2010

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entanglement is given by an analogue of the largest singular value of a matrix. ..... 0, there is a self-intersecting cyclic quadrilateral WXYZ with side-lengths.
Department of Mathematics

PREPRINT SERIES 2009/2010 NO: 10 TITLE: ‘MULTIPARTITE MULTIPARTITE ENTANGLEMENT AND HYPERMATRICES

AUTHOR(S):

Professor Anthony Sudbery Joseph Hilling, former PhD student at York

Multipartite entanglement and hypermatrices Joseph J. Hilling and Anthony Sudbery1 Department of Mathematics, University of York, Heslington, York, England YO10 5DD 1

[email protected]

Talk given at Quantum Theory: Reconsideration of Foundations 5, V¨axj¨o, Sweden, 17 June 2009 Abstract We discuss how the entanglement properties of a multipartite pure state can be described by extending conventional matrix theory to the hypermatrix formed by the coefficients of the state with respect to a product basis. In particular, we show that the geometric measure of entanglement is given by an analogue of the largest singular value of a matrix.

MULTIPARTITE ENTANGLEMENT A pure state of a composite quantum system is entangled if it is not a tensor product of states of the individual parts of the system. Thus one way of measuring the entanglement of the state (though there are many other independent types of entanglement [1]) is to measure how different the state is from any product state. If we consider only pure states, this can be expressed in geometrical terms as the distance from the closest pure product state. The space of pure states of a system described by a Hilbert space H is the projective space PH of one dimensional subspaces of H, and the closeness of two pure states ψ1 and ψ2 is measured by the modulus of the inner product |hψ1 |ψ2 i| where |ψ1 i and |ψ2 i are normalised vectors representing the states; this gives rise to a standard metric, the Fubini-Study metric [2], on the projective space PH. In the multipartite situation the Hilbert space is a 1

tensor product H = H1 ⊗· · ·⊗Hn , and the geometric measure of entanglement [3, 4] or Groverian measure [5] of a pure state |Ψi ∈ H is its Fubini-Study distance from the set of product states, namely 1 − g where g(|Ψi) = max |hΨ| (|φ1 i · · · |φn i) |2 , the maximum being taken over all single-system normalised state vectors |φ1 i ∈ H1 , . . . , |φn i ∈ Hn , and it is understood that |Ψi is normalised. GENERALISED SCHMIDT DECOMPOSITION The quantity g(|Ψi) occurs in the generalised Schmidt decomposition of the multipartite state |Ψi. For a bipartite state (n = 2) the Schmidt decomposition is well-known: it states that for any |Ψi ∈ H1 ⊗ H2 there is an orthonormal basis {|1i1 , . . . , |d1 i1 }, of H1 and an orthonormal basis {|1i2 , . . . , |d2 i2 } of H2 such that d1 X |Ψi = λk |ki1 |ki2 k=1

(assuming that d1 ≤ d2 ), where the coefficients λk are real and non-negative; g(|Ψi is the greatest of them. These coefficients are the singular values of the matrix (aij ) of coefficients of |Ψi with respect to an arbitrary orthonormal P product basis of H1 ⊗H2 . Thus we start with an expression |Ψi = aij |ii|ji which has d1 d2 non-zero terms, and change bases so as to obtain an expansion with only min(d1 , d2 ) non-zero terms. For general n, the corresponding statement [6] is that there are orthonormal bases of H1 , . . . , Hn such that the expansion of |Ψi has the minimal number of terms. These bases contain a first vector, denoted |0i, such that g(|Ψi) is the coefficient of |0i|0i · · · |0i in the expansion of |Ψi. For example, in the case of three qubits (n = 3, d1 = d2 = d3 = 2), the generalised Schmidt decomposition takes the form |Ψi = a|000i + b|011i + c|101i + d|110i + f |111i with a, b, c, d real and a ≥ b ≥ c ≥ d ≥ 0; then g(|Ψi = a. The general multipartite state |Ψi is specified (with respect to arbitrary bases) by d1 d2 . . . dn coefficients ai1 i2 ...in forming a hypermatrix. By analogy with the bipartite case, the function g(|Ψi) is called a singular value of this hypermatrix. We want to find an equation for this singular value, generalising the eigenvalue equation det[AA† − λ2 I] = 0

2

which is satisfied by the singular values λ of the matrix A = (aij ) in the bipartite case (A† denotes the hermitian conjugate of A). The definition of g as a maximum makes it susceptible to numerical calculation, but it is not so easy to treat it analytically. It has been determined in certain special cases [3, 4, 7, 8, 9, 10] but we would like to find a general method, reducing the problem to the solution of a polynomial equation. GENERALISED SINGULAR-VALUE EQUATIONS Given |Ψi ∈ H1 ⊗ · · · Hn , we want to find |φk i ∈ Hk , k = 1, . . . , n, which maximise the function f (φ1 , . . . , φn ) = |hΨ| (|φ1 i · · · |φn i) |2 (r)

subject to hφk |φk i = 1. Let |ei i (i = 1, . . . , dr = dim Hr ) be an orthonormal basis for Hr (r = 1, . . . , n), and write X X (r) (r) (1) (n) |Ψi = ai1 ...in |ei1 i . . . |ein i, |φr i = uir |eir i. i1 ...in

ir

The problem becomes: 2 (1) (n) Maximise f (u) = ai1 ...in ui1 . . . uin subject to

X

(r)

|uir |2 = 1 (r = 1, . . . n),

ir

where summation is understood over repeated lower indices i1 , . . . , in . Introducing Lagrange multipliers λr , we are led to the equations [6, 3] d (1) (r) (n) (r) ai1 ...in ui1 . . . uir . . . uin = λr uir ,

(1)

d (r) (r) where uir denotes that uir is to be omitted, and the overline denotes the (r) complex conjugate. Multiplying by uir and using the constraints, we find that the λr have a common value λ, which is the required extreme value of f (u). In the case n = 2 we can simplify the notation, writing (i, j) instead of (1) (2) (i1 , i2 ), and ui and vj instead of ui1 and ui2 . The equations become aij vj = λui

and

aij v j

or in matrix notation, Av = λu

and 3

A† u = λv.

We can choose the phases to make λ real. Then multiplying the second equation by A† and using the second shows that v is an eigenvalue of A† A with eigenvalue λ2 , i.e. λ is a singular value of A. DISCRIMINANTS AND HYPERDETERMINANTS If λ = 0 there is an established general theory for the equations (1), deriving from the theory of discriminants of homogeneous polynomials. If f (z1 , . . . , zN ) is a homogeneous function of N variables, its discriminant ∆f is a polynomial in the coefficients of f with the property that ∆f = 0 if and only if f has a critical point, i.e. if there is a non-zero solution z of the equations ∂f (z) = 0, k = 1, . . . , N. partialzk (There are N equations for N −1 independent variables (the ratios of the zk ), so the variables can be eliminated to give the equation ∆f = 0.) Examples of this are: 1. If f is quadratic, f (z) = aij zi zj = zT Az, then ∆f = det A. 2. If N = 2d, z = (x, y) where x and y are d-dimensional vectors, and f is a bilinear function f (x, y) = aij xi yj = xT Ay, then again ∆f = det A. 3. The case we are interested in: If N = d1 + · · · + dn , z = (u(1) , . . . , u(n) ) and f is multilinear, (1)

(n)

f (u(1) , . . . , u(n) ) = ai1 ...in ui1 · · · uin , then ∆f is the hyperdeterminant of the hypermatrix ai1 ...in (see [11]). The hyperdeterminant has appeared elsewhere in the theory of multipartite entanglement [12, 13]; for n = 3 and d1 = d2 = d3 = 2, its squared modulus is equal to the 3-tangle [14] of the 3-qubit state defined by the coefficients aijk . GENERALISING THE CHARACTERISTIC POLYNOMIAL In the case n = 2 the determinant function also gives the condition for the equations (1) to have a solution for any λ (not just λ = 0); the characteristic equation of a square matrix, and the singular-value equation of a rectangular matrix, are both given by determinants. In general, however, the characteristic equation of a hypermatrix is not so simply related to the hyperdeterminant function, and the theory needs to be extended. We have found a generalisation of the polynomial equation for the singular values of a matrix, as follows [15]: 4

Theorem. Let α : Cd1 × · · · × Cdn → C be a multilinear function of n vector variables:  (n) (1) α u(1) , . . . , u(n) = ai1 ...in ui1 . . . uin . For λ ∈ R, define the real polynomial function α ˜ (λ) : R2d3 × · · · × R2dn → R (identifying Cd with R2d ) by  α ˜ (λ) u(3) , . . . , u(n) = det[A† A − ku(3) k2 · · · ku(n) k2 I] where the d1 × d2 matrix A = A(u(3) , . . . , u(n) ) is defined by (3)

(n)

Ai1 i2 = ai1 i2 i3 ...in ui3 . . . uin . Suppose λ 6= 0. Then the equations d (n) (1) (r) (r) ai1 ...in ui1 . . . uir . . . uin = λuir

(2)

have a solution with all u(r) non-zero if and only if α ˜ (λ) has a real critical point. If this is so, λ satisfies the polynomial equation ∆α(λ) = 0. ˜

(3)

THREE-QUBIT STATES We will now apply the general theory of the previous section to the simplest case after the familiar bipartite (matrix) case, namely the case of three qubits. For a general three-qubit state X |Ψi = aijk |ii|ji|ki ijk

the equations (1) for the geometricPmeasure of entanglement, λ = g(|Ψi), and for the nearest separable state xi yj zk |ii|ji|ki, become aijk yj zk = λxi , aijk xi zk = λy j , aijk xi yj = λz k . with x† x = y† y = z† z = 1. In terms of the 2 × 2 matrix A(z)ij = aijk zk , A(z)† x = λy.

A(z)y = λx, 5

From these we obtain α ˜ λ (z, z) = det[A(z)† A(z) − λ2 ] = 0, identifying the function α ˜ (λ) of the Theorem, which we also write as α ˜ λ . It is a function of four variables, which we can take to be z1 , z2 , z 1 and z 2 . To obtain the equation for λ, we must find the discriminant of the function α ˜λ. This can be done in two steps [15]: first we find the discriminant of α ˜ λ (z, z) as a function of z, treating z as a constant; then we find the discriminant of the resulting function of z: ∆α(λ) = ∆z (∆z (˜ αλ ). ˜ In the first step, α˜λ is a quadratic function of z with coefficients which are quadratic functions of z: α ˜ λ (z) = F (z 0 , z 1 )z02 + G(z 0 , z 1 )z0 z1 + H(z 0 , z 1 )z12 . The discriminant of this function of z is a homogeneous quartic in z: ∆z (˜ αλ ) = G2 − 4F H = αz 40 + βz 3 z 1 + γz 20 z 21 + δz 0 z 31 + z 41

(4)

where α, β, γ, δ and  are quadratic functions of λ2 . The discriminant of the quartic is ∆ = 27B 2 − A3 where

A = α −

βδ γ 2 + , 4 12

B=

αγ αδ 2 + β 2  βγδ γ3 − + − . 6 16 48 216

Thus the characteristic equation ∆ = 0 has degree 12 in λ2 . THE ART OF THE SOLUBLE Thus the characteristic equation of a general 2 × 2 × 2 hypermatrix is dauntingly complicated. In the hope of finding a soluble problem, we will restrict attention to states in the generalised Schmidt form |Ψi = a|000i + b|011i + c|101i + d|110i + f |111i

(5)

with a, b, c, d real and positive, a ≥ b ≥ c ≥ d. This should be trivial. The first step in finding the Schmidt form for any state |Ψi is to find its injective tensor norm g(|Ψi); this gives the coefficient a. 6

The remaining real coefficients b, c, d are found by a further process of finding extrema of the same function whose maximum is g(|Ψi). Thus we might expect that if |Ψi is already in the form (5), its characteristic equation should simply have the solutions a, b, c, d; the problem should be like diagonalising a matrix which is already diagonal. Interestingly, it is not so simple; so much so that we shall need to simplify further by taking f = 0. If f = 0 the coefficients β and δ in (4) vanish, so that Dz α ˜ λ (z 0 , z 1 ) 2 2 becomes a quadratic form in (z 0 , z 1 ). The discriminant of the quartic becomes ∆(λ) = −

2 α 2 γ − 4α 16

= −16a2 b2 c2 d2 (λ2 − a2 )(λ2 − b2 )(λ2 − c2 )(λ2 − d2 )Q(λ)2 . Here Q(λ) is a quartic in λ2 which factorises completely: Q(λ) = λ4 [4S 2 λ2 − L2 ][4(S 2 − abcd)λ2 − (L2 − 2abcd)] where L and S are the symmetric functions of a, b, c, d introduced in [8]: L2 = (ab + cd)(ac + bd)(ad + bc), S 2 = (s − a)(s − b)(s − c)(s − d) (s = 21 (a + b + c + d)) 1 = 16 (−a + b + c + d)(a − b + c + d)(a + b − c + d)(a + b + c − d). (6) By Brahmagupta’s theorem, S is the area of the cyclic quadrilateral with sides a, b, c, d. SOLUTIONS OF THE CHARACTERISTIC EQUATION Thus for the tetrahedral hypermatrix whose only non-zero elements are a000 = a,

a011 = b,

a101 = c,

a110 = d,

(7)

the solutions of the characteristic equation ∆(λ) = 0 are λ = 0 (twice), a, b, c, d,

L L0 (twice) and (twice) 2S 2S 0

(8)

where L02 = L2 − 2abcd = (cd − ab)(bd − ac)(bc − ad), 1 S 02 = S 2 − abcd = 16 (a + b + c + d)(a + b − c − d)(a − b + c − d)(−a + b + c + d).

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GEOMETRY OF THE GEOMETRIC MEASURE Tamaryan et al. [8] have pointed out that the singular values (8) have a nice geometrical interpretation. If S 2 ≥ 0, the positive real numbers a, b, c, d satisfy quadrilateral inequalities (each of them is less than or equal to the sum of the others), and S is the area of the cyclic quadrilateral of which they are the side-lengths, while D = L/2S is the diameter of its circumcircle. If S 02 ≥ 0, there is a self-intersecting cyclic quadrilateral W XY Z with side-lengths a, b, c, d in which the vertices lie round the circle in the order W, Y, X, Z; the diameter of the circle is D0 = L0 /2S 0 , and S 0 can be interpreted in terms of signed areas, as follows. Y b X a W X W a b T R Y R0 c d d c

Z

Z

The areas S and S 0 are both given by the vector expression −−→ −−→ −−→ −→ W X × Y X + W Z × Y Z In the left-hand figure (S 2 > 0), this is the area of the quadrilateral W XY Z. In the right-hand figure (S 02 > 0), it is the difference between the areas of the triangles T W Z and T XY . GETTING THE RIGHT SOLUTION The geometric measure of entanglement of the state |Ψi = a|000i + b|011i + c|101i + d|110i is the largest singular value of the hypermatrix (7), which must be real and must satisfy the characteristic equation. Thus we have to order the solutions (8). This depends on the value of S 2 in (6), which is not necessarily positive. 8

• If S 2 < 0, there is no cyclic quadrilateral and no real solution D. The geometric measure is g(|Ψi) = max(a, b, c, d). • If 0 < S 2 < abcd, the solution D = L/2S is greater than a, b, c, d and also greater than D0 = L0 /2S 0 (if this is real), so g(|Ψi) = D. • If S 2 > abcd, the largest solution is either D or D0 . But this is not the end of the story. The largest real solution of the characteristic equation may not be a singular value: there are reality conditions that must be satisfied. If λ is a solution of the characteristic equation, we are guaranteed that the z-discriminant ∆z (˜ αλ (z)) is singular at some vector z. Fix this value of z and let z vary; then we are guaranteed that det[A(z)† A(z) − λ2 I] is singular at some vector z — but this may not be the complex conjugate of the vector that we fixed. We find [15] that this reality condition prevents D0 from ever being the geometric measure of the state |Ψi. As also found by Tamaryan et al. by different methods, the geometric measure is determined by the single quantity ra = 1−2a2 +2bcd/a, as follows: If ra ≤ 0, then g (|Ψi) = a;

(9) s

if ra ≥ 0, then g (|Ψi) = D =

(s − a)(s − b)(s − c)(s − d) . (ab + cd)(ac + bd)(ad + bc)

(10)

. MATRICES vs HYPERMATRICES It is interesting to compare our solutions for a tripartite state with the bipartite case. The singular values and singular column vectors of an m × n matrix A have the following properties: • The singular vectors and null vectors of A span Cn . • The singular values are all real. • Every solution of the characteristic equation is a singular value. • The singular vectors associated with a particular singular value λ form a vector subspace of Cn , whose dimension is the multiplicity of λ as a root of the characteristic polynomial.

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• Singular vectors associated with different singular values are orthogonal. The example we have studied shows that most of these properties do not extend to the multipartite case (i.e. to higher-dimensional hypermatrices). The first property does survive in our example; indeed, in some cases there are enough singular vectors that not only do the singular vectors |ui, |vi, |wi span the individual spaces, but their tensor products |ui|vi|wi span the tensor product space. This is related to the failure of the the orthogonality property of singular vectors associated with different singular values: not only are they not orthogonal, they may not even be independent. Thus there may be more singular vectors than in the bipartite case. In another sense, there may be fewer singular vectors: those associated with a particular singular value form a discrete set (apart from phase factors), rather than a vector subspace. This is a feature of the nonlinearity of the problem. Our example suggests that something may survive of the link between the multiplicity of a singular value (as a root of the characteristic polynomial) and the number of independent singular vectors. However, there can be roots of the characteristic polynomial which are not singular values; they may not be real, or they may be real but have no singular vectors associated with them.

Acknowledgements We are grateful to Sayatnova Tamaryan for an extensive email correspondence.

References [1] M. B. Plenio and S. Virmani. An introduction to entanglement measures. Quant. Inf. Comp. 7, 1 (2007). arXiv:quant-ph/0504163. [2] R.O.Wells. Differential Analysis on Complex Manifolds. Springer, 1980. [3] T.-C. Wei and P. M. Goldbart. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68, 042307 (2003). arXiv:quant-ph/0307219. [4] T.-C. Wei, M. Ericsson, P. M. Goldbart, and W. J. Munro. Connections between relative entropy of entanglement and geometric measure of entanglement. Quantum Inf. Comp. 4, 252 (2004). arXiv:quant-ph/0405002. 10

[5] O. Biham, M. A. Nielsen, and T. J. Osborne. An entanglement monotone derived from Grover’s algorithm. Phys. Rev. A 65, 062312 (2002). arXiv:quant-ph/0112097. [6] H. A. Carteret, A. Higuchi, and A. Sudbery. Multipartite generalisation of the Schmidt decomposition. J. Math. Phys. 41, 7932 (2000). arXiv:quant-ph/0006125. [7] O. Biham, Y. Shimoni and D. Shapira. Characterization of pure quantum states of multiple qubits using the Groverian entanglement measure. Phys. Rev. A 69, 062303 (2004). arXiv:quant-ph/0309062. [8] L. Tamaryan, D. Park, J.-W. Son, and S. Tamaryan. Geometric measure of entanglement and shared quantum states. Phys. Rev. A 78, 032304 (2008). arXiv:0803.1040. [9] E. Jung, M.-R. Hwang, D. Park, L. Tamaryan, and S. Tamaryan. Three-qubit groverian measure. Quant. Inf. Comp. 8, (2008). arXiv:0803.3311. [10] R. Orus, Universal geometric entanglement close to quantum phase transitions. Phys.Rev.Lett., 100, 130502 (2008), arXiv:0711.2556; Geometric entanglement in a one-dimensional valence bond solid state, Physical Review A, 78, 062332 (2008), arXiv:0808.0938; R. Orus, Q.-Q. Shi, J. O. Fjaerestad, and H.-Q. Zhou, Finite-size geometric entanglement from tensor network algorithms, arXiv:0901.2863. [11] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Discriminants, resultants and higher-dimensional determinants. Birkh¨auser, 1994. [12] A. Miyake. Classification of multipartite entangled states by multidimensional determinants. Phys. Rev. A, 67, 012108 (2003). arXiv:quant-ph/0206111. [13] A. Miyake and M. Wadati. Multipartite entanglement and hyperdeterminants. Quant. Inf. Comp. 2 (Special), 540 (2002). arXiv:quant-ph/0212146. [14] V. Coffman, J. Kundu, and W. K. Wootters. Distributed entanglement. Phys. Rev. A 61, 2306 (2000). arXiv:quant-ph/9907047. [15] J. J. Hilling and A. Sudbery. The geometric measure of multipartite entanglement and the singular values of a hypermatrix. arXiv:0905.2094.

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Department of Mathematics

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‘MULTIPARTITE MULTIPARTITE ENTANGLEMENT AND HYPERMATRICES’ HYPERMATRICES

Anthony Sudbery & Joseph Hilling