Tensor Representation of Spin States
aa r X i v : . [ qu a n t - ph ] F e b Tensor Representation of Spin States
O. Giraud , D. Braun , D. Baguette , T. Bastin , J. Martin LPTMS, CNRS and Université Paris-Sud, UMR 8626, Bât. 100, 91405 Orsay, France Institut für theoretische Physik, Universität Tübingen, 72076 Tübingen, Germany Institut de Physique Nucléaire, Atomique et de Spectroscopie, Université de Liège, 4000 Liège, Belgium (Dated: September 3, 2014 - Revised: February 26, 2015)We propose a generalization of the Bloch sphere representation for arbitrary spin states. Itprovides a compact and elegant representation of spin density matrices in terms of tensors that sharethe most important properties of Bloch vectors. Our representation, based on covariant matricesintroduced by Weinberg in the context of quantum field theory, allows for a simple parametrizationof coherent spin states, and a straightforward transformation of density matrices under local unitaryand partial tracing operations. It enables us to provide a criterion for anticoherence, relevant in abroader context such as quantum polarization of light.
PACS numbers: 03.65.Aa, 03.65.Ca
The concept of spin is ubiquitous in quantum the-ory and all related fields of research, such as solid-state physics, molecular, atomic, nuclear or high-energyphysics [1–5]. It has profound implications for the struc-ture of matter as a consequence of the celebrated spin-statistics theorem [6]. The spin of a quantum system,be it an electron, a nucleus or an atom, has also beenproven to be a key resource for many applications suchas in spintronics [7], quantum information theory [8] ornuclear magnetic resonance [9]. Simple geometrical rep-resentations of spin states [10] allow one to develop phys-ical insight regarding their general properties and evolu-tion. Particularly well studied is the case of a single two–level system, formally equivalent to a spin-1/2. In thiscase, the geometric representation is particularly sim-ple. Indeed, the density matrix can be expressed in abasis formed of Pauli matrices and the identity matrix,leading to a parametrization in terms of a vector in R .Pure states correspond to points on a unit sphere, theso-called Bloch sphere, and mixed states fill the inside ofthe sphere, the “Bloch ball”. The simplicity of this rep-resentation help visualize the action and geometry of allpossible spin- / quantum channels [11]. For arbitrarypure spin states, another nice geometrical representationhas been developed by Majorana in which a spin- j stateis visualized as j points on the Bloch sphere [12]. Thisso-called Majorana or stellar representation has been ex-ploited in various contexts (see, e. g., [11, 13–17]), butcannot be generalized to mixed spin states.Given the importance of geometrical representations,there have been numerous attempts to extend the previ-ous representations to arbitrary mixed states. The for-mer rely on a variety of sophisticated mathematical con-cepts such as su ( N ) -algebra generators [10, 18, 19], polar-ization operator basis [20–22], Weyl operator basis [23],quaternions [24], octonions [25] or Clifford algebra [26].In the present Letter we propose an elegant generalisa-tion to arbitrary spin- j of the spin-1/2 Bloch sphere rep-resentation based on matrices introduced by Weinberg inthe context of relativistic quantum field theory [27]. The main result of the paper is theorem 2, which allows us toexpress any spin- j density matrix as a linear combinationof matrices with convenient properties. The remarkablefeatures of our representation are especially reflected inthe simple coordinates of coherent states, transformationunder SU(2) operations, and the simplicity of the repre-sentation of reduced density matrices. To illustrate theusefulness of such a representation, we show that it allowsus to give an easy characterisation of anticoherent spinstates. Such states have been studied in various contexts,such as quantum polarization of light (see e.g. [28, 29]),spherical designs [30], as well as in the search for maxi-mally entangled symmetric states [31]. We believe thatour representation should prove useful in many of thecontexts where the spin formalism is used.We construct this parametrization of a spin- j den-sity matrix ρ from the set of j covariant matrices [27].Defining the 4-vector q = ( q , q , q , q ) ≡ ( q , q ) , Wein-berg’s covariant matrices S µ µ ...µ j , with µ i ,are constructed from products of components of J =( J , J , J ) with J a ( a ), the usual (2 j + 1) -dimensional representations of angular momentum oper-ators. They can be obtained by expanding the squareof the (2 j + 1) -dimensional matrix corresponding to the ( j, representation of a Lorentz boost in direction q ,which can be put in the form [27] Π ( j ) ( q ) ≡ ( q − | q | ) j e − η q ˆ q · J (1)with η q = arctanh( −| q | /q ) and ˆ q = q / | q | . Matri-ces S µ µ ...µ j are defined in [27] by identifying the co-efficients of the multivariate polynomial with variables q , q , q , q in (1) with those of the polynomial Π ( j ) ( q ) = ( − j q µ q µ . . . q µ j S µ µ ...µ j (2)(we use Einstein summation convention for repeated in-dices). An explicit expression for Π ( j ) ( q ) is given in [27]as Π ( j ) ( q ) = ( q − q ) j + j X k =1 ( q − q ) j − k (2 k )! (2 q · J ) k − Y r =1 [(2 q · J ) − (2 r q ) ] ! (2 q · J + 2 kq ) (3)for integer j , and Π ( j ) ( q ) = ( q − q ) j − / ( − q − q · J ) − j − / X k =1 ( q − q ) j − / − k (2 k + 1)! k Y r =1 [(2 q · J ) − ((2 r − q ) ] ! (2 q · J +(2 k +1) q ) (4)for half-integer j . The identity matrix is implicit in front of constant terms. For instance, identifying the coefficientof q j in these expressions we get that S ... is the (2 j + 1) -dimensional identity matrix, j +1 .The matrices S µ µ ...µ j are Hermitian matrices, invari-ant under permutation of indices, and they obey the fol-lowing linear relation: g µ µ S µ µ ...µ j = 0 , (5)where g ≡ diag( − , + , + , +) .Let us briefly consider the simplest examples. From(4), the explicit expression of Π ( j ) ( q ) for spin-1/2 reads Π (1 / ( q ) = − q − q · J (6)where J a are spin-1/2 representations of the angular mo-mentum operators. Identifying with (2) directly gives S = σ and S a = 2 J a = σ a where σ is the × iden-tity matrix and σ a are the usual Pauli matrices. Theusual Bloch sphere representation for an arbitrary spin-1/2 density matrix ρ = σ + x · σ can then be expressedin terms of the S µ ( µ ) as ρ = 12 x µ S µ (7)with the Bloch vector x = tr( ρ σ ) and x = 1 .For j = 1 , the equality between expressions (2) and(3) for Π (1) ( q ) reads ( q − q ) + 2 q · J ( q · J + q ) = q µ q µ S µ µ . (8)Identifying coefficients of this quadratic form yields S = J , S a = J a and S ab = J a J b + J b J a − δ ab J with J the × identity matrix. Again, the set of S µ µ matricescan serve to express any spin-1 density matrix ρ as ρ = 14 x µ µ S µ µ (9)with coordinates x µ µ = tr( ρ S µ µ ) . (10)Expressions (3)–(4) can be used to generalize this ex-pansion to arbitrary j , as we will show in Theorem 2.The main property of the covariant matrices is given byTheorem 1 below. We first give a useful lemma. Lemma 1.
Let | α i be a spin- j coherent state, defined for α = e − iϕ cot( θ/ with θ ∈ [0 , π ] and ϕ ∈ [0 , π [ by | α i = j X m = − j s(cid:18) jj + m (cid:19) (cid:2) sin θ (cid:3) j − m (cid:2) cos θ e − iϕ (cid:3) j + m | j, m i (11) in the standard angular momentum basis {| j, m i : − j m j } , and let n = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) . Then h α | Π ( j ) ( q ) | α i = ( − j ( q + q · n ) j . (12)The proof of this lemma is based on the SU(2) disen-tangling theorem and can be found in the SupplementalMaterial. One of its consequences is that, by identifyingcoefficients of the polynomial in q µ in (12), we get h α | S µ µ ...µ j | α i = n µ n µ . . . n µ j , (13)with n = 1 .In the Majorana representation, any pure spin- j stateis viewed as a permutation symmetric state of a system of N ≡ j spin- / , or equivalently as an N -qubit symmet-ric state. The Hilbert space H ≡ C N of an N spin- / system has dimension N but its symmetric subspace H S has only dimension N + 1 = 2 j + 1 . It is spanned by thesymmetric Dicke states | D ( k ) N i = N X π | ↓ . . . ↓ | {z } N − k ↑ . . . ↑ | {z } k i , k = 0 , . . . , N, (14)where the sum runs over all permutations of the stringwith N − k spin down and k spin up, and N is the nor-malization constant. The Dicke state | D ( k ) N i correspondsto | j, m i with j = N/ and m = k − N/ .Let L ( H ) be the Hilbert space of linear operators act-ing on the finite-dimensional space H . An operator basisfor L ( H ) equipped with the standard Hilbert-Schmidtinner product is given by the set of the N generalizedPauli matrices defined as the N -fold tensor products ofthe × matrices σ , σ , σ , σ [8], σ µ µ ...µ N = σ µ ⊗ σ µ ⊗ . . . ⊗ σ µ N . (15)These Hermitian operators verify the relations tr( σ µ µ ...µ N σ ν ν ...ν N ) = 2 N δ µ ν δ µ ν . . . δ µ N ν N and thus form an orthogonal basis. Any state ρ of N spin- / can be expanded in this basis as ρ = 12 N x µ µ ...µ N σ µ µ ...µ N , (16)where x µ µ ...µ N are real coefficients given by x µ µ ...µ N = tr( ρ σ µ µ ...µ N ) . (17)We can now prove the following theorem: Theorem 1.
The Weinberg covariant matrices definedin Eq. (2) are given by the projection of tensor productsof Pauli matrices into the subspace H S of states that areinvariant under permutation of particles. Namely, de-noting by P S ≡ P Nk =0 | D ( k ) N ih D ( k ) N | the projector onto H S , the S µ µ ...µ N matrix corresponds to the ( N + 1) -dimensional block spanned by the | D ( k ) N i of the matrix P S σ µ µ ...µ N P † S , i.e., in terms of matrix elements h D ( k ) N | S µ µ ...µ N | D ( ℓ ) N i = h D ( k ) N | σ µ µ ...µ N | D ( ℓ ) N i , (18) with k, ℓ N .Proof. Let ˜ S µ µ ...µ N = P S σ µ µ ...µ N P † S . Any spin- j co-herent state | α i defined by Eq. (11) can also be writ-ten as the tensor product of identical spin-1/2 coher-ent states. As a symmetric state, | α i is invariant un-der P S , i.e., | α i = P S | α i , so that h α | ˜ S µ µ ...µ N | α i = h α | σ µ µ ...µ N | α i = n µ n µ . . . n µ N . Using Eq. (13), wethus have h α | ˜ S µ µ ...µ N | α i = h α | S µ µ ...µ N | α i (19)for all α , i.e., the Husimi functions of the two operatorsare identical. Therefore S µ µ ...µ N and ˜ S µ µ ...µ N coin-cide in H S .In other words, instead of obtaining the Weinberg ma-trices from the expansion of the rather complicated ex-pressions (3)–(4), we can construct them simply by pro-jecting the corresponding tensor product of Pauli opera-tors into the symmetric subspace. In order to fully ex-ploit the consequences of this fact, we need some basicnotions of frame theory [32].A family of vectors | φ i i , i ∈ { , . . . , M } , is called aframe for a Hilbert space H with bounds A, B ∈ ]0 , ∞ [ , if A || ψ || M X i =1 |h ψ | φ i i| B || ψ || , ∀ | ψ i ∈ H . (20)If A = B , then the frame is called an A -tight frame.Orthonormal bases are a special case of A -tight frames.In particular, the generalized Pauli matrices (15) form –up to normalization – an orthonormal basis of L ( H ) , andare in fact an A -tight frame, which verifies Eq. (20) with A = B = 2 N and M = 4 N . According to proposition22 in [32], a frame of a Hilbert space H with bounds A, B that is orthogonally projected to a subspace P H is a frame of P H with the same bounds A, B . Therefore wehave as a corollary of Theorem 1 that the set of covariantmatrices S µ µ ...µ N forms a N -tight frame for L ( H S ) .Tight frames are in a sense a generalization of or-thonormal bases, as they allow an expansion over theelements of the frames with the same formulas as foran orthonormal basis, i.e., for all | ψ i ∈ H , we have | ψ i = A − P Mi =1 h φ i | ψ i| φ i i (proposition 20 in [32]). Thisimmediately entails the following result, which providesa generalization of the Bloch sphere representation forspin-1/2, Eq. (7), to any spin: Theorem 2.
For general spin- j , the N Hermitian ma-trices S µ µ ...µ N (with N ≡ j ) provide an overcompletebasis (more precisely, a N -tight frame) over which ρ canbe expanded, that is, any state can be expressed as ρ = 12 N x µ µ ...µ N S µ µ ...µ N , (21) with coefficients x µ µ ...µ N = tr( ρ S µ µ ...µ N ) (22) real and invariant under permutation of the indices. Since S ... is the identity matrix, the condition tr ρ =1 for density matrices is equivalent to x ... = 1 . Thetight frame property allows one to write the Hilbert-Schmidt scalar product of any two Hermitian operators ρ and ρ ′ with coordinates x µ µ ...µ N and x ′ µ µ ...µ N as thescalar product of coordinates, more precisely tr( ρρ ′ ) = 12 N x µ µ ...µ N x ′ µ µ ...µ N . (23)The condition tr ρ that every state must sat-isfy translates into P µ . . . P µ N x µ µ ...µ N N . Notethat from Eq. (22) and the definition of S µ µ ...µ N ,the coordinates x µ µ ...µ N appear as the coefficients of ( − N h Π ( j ) ( q ) i , which is a multivariate polynomial invariables q , q , q , q , ( − N h Π ( j ) ( q ) i = x µ µ ...µ N q µ . . . q µ N . (24)Due to the overcompleteness of the S µ µ ...µ N the coor-dinates x µ µ ...µ N in (21) are so far not unique. However,for a given spin- j density matrix ρ , (22) is the uniquechoice of coordinates x µ µ ...µ N such that these coordi-nates are real numbers, invariant under permutation ofthe indices, and verifying the condition g µ µ x µ µ ...µ N =0 (see Proposition 1 in the Supplemental Material).The generalized Bloch representation (21) shares withthe Bloch representation of a spin-1/2 several crucialproperties. First of all, using Eqs. (13) and (22), wesee that coordinates of a coherent state are simply givenby the product of components of the 4-vector n = (1 , n ) ,namely x µ µ ...µ N = n µ n µ . . . n µ N . This generalizes thefact that the Bloch vector representing a spin-1/2 statepoints in the direction given by the angles defining thecoherent state. Secondly, under any SU(2) transforma-tion, the Bloch vector of a spin-1/2 simply rotates, i.e.,transforms according to x a → R ab x b , where R is a ro-tation matrix. Similarly, for higher spins the tensor ofcoordinates of an arbitrary state transforms according to x µ ...µ N → R µ ν . . . R µ N ν N x ν ...ν N , with R ab the × rotation matrix and R µ = R µ = δ µ . This is a conse-quence of a more general covariance property of the ba-sis matrices S µ µ ...µ N . Indeed, they were constructed insuch a way that for any element Λ of the Lorentz group,with D ( j ) [Λ] the (2 j + 1) -dimensional matrix associatedwith Λ in the ( j, representation, D ( j ) [Λ] S µ µ ...µ N D ( j ) [Λ] † = Λ ν µ . . . Λ ν N µ N S ν ν ...ν N (25)in the covariant-contravariant notation of [27]. FromEq. (22) this property translates to coordinates x µ µ ...µ N . For rotations R µν , the distinction betweenupper and lower indices becomes irrelevant.In addition to the shared advantages of a Bloch vec-tor, our generalized Bloch sphere representation (21) en-joys additional convenient properties relevant for systemsmade of many spin- / or qubits. For instance, coordi-nates of the spin- k reduced density matrix obtained bytracing the spin- j matrix over j − k spins are simply givenby x µ ...µ k = x µ ...µ k ... (26)(see Proposition 3 in the Supplemental Material). Notethat in [33] a similar property was observed for thecoefficients in the expansion of ρ over generalized Paulimatrices, and a formal Lorentz invariance of that ex-pansion was used very recently to generalize monogamyrelations of entanglement [34].We now consider a few examples of states and givetheir coordinates in our representation. The maximallymixed state ρ = j +1 j +1 has coordinates x µ µ ...µ j given by x µ µ ...µ j q µ . . . q µ j = j X k =0 (cid:0) j k (cid:1) k + 1 q j − k )0 | q | k (27)(see Proposition 2 in the Supplemental Material). An-other example is given by the Schrödinger cat states | ψ ( j )cat i = ( | j, − j i + | j, j i ) / √ . By linearity of the ex-pansion (21) and of the trace, they have coordinates x cat µ ...µ N = 12 N Y i =1 n ( − , − ) µ i + N Y i =1 n ( , ) µ i ! +Re " N Y i =1 n ( − , ) µ i (28)where n ( ± , ± ) = (1 , , , ± are the coordinatesof the coherent states | , ± ih , ± | and n ( − , ) =(0 , , − i, are the coordinates of the non-Hermitian op-erator | , − ih , | .While the complete characterization of the set of co-ordinates for which ρ is positive is difficult in any repre-sentation [18, 21, 23], our representation (21) allows one to solve this problem explicitely for j = 1 . The set of allspin-1 states is characterized by 8 real parameters. Thetransformation of tensor x µν by rotation matrices underSU(2) operations allows one to diagonalize the 3 × x ab ( a, b ), and Eq. (5) imposes P i =1 µ i = 1 forthe eigenvalues µ i , leaving five real parameters µ , µ , x ≡ ( x , x , x ) . In this case, x coincides with u inthe representation found in [35]. We therefore immedi-ately obtain that up to two special cases of measure zerothe set of all spin-1 states can be represented as a two–parameter family of ellipsoids in the space of vectors x (Eq. (21) in [35] with u = x and w ab = x ab ), thus pro-viding a simple geometrical picture of all spin-1 states.As a direct application of our formalism, we give a sim-ple necessary and sufficient criterion for anticoherence ofspin states. Spin states are said to be anticoherent toorder t if h ( n · J ) k i is independent on the unit vector n for any k with k t [36]. Various characteri-sations have been given [37]. Very recently the case ofpure but not necessarily symmetric states was consideredin [33, 38]. The definition of matrices S µ µ ...µ N via (1)–(2) as a function of J makes them most convenient forthe characterisation of anticoherent states. One can showthe following result: Theorem 3.
A spin- j state ρ , pure or mixed, is antico-herent to order t if and only if its spin- ( t/ reduced den-sity matrix is the maximally mixed state ρ = t +1 t +1 . The proof (see Supplemental Material for more detail)relies on the calculation of h Π ( j ) ( q ) i for an anticoher-ent state, using the expansion (3)–(4) and identifyingterms up to order t with the expansion (27) of the max-imally mixed state. For instance, spin- j anticoherentstates to order 1 are characterized by h S µ ... i = δ µ while anticoherent states to order 2 are characterized by h S µν ... i = diag(1 , / , / , / . From the characteri-zation of anticoherence given by Theorem 3, one can eas-ily obtain another characterization based on coefficientsof the multipolar expansion of the density matrix. For aspin- j density operator ρ , the expansion reads ρ = j X k =0 k X q = − k ρ kq T ( j ) kq (29)with ρ kq = tr (cid:0) ρ T ( j ) kq † (cid:1) , where T ( j ) kq are the irreducibletensor operators [20] T ( j ) kq = s k + 12 j + 1 j X m,m ′ = − j C jm ′ jm,kq | j, m ′ ih j, m | , (30)and C jm ′ jm,kq are Clebsch-Gordan coefficients. The follow-ing corollary of Theorem 3 can now be stated (see Sup-plemental Material for a proof). Corollary 1.
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In this supplemental material, we provide the proofs of the propositions, lemmas and theorems 1 and 3 stated inthe main text. We recall that covariant matrices S µ µ ...µ j are obtained by identifying coefficients of Π ( j ) ( q ) = ( − j q µ q µ . . . q µ j S µ µ ...µ j (1)with the expansion of the polynomials (multivariate in the q µ ) Π ( j ) ( q ) = ( q − q ) j + j X k =1 ( q − q ) j − k (2 k )! (2 q · J ) k − Y r =1 [(2 q · J ) − (2 r q ) ] ! [2 q · J + 2 kq ] (2)for integer j , and Π ( j ) ( q ) = ( q − q ) j − / ( − q − q · J ) − j − / X k =1 ( q − q ) j − / − k (2 k + 1)! k Y r =1 [(2 q · J ) − ((2 r − q ) ] ! (2 q · J +(2 k +1) q ) (3)for half-integer j , with q = ( q , q , q , q ) ≡ ( q , q ) . We denote N = 2 j . The operator Π ( j ) ( q ) is proportional to thesquare of the Hermitian operator associated to a Lorentz boost in direction q for a particle of mass m , Π ( j ) ( q ) = m N exp ( − η q ˆ q · J ) , (4)where ˆ q = q / | q | , and η q and m are defined by q = − m cosh η q , (5) | q | = m sinh η q . (6)We recall that the S µ µ ...µ N are linked by a linear relation, given by g µ µ S µ µ ...µ N = 0 , (7)where g ≡ diag( − , + , + , +) . Theorem 2 which was proved in the paper states that for general spin– j , the Hermitianmatrices S µ µ ...µ N provide an overcomplete basis over which ρ can be expanded, that is, any state can be expressedas ρ = 12 N x µ µ ...µ N S µ µ ...µ N , (8)with real coefficients given by x µ µ ...µ N = tr( ρS µ µ ...µ N ) . (9)We now turn to the proofs. Lemma 1.
Let | α i be a spin– j coherent state, defined for α = e − iϕ cot( θ/ with θ ∈ [0 , π ] and ϕ ∈ [0 , π [ by | α i = j X m = − j s(cid:18) jj + m (cid:19) (cid:2) sin θ (cid:3) j − m (cid:2) cos θ e − iϕ (cid:3) j + m | j, m i , (10) = 1(1 + | α | ) j e αJ + | j, − j i , (11) in the standard angular momentum basis {| j, m i : − j m j } , and let n = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) . Then h α | Π ( j ) ( q ) | α i = ( − N ( q + q · n ) N . (12) Proof.
We use the SU(2) disentangling theorem [33]: for a + , a − , b ∈ C , exp( a + J + + bJ z + a − J − ) = e b − J − e b z J z e b + J + (13)with constants given by b ± = 2 a ± sinh( δ/ δ cosh( δ/
2) + b sinh( δ/ (14) b z = 2 ln (cid:18) δ cosh( δ/
2) + b sinh( δ/ δ (cid:19) , (15)where δ = p b + 4 a + a − . From its definition (4), Π ( j ) ( q ) can be written as in (13) with a ± = − η q (ˆ q x ∓ i ˆ q y ) and b = − η q ˆ q z . Thus, using (11) and (13) we get h α | Π ( j ) ( q ) | α i = m N h ψ | e b z J z | ψ i (16)with | ψ i = 1(1 + | α | ) j e ( α + b + ) J + | j, − j i . (17)If we parametrize ˆ q = (sin γ cos φ, sin γ sin φ, cos γ ) , we get from (14)–(15) that the constants are given by δ = 2 η q and b ± = − sin γe ∓ iφ coth η q − cos γ , (18) b z = 2 ln (cosh η q − sinh η q cos γ ) . (19)We see that, up to normalization, | ψ i is of the form (11). Namely, we have | ψ i = (cid:18) | α + b + | | α | (cid:19) j | α + b + i . (20)From Eq. (10) we get for a coherent state | α i and a complex number t the identity h α | e tJ z | α i = (cosh t + sinh t cos θ ) N . (21)Applying (21) to the coherent state | α + b + i gives h α | Π ( j ) ( q ) | α i = m N (cid:18) | α + b + | | α | (cid:19) N (cid:18) cosh b z | α + b + | − | α + b + | + 1 sinh b z (cid:19) N . (22)On the right-hand side of (12) we have the N th power of − q − q · n = m (cosh η q − sinh η q (sin θ sin γ cos( ϕ − φ ) + cos γ cos θ )) . (23)Using the explicit expressions of α , b + and b z in Eq. (22), Eq. (12) is now equivalent to a trigonometric identity thatis easily verified. Proposition 1.
For a given density matrix ρ , Eq. (9) is the unique choice of coordinates which are real numberssymmetric under permutation of the indices and verifying g µ µ x µ µ ...µ N = 0 (24) Proof.
The fact that Eq. (24) is verified follows immediately from Eqs. (7) and (9), and by linearity of the trace.It remains to show the uniqueness of this choice. Matrices S µ µ ...µ N are not linearly independent since they areinvariant under permutation of indices and related through Eq. (7). The number of distinct sets ( µ , . . . , µ N ) up topermutation of indices is (cid:0) N +33 (cid:1) , from which one has to subtract the (cid:0) N +13 (cid:1) relations (7). That leaves (cid:18) N + 33 (cid:19) − (cid:18) N + 13 (cid:19) = ( N + 1) (25)independent basis matrices, which coincides with the number of independent parameters in a ( N + 1) × ( N + 1) Hermitian matrix (here we disregard the fact that tr ρ = 1 , which would just correspond to imposing x ... = 1 ). Thismeans that there cannot be any other relation between the S µ µ ...µ N than the permutation-symmetry relations andthe relations (7) (otherwise there would not be enough parameters to describe all matrices ρ ). In particular, if wefix µ , . . . , µ N , the vector space V µ ...µ N generated by matrices S ννµ ...µ N , ν , is of dimension , and fromEq. (7) we get that V µ ...µ N is of dimension . So the only additional relation one can impose between the x µ µ ...µ N is between the x ννµ ...µ N . Choosing Eq. (24) thus defines coordinates of a density matrix in a unique way. Proposition 2.
Coordinates of the maximally mixed state ρ = N +1 N +1 (with N +1 the ( N + 1) -dimensionalidentity matrix) are given by x µ µ ...µ N q µ . . . q µ N = j X k =0 (cid:0) N k (cid:1) k + 1 q j − k )0 | q | k . (26) Proof.
We use the expansion of the identity in terms of coherent states, ρ = 14 π Z dα | α ih α | . (27)According to the main text, the coordinates x µ µ ...µ N are the coefficients of the multivariate polynomial ( − N h Π ( j ) ( q ) i . Using (27) we get tr( ρ Π ( j ) ( q )) = 14 π Z dα h α | Π ( j ) ( q ) | α i = ( − N π Z dα ( q + q · n ) N , (28)where the last equality comes from Lemma 1. Since the integral runs over the whole sphere, one can take q = (0 , , | q | ) and rewrite the integral as ( − N Z π dθ ( q + | q | cos θ ) N sin θ = ( − N N X k =0 (cid:18) Nk (cid:19) q N − k | q | k − k k + 2 . (29)Using Eq. (1), we get the result. Proposition 3.
Coordinates of the spin– k reduced density matrix obtained by tracing the spin– j matrix over j − k spins are given by x µ ...µ k = x µ ...µ k ... (30) Proof.
Let ρ be a density matrix. Expanding ρ over coherent states as ρ = Z dαP ( α ) | α ih α | , (31)we get tr( ρ Π ( j ) ( q )) = Z dαP ( α ) h α | Π ( j ) ( q ) | α i . (32)Using Lemma 1 and the expansion (1), Eq. (32) gives q µ . . . q µ j tr( ρS µ µ ...µ j ) = Z dαP ( α )( q + q · n ) j , (33)where n is the unit vector associated with | α i . In the expansion (31), coherent states | α i are a j –fold tensor productof spin– / coherent states | α (1 / i . The trace of | α ih α | over j − k spins is the k –fold tensor product of the projectoron | α (1 / i . From Eqs. (33) and (9), we get x µ µ ...µ j q µ . . . q µ j = Z dαP ( α )( q + q · n ) j . (34)and the coordinates x µ µ ...µ k of the spin– k reduced density matrix are thus given by x µ µ ...µ k q µ . . . q µ k = Z dαP ( α )( q + q · n ) k . (35)The x µ µ ...µ k defined by (35) can then be directly read off Eq. (34), which yields the result. Theorem 3.
A spin– j density matrix is anticoherent to order t if and only if its spin– ( t/ reduced density matrixis the maximally mixed state ρ = t +1 t +1 .Proof. Let us first consider the case t = 2 j . The matrix ρ is entirely characterized by the quantities h β | ρ | β i , where | β i runs over coherent states. If one expands ρ as in (31), then h β | ρ | β i = 1 N + 1 = Z dαP ( α )(1 + n ′ · n ) N , (36)where n ′ denotes the unit vector corresponding to | β i . The maximally mixed state is thus characterized by the factthat for any n ′ its P –function verifies the right-hand equality in Eq. (36). Any state such that the right-hand side of(36) is independent of n ′ is thus proportional to ρ .Anticoherence to order j means that h ( n · J ) k i is independent of n for k up to j . The operator n · J has eigenvalues − j, − j + 1 , . . . , j , and since its characteristic polynomial is also a minimal polynomial, one has j Y m = − j ( n · J − m ) = 0 , (37)which, by the way, is the reason why (4) can be expanded into a finite sum (2)–(3). From Eq. (37) this means that h ( n · J ) k i is in fact independent of n for any k , which in turn implies that h Π ( j ) ( q ) i is independent of ˆ q . If P is the P –function associated with ρ as in (31), Eqs. (32)–(33) imply that Z dαP ( α )( q + q · n ) N (38)is independent of ˆ q . In particular, for q = 1 and ˆ q = n ′ one recovers the condition (36) which characterizes ρ (up toa multiplicative constant, which is then fixed by the normalization condition tr ρ = 1 ). Thus anticoherence to order j implies that ρ = ρ . The converse is true: since from Proposition 2 the coordinates of ρ do not depend on ˆ q ,Eq. (23) of the paper, ( − N h Π ( j ) ( q ) i = x µ µ ...µ N q µ . . . q µ N , (39)implies that h Π ( j ) ( q ) i does not depend on ˆ q , and thus the coefficients of its series expansion in powers of η q , obtainedfrom (4), do not depend on ˆ q either.Let us now consider a state ρ anticoherent to order t j , and let x µ ...µ N be its coordinates. From (9) andproposition 3, the spin– ( t/ reduced density matrix of ρ has coordinates given by x µ ...µ t ... . Following Proposition2, we thus want to show that a state is anticoherent to order t if and only if its coordinates are such that x µ µ ...µ t ... q µ . . . q µ t = t/ X k =0 (cid:0) t k (cid:1) k + 1 q t − k | q | k , (40)or, equivalently, x µ µ ...µ t ... q µ . . . q µ t q N − t = t/ X k =0 (cid:0) t k (cid:1) k + 1 q N − k | q | k . (41)From the form (2), expanding the powers of q − q , one has, for integer j , h Π ( j ) ( q ) i = j X s =0 (cid:18) js (cid:19) q j − s )0 | q | s + j X k =1 j − k X i =0 (cid:18) j − ki (cid:19) q j − k − i )0 | q | i (2 k )! D (2 q · J ) k − Y r =1 [(2 q · J ) − (2 r q ) ] ! (2 q · J + 2 kq ) E . (42)Following Eq. (39), the x µ µ ...µ t ... are obtained by considering the terms containing a factor q k with k ≥ N − t in ( − N h Π ( j ) ( q ) i . The term in q j − s )0 in (42) reads q j − s )0 ((cid:18) js (cid:19) | q | s + s X k =1 (cid:18) j − ks − k (cid:19) | q | s − k ) (2 k )! D (2 q · J ) k − Y r =1 [(2 q · J ) − (2 r q ) ] ! (2 q · J ) E) (43)and the term in q j − s )+10 in (42) reads q j − s )+10 ( s X k =1 (cid:18) j − ks − k (cid:19) | q | s − k ) (2 k − D (2 q · J ) k − Y r =1 [(2 q · J ) − (2 r q ) ] ! E) . (44)The largest power of q · J in (43) corresponds to k = s and r = k − , which gives a power ( q · J ) s . From thedefinition of an anticoherent state of order t , we have that for s t/ , all powers of q · J appearing in (43) aresuch that their average does not depend on ˆ q . A similar reasoning holds for Eq. (44). One can thus rewrite (43) and(44) respectively as q N − s | q | s s X k =0 (cid:0) j − ks − k (cid:1) (2 k )! D k − Y r =0 [(2ˆ q · J ) − (2 r ) ] E (45)and q N − s +10 | q | s − s X k =1 (cid:0) j − ks − k (cid:1) (2 k − D (2ˆ q · J ) k − Y r =1 [(2ˆ q · J ) − (2 r ) ] E , (46)where any h (ˆ q · J ) k i can be replaced by h ( J z ) k i .When summing over µ , . . . , µ t in the left-hand side of (41), the coefficient of a given q µ . . . q µ t q N − t is x µ µ ...µ t ... multiplied by the number of permutations of { µ µ , . . . , µ t } , while the coefficient of q µ . . . q µ t q N − t in (42), or in (45)–(46), is obtained from (1) and (9) as x µ µ ...µ t ... multiplied by the number of permutations of { µ , µ , . . . , µ t , , . . . , } .Let us group terms according to the number p ν of indices { µ , µ , . . . , µ t } equal to ν . In order to identify termscontaining a power q p + N − t on both sides of Eq. (41) we must consider terms such that k = ( t − p ) / when t − p is even (the contribution is 0 for odd t − p ). Similarly we must take terms such that s = ( t − p ) / in (45) when t − p is even, and terms such that s = ( t − p + 1) / in (46) when t − p is odd. In order to show equality betweenthe x µ µ ...µ t ... defined from (41) and those defined from (45) for t − p even, one must show that the coefficient of q N − s | q | s in (45) with s = ( t − p ) / is equal to the coefficient of q N − k | q | k with k = ( t − p ) / , multiplied by perm. { µ µ . . . µ t . . . } perm. { µ µ . . . µ t } = (cid:0) Np + N − t,p ,p ,p (cid:1)(cid:0) tp ,p ,p ,p (cid:1) = N ! p !( p + N − t )! t ! = (cid:0) N k (cid:1)(cid:0) t k (cid:1) , (47)where (cid:0) nn ,n ,...,n k (cid:1) stands for the multinomial coefficient n ! n ! n ! ...n k ! . From the right-hand side of (41), the coefficientof q N − k | q | k multiplied by the combinatorial factor (47) is readily seen to be (cid:0) N k (cid:1) k + 1 (48)for ( t − p ) / integer. The equality between (48) and the coefficient obtained from (45) has to be shown for all s from0 to t/ . But note that t has been eliminated both from (45) and (48), so that for fixed s or k these equations arethe same as the ones between coefficients of a spin– j state anticoherent to order j (the only difference being that inthe latter case Eq. (45) would hold for all s up to s = j ). Since the case t = 2 j has already been proved, the resultensues. The same argument applies to the case t − p odd. Finally, the result for half-integer j can be derived alongthe same line of reasoning. Corollary 1.
A spin– j state ρ is anticoherent to order t if and only if ρ kq = 0 , ∀ k t , ∀ q : − k q k .Proof. Let ρ be a spin– j density matrix with multipolar expansion ρ = j X k =0 k X q = − k ρ kq T ( j ) kq (49)with ρ kq = tr (cid:0) ρ T ( j ) kq † (cid:1) , where T ( j ) kq are the irreducible tensor operators T ( j ) kq = s k + 12 j + 1 j X m,m ′ = − j C jm ′ jm,kq | j, m ′ ih j, m | , (50)and C jm ′ jm,kq are Clebsch-Gordan coefficients. For an anticoherent state to order t , ρ kq = 0 ∀ k t follows from thefact that coefficients ρ kq are proportional to coefficients R kq of the expansion of h α | ρ | α i over spherical harmonics (seee.g. [20]). From Eq. (25) of the paper, tr( ρρ ′ ) = 12 N x µ µ ...µ N x ′ µ µ ...µ N , (51)one gets h α | ρ | α i = X k,q R kq Y kq ( θ, ϕ ) = 12 N x µ µ ...µ N n µ n µ . . . n µ N (52)with n = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) .If a state ρ is such that its spin– ( t/ reduced density matrix is the maximally mixed state then from (3) itscoefficients x µ µ ...µ t ... are given by (26), and all terms x µ µ ...µ t ... n µ . . . n µ t on the right-hand side of (52) can beresummed to a constant independent of θ and ϕ given by x µ µ ...µ t ... n µ . . . n µ t = t/ X k =0 (cid:0) t k (cid:1) k + 1 n t − k | n | k = 2 t t + 1 . (53)What remains in the sum (52) are the terms x µ µ ...µ N n µ n µ . . . n µ N with at least N − t non-vanishing indices µ i ,yielding trigonometric polynomials in θ and ϕ of order at least t + 1 and thus all coefficients R kℓ with k t (apartfrom R ) vanish, and so do coefficients ρ kq .As an illustration, let us consider specific examples. Spin–1 anticoherent states to order 1.
The most general form of spin–1 states that are anticoherent to order1 is obtained by setting x = 1 , x = x = x = 0 . The condition x = 1 imposes unit trace of ρ . The remainingconditions x = x = x = 0 imply, according to Eqs. (26) and (3), that the spin– / reduced density matrix ismaximally mixed. This yields a density matrix in the | j, m i basis of the form ρ = + a β γβ ∗ − a − βγ ∗ − β ∗ + a (54)with non-zero coordinates (up to permutations) x = 1 x = 2[Re( γ ) − a ] , x = 2 Im( γ ) , x = − √ β ) x = − γ ) + a ] , x = 2 √ β ) , x = 4 a + 1 (55)where β, γ ∈ C and a ∈ R . Positivity of ρ is however not yet guaranteed and imposes additional constraints on thevalues of β, γ and a . If we ask that all principal minors of ρ are nonnegative, which translates into the conditions a (1 + 2 a ) −| β | a (cid:18) | γ | − | β | − − a (1 + a ) (cid:19) > | β | + 2 Re[ γβ ∗ ] , (56)then the matrix ρ is positive semi-definite and represents a possible spin–1 state anticoherent to order 1. Conversely,every spin–1 state anticoherent to order 1 has the form (54) with β, γ ∈ C and a ∈ R verifying conditions (56). Anexpression for spin– / anticoherent state to order 2 has been given in [34]. Anticoherent states to order 2.
From Theorem 3 and Eqs. (9) and (26), spin- j state anticoherent to order 2 ischaracterized by the fact that the matrix A µν ≡ h S µν ... i is given by A =
00 0 0 ..