states are also distinguished by our covariants. Finally, the investigation of entanglement measures requires an understanding of invariants under local unitary transformations (LUT) [24]. The subgroup K = SU(2)⊗5 has many more invariants than G. We plan to explain in a forthcoming paper how to construct them from a basis of G-covariants. References [1] Luque J-G and Thibon J-Y 2003 J. Phys. A: Math. Gen. 36 5267–92 [2] Luque J-G and Thibon J-Y 2004 Mol. Phys. 102 1351–9 [3] Brennen G K 2003 Quantum Inf. Comput. 3 619 [4] Meyer D A and Wallach N R 2002 J. Math. Phys. 43 4273 [5] Emary C 2004 Preprint quant-ph/0405049 [6] Verstraete F, Dehaene J, De Moor B and Verschelde H 2002 Phys. Rev. A 65 052112 [7] Briand E, Luque J-G and Thibon J-Y 2003 J. Phys. A: Math. Gen. 36 9915 [8] Klyachko A A 2002 Preprint quant-ph/0206012 [9] Klyachko A A and Shumovsky A S 2003 J. Opt. B 5 S322 [10] Peres A 1996 Phys. Rev. Lett. 77 1413 [11] Horodecki H, Horodecki P and Horodecki R 1996 Phys. Lett. A 223 1 [12] Du¨r W, Vidal G and Cirac J I 2000 Phys. Rev. A 62 062314 [13] Le Paige C 1881 Bull. Acad. R. Sci. Belg. 3 2 40 [14] Osterloh A and Siewert J 2005 Preprint quant-ph/0506073 [15] Le´vay P 2005 Preprint quant-ph/0507070 [16] Xin G 2004 Electron. J. Combin. 11 R58 (electronic math.CO/0408377, associated Maple package: ELL.mpl) [17] Hamermesh M 1962 Group Theory and its Applications to Physical Problems (Reading, MA: Addison-Wesley) [18] Vinberg E B and Popov V L 1994 Invariant theory Algebraic Geomerty IV (Encyclopaedia of Mathematical Sciences vol 55) ed I R Shafarevich (Berlin: Springer-Verlag) pp 284 [19] Olver P J 1999 Classical Invariant Theory (Cambridge: Cambridge University Press) [20] Peano G 1882 G. Mat. 20 79 [21] Cayley A 1846 J. Reine u. Angew. Math. J. 30 1–37 [22] Luque J-G and Thibon J-Y 2003 Phys. Rev. A 67 042303 [23] Verstraete F, Dehaene J and De Moor B 2001 Preprint quant-ph/0105090 [24] Grassl M, Ro¨tteler M and Beth T 1998 Phys. Rev. A 58 1833–9 1. Introduction 2. Hilbert series 3. The simplest invariants 3.1. Transvectants and Cayley's Omega process 3.2. Degree 4 3.3. Degree 6 3.4. Degree 8 3.5. Algebraic independence 4. Conclusion References