A Complete Set of Invariants for LU-Equivalence of Density Operators

We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature. This implicitly gives a finite complete set of invariants for local unitary equivalence. This is done by showing that local unitary equivalence of density operators is equivalent to local ${\rm GL}$ equivalence and then using techniques from algebraic geometry and geometric invariant theory. We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of $n$-qubit pure states. Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC.


Introduction
Consider the local unitary group U d :"ˆn i"1 U`C d i˘, a product of unitary groups where d " pd 1 , . . . , d n q are positive integer dimensions. Let V i be a d i -dimensional complex Hilbert space and V " b n i"1 V i . Then U d acts on the vector space EndpV q " Â n i"1 EndpV i q, dimpV i q " d i , by linear extension of the action n i"1 g i .ˆn This in turn can be naturally extended to an action on EndpV q 'm by simultaneous conjugation.
This action on density operators is important for understanding entanglement of quantum states [3,14,15,16,21,25,32,33,35]. Many of the most important notions of entanglement are invariant under the action of U d :"ˆn i"1 U`C d i˘ [ 11,34]. Entanglement in turn relates to quantum computation [38,42], quantum error correction [38], and quantum simulation [31]. Two density operators in the same U d orbit are said to be local unitary (LU)-equivalent.
When considering the local unitary equivalence of two mixed quantum states, one can either take two views: the first is that the entire system as a whole is related by a local unitary change of basis. In this case we look at a single density operator acted on by U d . The second is that by considering the same change of basis on each pure state in the mixture, one can take one mixed system to the other. In the latter case, we are looking at local unitary group acting in a simultaneous fashion on the m pure states in the mixed state. Furthermore, our proofs are simplified by considering the problem of classifying the invariants of EndpV q 'm for all m simultaneously.
In this paper, we concern ourselves with the problem of finding a complete set of invariants for density operators. By this we mean a set of U d -invariant functions f 1 , . . . , f s such that two density operators Ψ 1 and Ψ 2 are in the same U d orbit if and only if f i pΨ 1 q " f i pΨ 2 q for all i. In the first part of this paper, we will restrict our attention to polynomial invariants of this action. Remark 1.1. As a caveat: throughout this paper, when we say polynomial invariants, we mean those invariants that are polynomials in the ring Crv 1 , . . . , v n s where the v i are a basis for space EndpV q viewed as a complex vector space. Quite frequently in the physics literature, the term polynomial invariant refers to polynomials in the basis of EndpV q as a real vector space. This allows for invariants such as the Hermitian form. It is known that the set of all polynomial invariants found by viewing EndpV q as a real vector space is complete [39]. It is an interesting consequence of our main theorem, however, that this larger set of polynomial invariants is not necessary for finding a complete set of invariants, which is important if we wish to find minimal complete sets of invariants.
We denote the ring of invariants for G ñ V , V a vector space over a field k, by krV s G . We recall that krV s is to be interpreted as the polynomial ring krv 1 , . . . , v n s where v 1 , . . . , v n form a basis for V . This paper focuses on the completeness of these invariants; finiteness results have been found previously by exhibiting degree bounds on generators and we do not make further contributions in this regard. We show that for density operators in EndpV q, polynomial invariants of degree at most Throughout this paper, whenever possible, our theorems hold for the invariant ring krEndpV qs GL d , where k is an algebraically closed field of characteristic zero which has a Hilbert space structure. Otherwise, k " C. We wish to find a finite (and preferably small) generating set of invariants. We consider the constant β G pV q :" min d | krV s G is generated by polynomials of degree ď d ( .
Upper bounds for this constant have been studied in previous works. We discuss the specific upper bounds for β U d pEndpV q 'm q that arise from general bounds given in the literature, thus giving a finite set of invariants that we show is complete. We now give a brief example to show why completeness of invariants is a non-trivial phenomenon requiring proof. Indeed, it is far from obvious that one cannot find two density operators that are not in the same local unitary orbit but take the same value for every polynomial invariant evaluated on them.
Example 1.2. Consider C 2 being acted upon by the group Cˆin the following manner: λ.px, yq :" pλx, λ´1yq. It is clear that the only invariant is xy. However, if xy " 0, then there are three distinct orbits that px, yq could be in: X :" tpx, 0q | x P Czt0uu, Y :" tp0, yq | y P Czt0uu, or the origin. So we say that these three orbits, while distinct, cannot be separated (or distinguished) by invariants. This problem can be seen in this example in the following way: most orbits are hyperbolas defined by xy " c for c ‰ 0. Therefore each of these orbits is a Euclidean closed subset.
However, for the three problematic orbits, two of them are not closed and contain the origin in their closure. As such, given any continuous function constant on Y , it is also constant on the whole y-axis. Similarly for the functions constant on X. Given any continuous function that is constant on orbits, we see that it must take the same value on X and Y since it is constant on the entire x-axis and constant on the entire y-axis and these two sets intersect.
The goal of this paper is to show that such a phenomenon does not occur if we restrict our attention to density matrices under the local unitary action.
The above example contained orbits that could not be distinguished even by all continuous invariants (as opposed to just the polynomial invariants) and thus we could use the Euclidean topology to understand the problem. However, since we are interested in polynomial invariants, the more natural topology is the Zariski topology. We wish to show that the Zariski closure of two U d orbits of two inequivalent density operators do not intersect. Throughout the paper, we will assume that we are working in the Zariski topology. When we say the closure of a set X, which we will denote X, we will mean the Zariski closure.
We remind the reader that the Zariski closure of a set X is the largest set X, containing X, such that every polynomial that vanishes identically on X must also vanish identically on X. If X " X, we say that X is Zariski closed. We call X Zariski dense in Y if every polynomial that vanishes identically on X must vanish identically on Y .
We wish to use techniques from classical invariant theory and algebraic geometry. The group U d does not satisfy the necessary conditions for the theorems we wish to use (it is not reductive). So instead, we consider the group GL d :"ˆn i"1 GLpC d i q, which is reductive (over C, this means that all of its rational representations are semi-simple). We shall see that for this group action, the Zariski closure of the orbits will actually coincide with its Euclidean closure. This simplifies the problem greatly. We note that throughout the paper, a GL d orbit or set is not assumed to be closed unless explicitly stated.
We say that a group G acts on a vector V rationally, or equivalently, is a rational representation if the map G Ñ EndpV q is given in every coordinate by a rational function that is well-defined everywhere on G. The following two propositions tell us that studying GL d is sufficient. Rational functions are continuous maps with respect to the Zariski topology and so send Zariski dense subsets to Zariski dense subsets. Proposition 1.3. If H is a Zariski dense subgroup of G and ρ is a rational representation of G acting on a vector space V , krV s G " krV s H .
Proof . The representation ρ is a continuous map from G Ñ GLpV q with respect to the Zariski topology by assumption of the rationality of the representation. For every v P V , consider the map ϕ v : G Ñ G.v given by g Þ Ñ g.v. This is also a continuous map and it implies that for every v P V , H.v is dense in G.v since the continuous image of dense sets are dense. The invariant ring is the ring of polynomials which are constant on orbit closures. Since the orbit closures of H and G coincide, their invariant rings must be the same.
It is well known that UpC d i q is a Zariski dense subgroup of GLpC d i q, a fact sometimes known as Weyl's trick. This implies that U d is Zariski dense in GL d , so CrEndpV q 'm s U d " CrEndpV q 'm s GL d . Furthermore, the action GL d ñ EndpV q 'm is not faithful since conjugating a matrix M by αI for α P C leaves M fixed. Therefore, we have that CrEndpV q 'm s SU d " CrEndpV q 'm s SL d " CrEndpV q 'm s GL d . Proof . Consider the polar decomposition of b n i"1 g i " pb n i"1 p i qpb n i"1 u i q where the p i are invertible Hermitian matrices and the u i are unitary. We can assume without loss of generality that all u i " id since it does not change the U d orbit we are in. So note that P " b n i"1 p i is a Hermitian matrix. Let H be Hermitian and suppose that P HP´1 is Hermitian. Then P HP´1 " pP HP´1q : " P´1HP , implying that P 2 HP´2 " H. This implies that either P commutes with H, and thus P HP´1 is in the same U d orbit as H, or P 2 " P P : " id, implying that P was unitary.
By restricting the invariant functions we study to be polynomials, Propositions 1.3 and 1.4 tell us that we can focus our attention instead on the ring CrEndpV qs GL d . However, we may run into the problem that two density operators are in distinct GL d orbits but cannot be distinguished by invariant polynomials. We show in Section 4 that GL d orbits of density operators can always be separated by invariant polynomials.

Background
Previous work on LU-equivalence includes both the invariant theory and normal form approaches. Invariants for LU-equivalence are studied in [15] and much work has been done to understand the invariant rings especially in the case V i -C 2 [49,51,52].
Many polynomial invariants (as well as other invariants) have been identified for this group action. In fact, all polynomial invariants have been found, however this fact has not been proven. We do so in this paper. Invariant based approaches are sometimes criticized because of the difficulty of interpreting the invariants [29,48].
A necessary and sufficient condition for LU-equivalence of a generic class of multipartite pure qubit states is given by Kraus in [25] using a normal form. In [50] the non-degenerate mixed qudit case is covered. Finally a necessary and sufficient condition for LU-equivalence of multipartite mixed states, including degenerate cases, is given by Zhang et al. in [49], also based on a normal form. A similar normal form is given in [29,30] based on HOSVD. The mixed case is treated by purification, so ρ " ρ if and only if Ψ ρ " Ψ ρ .
The normal form approaches work by locally diagonalizing the density operator. They require that the coefficients of the pure or mixed states be known precisely and explicitly so that the normal forms may be computed. However, given two quantum states in the laboratory, determining the density operators Ψ 1 and Ψ 2 is not necessarily feasible.
Nevertheless, computing the values of invariant polynomials for a density operator may not require such knowledge. Given a bipartition A : B of V , where A and B are complementary subsystems, and a density operator ρ, we then note the following equality which is a polynomial for q a natural number. The Rényi entropies [2,3,4,12,44] are a wellstudied measurement of entanglement. Positive integral (q P Z ě1 ) Rényi entropies can be measured experimentally without computing the density operators explicitly [1,7,9,41,45]. This suggests that it may be possible to compute the value of Ψ 1 on an invariant without computing Ψ 1 . This would mean that the invariant polynomials can be expressed as a series of measurements that can be carried out on a quantum state in the laboratory. However, whether or not this is true is still unresolved.

Organization of the paper
In Section 2, we cover the preliminaries of invariant theory we shall need. In Section 3, we classify the invariants of GL d acting EndpV q 'm ; Theorem 3.5 gives the result. In Section 4 we prove the title result. Theorem 4.7 and Corollary 4.8 show that density operators can be distinguished by polynomial invariants. We then draw on results from different sources to find finite sets of polynomial invariants that are complete. Lastly, in Section 5, we discuss a related problem in the study of quantum entanglement. Given the group SL d :"ˆn i"1 SLpC d i q, there is an action on V by pg 1 , . . . , g n q.v :" pb n i"1 g i qv. There has been much research done on computing invariants of this action, known as SLOCC. An algorithm was given that computes all such invariants [14]. For small numbers of qubits (up to four), finite generating sets are explicitly known [40,47] (although there was a misprint in [47] that was corrected in [8]). Work has been done for higher numbers of qubits [15,16,33]. In Theorem 5.6, we classify all invariants for this action for any number of qubits.

Preliminaries
In this section, we state the necessary definitions and theorems we shall need for the rest of this paper.
The result is a multihomogeneous function.
The notion of restitution simply makes formal the idea that if one is given a multilinear function f pX 1 , . . . , X m q, then one may force some of the variables to be equal and the resulting function is no longer multilinear. For example, the function TrpXY 2 q is not multilinear in the variables X and Y . However, it may be seen as the multilinear function TrpXY Zq where we have imposed the restriction that Y " Z. Thus TrpXY 2 q is a multihomogeneous function that is a restitution of the multilinear function TrpXY Zq.
By taking restitutions of multilinear invariants, we can recover generators for the ring of all invariants. An important observation that we shall use later is that if two representations have the same multilinear invariants, then their invariant rings coincide.
Invariant rings can always be generated by multihomogeneous polynomials. The reason for this is that the action of a linear group does not change the degree of the polynomials since it only involves a linear change of variables.
So while it is not true that every invariant is the restitution of a multilinear invariant, the restitutions of multilinear invariants will generate the invariant ring. Furthermore, this ring is finitely generated for certain kinds of groups.
Theorem 2.4 ( [17,18]). If W is a G-module and the induced action on krW s is completely reducible, the invariant ring krV s G is finitely generated.
So we know by the above Theorems that krEndpV q 'm s GL d is always finitely generated.
Definition 2.5. The null cone of an action G ñ V is the set vectors v such that 0 P G.v. We denote it by N V . Equivalently, N V are those v P V such that f pvq " f p0q for all invariant polynomials f . When studying orbit closures, the following theorem is a powerful tools when dealing with reductive groups. It gives a picture of which orbits cannot be distinguished from each other by means of polynomial invariants.
Theorem 2.6 ( [6,36]). Given an action of an algebraic group G ñ V , the orbit closure G.x is the union of G.x and orbits of strictly smaller dimension. An orbit of minimal dimension is closed, thus every closure G.x contains a closed orbit. Furthermore, this closed orbit is unique.
The following theorem gives us a way to reason about points in the orbit closure of a reductive group action that are not in the orbit. Indeed, as it turns out, all such boundary points can be found as endpoints of a path inside of the orbit. This, combined with the fact that every Zariski closed set is Euclidean closed, implies that for reductive group actions, the Zariski closure and Euclidean closure of an orbit coincide.
Theorem 2.7 (the Hilbert-Mumford criterion [22]). For a linearly reductive group G acting on a variety V , if G.wzG.w ‰ ∅, then there exists a v P G.wzG.w and a 1-parameter subgroup por cocharacterq λ : kˆÑ G pwhere λ is a homomorphism of algebraic groupsq, such that lim tÑ0 λptq.w " v.
Note that for the action of GL d ñ EndpV q, if G.w is not closed, then for any v P G.wzG.w, there is a cocharacter λptq such that lim there is some v 1 and cocharacter µptq such that lim Then note that if we define λptq " g´1µptqg, we get a cocharacter of GL d sending w to v as desired.
So we have that every orbit class has a unique representative given by a closed orbit and every closed orbit trivially lies in some orbit class. This motivates the definition of different types of points in V with respect to an action of G.
These definitions have been reinterpreted in terms of the study of entanglement of pure states by Klyachko [23]. For example, every stable point is in the orbit of a completely entangled state and entangled states are simply the semistable points.
Given an action of a reductive group G ñ V , there is a way to write every vector that highlights whether or not its orbit is closed and a representative in the closed orbit its orbit closure contains. Definition 2.9. Given an action G ñ V , a Jordan decomposition of a point v is given by v " v s`vn where v s is a polystable point and v n is an unstable point.
For a rational representation of a reductive group G ñ V , such a Jordan decomposition always exists, although it is not unique. This is well known (cf. [27]), but we include a proof for completeness.
Theorem 2.10. For a reductive group action ϕ : G Ñ GLpV q a Jordan decomposition always exists.
Proof . By Theorem 2.6, ϕpGqv contains a polystable point v s , and by the Hilbert-Mumford criterion (Theorem 2.7), there exists a cocharacter λptq : kˆÑ G such that lim tÑ0 ϕpλptqqv is polystable. Since ϕpλptqq is diagonalizable, there is some g P GLpV q such that lim tÑ0 gϕpλptqqg´1gv " gv s for some v s P V . Now if gϕpλptqqg´1 is diagonal, then gϕpλptqqv is the vector gv with every entry multiplied by a some non-negative power of t (since the limit exists). The unstable part of gv, denoted gv n , is the all zero vector except for those entries of gv that get multiplied by a positive power of t. The stable part is gv s " gv´gv n . Then we see that lim tÑ0 gϕpλptqqg´1gv s " gv s and so lim tÑ0 ϕpλptqqv s " v s . Then we let v n " v´v s . We quickly see that lim tÑ0 ϕpλptqqv " v s and thus lim tÑ0 ϕpλptqqv n " 0. Then v " v s`vn is the Jordan decomposition.
3 Describing the ring krEndpV q 'm s GL d In this section, we describe the invariant ring krEndpV q 'm s GL d by giving a description of all multihomogeneous elements of said ring. We follow Kraft and Procesi's (specifically Chapter 4 in [24]) treatment of the fundamental theorems, generalizing to local conjugation by GL d ; see also Leron [28].
Let us consider the representation of GL d given by µ : v iσ´1 i pjq extended linearly. We will show that the centralizer of this action of GL d is precisely the described action of S n m . In the case of n " 1, the group algebra of S m is precisely the centralizer of GLpV q acting on this space. Furthermore, over an algebraically closed field, the centralizer of the centralizer of an algebra is the original algebra. This a classical theorem called the double centralizer theorem (cf. [26]).
Given a representation ϕ : G Ñ EndpV bm q, denote by xGy ϕ the linear span of the image of G under the map ϕ. We denote the centralizer of the image of µ by End µ GL d pV bm q and the centralizer of the image of ρ by End ρ Sm pV bm q. The following result has appeared before frequently in the literature (for example [15]) but we know of no place where a proof is written down.
Theorem 3.1. Given the described representations µ and ρ, then Proof . Part (b) follows from part (a) by the double centralizer theorem. Now consider the isomorphism ϕ : EndpV q bm -EndpV bm q given by We want to find those elements of EndpV bm q which commute with S n m . So let σ " pσ 1 , . . . , σ n q P S n m and consider The map ϕ induces an isomorphism from End ρ S n m pV bm q to the subalgebra Σ d of EndpV q bm that is S n m invariant under the induced action. We look at its decomposition as a S n m module. Since S n m acts trivially on it, every non-zero irreducible submodule will be one dimensional. Every irreducible representation of S n m is the tensor product of n irreducible S m modules. So we see that an irreducible S n m submodule of Σ d is spanned by a vector s 1 b¨¨¨b s n where each s i is a symmetric tensor in EndpV i q bm since it is invariant under S m .
So we see that Σ d " Â n i"1 Σ i m where Σ i m are the symmetric tensors of EndpV i q bm . However, it is known that Σ i m is generated as an algebra by elements of the form b m i"1 g i for g i P GLpV i q, i.e., Σ i m " xGLpV i qy µ i , where µ i is the restriction to GLpV i q ñ V bm i . This fact is the classical case of the centralizer algebra of the general linear group [5].
So we get Σ d " Â n i"1 xGLpV i qy µ i . However, this algebra is clearly generated as an algebra by elements of the form g bm 1 b¨¨¨b g bm n and so we get that Σ d -xGL d y µ . So we get the equality End ρ S n m pV bm q " xGL d y µ . We now define a set of multilinear polynomials that generalize the trace powers that appear in the classical setting.
Definition 3.2. For σ " pσ 1 , . . . , σ n q P S n m , let σ i " pr 1¨¨¨rk qps 1¨¨¨sl q¨¨¨be a disjoint cycle decomposition. For such a σ P S n m , define the trace monomials by Tr σ " T σ 1¨¨¨T σn on EndpV q 'm , where Proof . Let F denote the space of multilinear functions from EndpV q 'm -pV b V˚q 'm Ñ k.
We caution that F is not the set of linear functions from EndpV q 'm to k, but the set of functions f pM 1 , . . . , M m q from EndpV q 'm to k that is multilinear, i.e., linear in each of the m arguments. We recall that the universal property of tensor products states that the set of functions from V ' W to k that are linear in both arguments is isomorphic to the space pV b W q˚. Extending this, we can identify F with rpV b V˚q bm s˚by the universal property of tensor product. We note that there is an GL d -equivariant isomorphism β : rpV b V˚q bm s˚» Ý Ñ rV bm b pV bm q˚si nduced by rearranging the order of the tensor product in the obvious way and the canonical isomorphism pV˚q bm » Ý Ñ pV bm q˚. We also have an isomorphism of the spaces iven by αpAqpv b φq " φpAvq and extending linearly, which is GLpV bm q-equivariant. Since GL d is a subgroup of GLpV bn q, we get a GL d -equivariant isomorphism EndpV bm q » Ý Ñ F by the map β´1˝α. This induces an isomorphism Note that the following holds for the isomorphism α i : (a) Trpα´1 i pv b ϕqq " ϕpvq, We explain these two equalities in more familiar terms. Equality (a) is the statement that Trpvu T q " u T v " xu, vy for u, v in some vector space U and x¨,¨y the usual inner product.
Since End µ GL d pV bm q -F GL d , by Theorem 3.1, the images of σ P S n m under α are the generators of F GL d . For σ " pσ 1 , . . . , σ n q, we have where the first equality is a consequence of equality (a) and the second equality is a consequence of equality (b) above.
Definition 3.4. Given a vector P " pp 1 , . . . , p |P | q with all p i P rms, and σ P S n |P | , define the polynomials on EndpV q 'm by their action on simple tensors in Â n i"1 EndpV i q, Tr P σ " Tr σˆn Note that Definition 3.4 differs from Definition 3.2 in that it allows for repetition of a matrix in the arguments. So we see that it is precisely a restitution of the multilinear invariants given in Definition 3.4. We now prove this formally.
Theorem 3.5. The ring of GL d -invariants of EndpV q 'm is generated by the Tr P σ . Proof . We observed previously that the multihomogeneous invariants generate all the invariants. Let W " EndpV q. Consider a multihomogeneous invariant function of degree α " pα 1 , . . . , α m q (where some of the α i might be zero) in krW 'm s. It is the restitution of a multi- By Proposition 2.3, we need only look at the restitutions of Tr σ , for σ P S n |α| . What we get is the following:  We can visualize the invariants Tr P σ in an intuitive way. For those familiar with tensor networks, they will recognize the following diagrams. For those unfamiliar, for this particular situation, the rules are very simple. Those interested in knowing more about these invariants as tensor networks can see [3].
We represent the matrix M i P EndpV q 'm by the following picture: . . . M i . . .
In the picture, there are n wires on both sides of the box. Each wire represents one of the vector spaces in V " Â n i"1 V i . The following picture describes how to represent the multiplication M i M j : Given a matrix M P EndpV q, we can take a partial trace relative to one of its subsystems. Suppose we trace out the subsystem V 1 . In the diagram, this would look like the following: Every invariant can be built up by combining these two procedures in any way possible until there are no more "hanging" wires. The resulting picture is a series of loops aligned in n rows. The loops are given by the disjoint cycle decomposition of some permutation and so each invariant is specified by some element in S n m as we saw before.
Example 3.6. We consider a specific invariant for pM 1 , M 2 q P EndpV 1 b V 2 q '2 : Tr p1,1,2q The disjoint cycle decomposition of the first permutation is p1qp23q telling us that in the top row the first box receives a loop and the next two boxes receive a joint loop. Similarly in the bottom row, we see that p12qp3q tells that the first two boxes receive a joint loop and last box a loop on its own. The vector p1, 1, 2q tells us that the boxes are labeled M 1 , M 1 , and M 2 in that order.

Restrictions on the Tr P σ
Much is known about the ring of invariants of EndpV q 'm under the adjoint representation of GLpV q including that it is Cohen-Macaulay and Gorenstein [19]; see Formanek [13] for an exposition.
The following theorem about generators of this invariant ring is classical [24, Section 2.5].
Theorem 3.7 ( [24]). The ring krEndpV q 'm s GLpV q is generated by Furthermore, it is well known that krEndpV qs GLpV q is generated by the polynomials TrpM k q for 1 ď k ď dimpV q and that furthermore, these polynomials are algebraically independent pcf. [24]).
Note that the degree of Tr P σ as a polynomial in the matrix entries equals |P |. Theorem 3.7 does not provide a bound on the generating degree for the invariant ring of the local action krEndpV q 'm s GL d . The reason is that some trace monomials do not factorize into trace monomials of smaller degree, for example see Example 3.6. If it could, we could separate it as two separate invariants placed adjacent to each other.
It is an interesting question to know if one can determine when such an invariant can be factorized. Unfortunately, this problem is NP-complete as we will show by reducing to the following problem. Suppose we are given n multisets S 1 , . . . , S n . Define ΣpS i q :" ř jPS i j. Now suppose ΣpS i q " ΣpS j q for all i, j. Then we want to know if every set admits a partition S j " A j \ B j such that ΣpA j q " ΣpA i q for all i, j and likewise for the sets B i . Deciding this problem is NP-complete if n ą 1 [46]. Proof . The containment of this decision problem in NP is clear. We simply need to prove hardness. Suppose we could decide this problem, then we could decide it for Tr P σ pM q, the case when m " 1. Then define the set S i to be the cycle lengths in the disjoint cycle decomposition in σ i . We see that ΣpS i q " ΣpS j q for all i, j. Furthermore, we see that Tr P σ pM q factors if and only if every set S i admits a partition S i " A i \ B i such that ΣpA i q " ΣpA j q for all i, j and likewise for the sets B i . Proposition 3.8 cautions us about the wisdom of trying to find minimal complete sets of invariants by simply enumerating them and checking to see if they are redundant. This approach will involve solving many instances of an NP-complete problem. However, such an enumeration procedure was recently proposed in [14] for SLOCC invariants. We will see later, that such invariants for n-qubit systems are of the form Tr P σ where the inputs are matrices of restricted form.
Theorem 3.7 does allow us to restrict the functions Tr P σ that act as candidates for generators for the ring krEndpV qs GL d (Proposition 3.11).
Definition 3.9. The size of T P σ i is defined to be the size of the largest cycle in the disjoint cycle decomposition of σ i . Definition 3.10. Given a minimal set of generators, the girth of krEndpV q 'm s GL d is a tuple pw 1 , . . . , w n q where w i is the maximum size of any T P σ i appearing in a generator. The girth of a function Tr P σ is a tuple ps 1 , . . . , s n q, where s i is the size of T P σ i .
Note that the girth of the simple case krEndpV qs GLpk d i q is simply the minimum such that the functions tTrpM i 1¨¨¨M i q : 1 ď i 1 , . . . , i ď mu generate it. We put a partial ordering on girth as follows: pw 1 , . . . , w n q ă pw 1 1 , . . . , w 1 n q if there exists i such that w i ă w 1 i and for no j do we have w 1 j ă w j . The girth is bounded locally by the square of the dimension.
Proposition 3.11. If pw 1 , . . . , w n q is the girth of krEndpV q 'm s GL d , then w i ď y i , where y i is the girth of krEndpV i q 'm s GLpk d i q . In particular for V " V 1 b¨¨¨b V n , the girth of krEndpV q 'm s GL d is bounded by pd 2 1 , . . . , d 2 n q. If d i ď 3, then the girth is bounded by``d 1`1 2˘, . . . ,`d n`1

2˘˘.
Proof . First note that T P σ i lies in the invariant ring R i " krEndpV i q 'm s GLpk d i q . Thus it has size at most y i , where y i is the girth of R i . Now apply Theorem 3.7.

Closed orbits
We first give an a sufficient condition for pM 1 , . . . , M m q P EndpV 'm q to have a closed GL d orbit, where V is a Hilbert space throughout this section. We show that, in particular, tuples of normal matrices over C satisfy the given properties. Since density operators are Hermitian, they are immediately normal. So we seek to show that normal matrices have closed orbits. This will show that polynomial invariants serve as a complete set of invariants when restricted to density operators. As we noted before, the Zariski closures and Euclidean closures of orbits coincide for reductive groups acting rationally. As such, Theorem 4.1 implies that two closed orbits are distinguishable by continuous invariants if and only if they are distinguishable by polynomial invariants. Returning to Remark 1.1, this implies that we need not consider the more general notion of polynomial invariants as often defined in the literature in order to find a complete set of invariants.
, is said to be separable if there exists a cocharacter of GL d , λptq such that @ w P W , lim tÑ0 λptqw " 0, and @ w P W K , w ‰ 0, lim tÑ0 λptqw ‰ 0. We call λptq a separating subgroup of the decomposition (this group is not unique).
Caveat: The definition of a separable decomposition depends on the order in which the summands are written. If V " W ' W K is a separable decomposition, it is not necessarily the case that W K ' W is also a separable decomposition.
Given an arbitrary cocharacter of GL d , it is not clear that there is necessarily a separable decomposition that one can associate to it. The following lemma allows us to replace a cocharacter by one that does have a separable decomposition associated to it that does not affect limits. µptqM µptq´1 for all M P EndpV q such that the limit exists, pbq µp0q :" lim tÑ0 µptq exists, pcq unless λptq " t α id, then µp0q has two nontrivial eigenspaces with eigenvalues 0, 1.
Proof . We can diagonalize λptq by some element g P GL d . Thus it suffices to prove the aboves statements for diagonal cocharacters. If λptq is a diagonal cocharacter, the diagonal entries are of the form t α i , α i P Z (cf. [24]). Let α m be the most negative exponent, or if all α i are strictly positive, then let α m be the smallest positive exponent. Then let µptq " t´α m λptq. We see that for any M P EndpV q, λptqM λptq " µptqM µptq´1. Therefore lim tÑ0 λptqM λptq´1 " lim tÑ0 µptqM µptq´1 whenever the limit exists.
Furthermore, we see that µptq has diagonal entries all non-negative powers of t. Therefore, lim tÑ0 µptq exists and is in fact equal to µp0q. Furthermore, unless µptq " t α id, µp0q will have both zeros and ones on the diagonal. Thus it will have to non-trivial eigenspaces with eigenvalues 0, 1.
We now show how to construct separable decompositions as it is not clear that they necessarily exist. We must use cocharacters of the form as in Lemma 4.3.
Lemma 4.4. Given a cocharacter as in Lemma 4.3, except for λptq " t α id, we can associate it to a separable decomposition for which it is the separating subgroup.
Proof . Let µptq be a cocharacter as in Lemma 4.3. Then we know that µp0q :" lim tÑ0 µptq exists and is a matrix. Then µp0q has two eigenspaces, one attached to eigenvalue 1 and the other to eigenvalue 0. Let W be the null space of µp0q. Then consider the decomposition V " W ' W K . Then @ w P W , lim tÑ0 µptqW " µp0qW " 0, and @ w P W K then lim tÑ0 µptqw " µp0qw, which projects W K onto the eigenspace attached to the eigenvalue 1. This means that the only v P W K such that µp0qv " 0 is v " 0. So this a separable decomposition for which µptq is the separating subgroup.
Let us analyze which decompositions are separable. Let us first analyze the case that λptq " Â n i"1 λ i ptq is as in Lemma 4.3 and is diagonal. Then λ i ptq is diagonal and can be taken to have diagonal entries with all non-negative powers of t. Thus, for every i, we can decompose λptqw " 0 for all w P W i and λptqw " w for all w P W K i . Then pW K 1 b¨¨¨b W K n q K gets sent to zero by λptq. It is easy to see that every separable decomposition for a diagonal cocharacter is of the form From here, it is easy to see that every separable decomposition is of the same form by taking the GL d orbits of diagonal cocharacters. Given a matrix M P EndpV q, we are interested in separable decompositions W ' W K such that M pW q Ď W . Let P W and P W K be the projection operators onto each of the two subspaces. Then define M | W :" P W pM q and M | W K :" P W K pM q. Proof . We can write M as We know that W " pW K 1 b¨¨¨b W K n q K for subspaces W i Ď V i . Then we let λptq " Then we see that where Qptq is a diagonal matrix with non-zero entries being non-negative powers of t. In particular, it is invertible. Then we have that Letting W be the kernel of λp0q, we see that V " W ' W K is a separable decomposition. Let w P W . We note that λptqM w " λptqM λptq´1λptqw. We know that λptqM λptq´1 is a matrix in which only non-negative powers of t appears. Furthermore, every entry of λptqw is scaled by some positive power of t. Therefore every element of λptqM w is scaled by a positive power of t, so lim tÑ0 λptqM w " 0. Therefore M pW q Ď W .
Notice that a similar argument shows that M s pW q Ď W and therefore we can write However, by Proposition 4.5, we can assume that B " 0. That is to say, M s pW K q Ď M s pW K q. If u P W K , then lim tÑ0 λptqu lies in the eigenspace of λp0q attached to the eigenvalue of 1 (it may not be the case that this eigenspace is orthogonal to the kernel of λp0q). However, we note that λptqM n λptq´1 has every entry scaled by a positive power of t, and thus λptqM λptq´1λptqu has all entries scaled by some positive power of t and thus lim tÑ0 λptqM n u " 0. This implies that M n u is in W and therefore, and since M s puq P W K , W K is not an invariant subspace.
We can show that matrices that respect orthogonal decompositions have closed orbits. The prime example are normal matrices as these are precisely the matrices with an orthogonal basis by the spectral theorem. Proof . It suffices to show that for GL d ñ EndpV q, matrices with an orthogonal eigenbasis have closed orbits. Then the result follows from the fact that, if such a pM 1 , . . . , M m q acted on by GL d did not have a closed orbit, then projecting onto some coordinate, say i, would induce a non-trivial limit point, implying that the matrix M i did not have a closed orbit.
Let M have an orthogonal eigenbasis. Then let V " W ' W K be a separable decomposition such that M pW q Ď W . It must be that W is a direct sum of eigenspaces of M (here, by eigenspace, we mean any subspace which M acts on by scaling). Since the eigenspaces of M are orthogonal (in the sense that given two vectors in two different eigenspaces, they are orthogonal), we immediately have that W K is a direct sum of eigenspaces. Thus W K is an invariant subspace of M . Then applying Theorem 4.6, we get that M has a closed orbit.
Corollary 4.8. The GL d orbits of tuples of density matrices are closed, so they can be separated by polynomial invariants. Moreover, two Hermitian matrices are in the same GL d orbit if and only if they are in the same U d orbit.
Proof . We know from Proposition 1.4 that two density operators are in the same GL d orbit if and only if they are in the same U d orbit. We know from Theorem 4.7 that tuples of density operators have closed orbits. We know from Theorem 4.1 that two closed orbits can be distinguished by invariants if and only if they are distinct.
Corollary 4.9. The functions Tr P σ form a complete set of invariants for tuples of density operators under the action of U d .
Proof . This follows from Corollary 4.8 and Theorem 3.5.
So we know that two tuples of density operators are not in the same U d orbit if and only if there is some Tr P σ on which they take different values. We know from Theorem 2.4, that there exists a finite set of functions Tr P σ that forms a complete system of invariants. This theorem does not tell us what such a finite set may be. However, we have a bound given by the following result. . . , f be homogeneous invariants, with maximum degree γ, such that their vanishing locus is N V . Then Furthermore, γ is bounded by CA m where C is the degree of G as a variety and m " dimpρpGqq.
Since ρ is a rational map, it can be viewed as a vector valued function with a rational function in each coordinate. Then A is defined to be the maximum degree of any of these coordinate rational functions.
As we noted earlier, GL d can be replaced by SL d :"ˆn i"1 SLpV i q since this group action has the same invariant ring. Proof . The first part of the statement follows from Proposition 3.11. The degree bound comes from Theorem 4.10 and the following facts. SL d is defined by equations of degrees d i since SL d consists of tuples of matrices each of determinant one, so C ď max d i . It is easy to see that A " 2n as taking the Kronecker product of n matrices gives monomials of degree n in the entries of the original matrices and conjugation is a quadratic action. Since the representation of SL d is faithful dimpρpSL d qq " dimpSL d q " n ř i"1 pd i´1 q. Lastly, we note that dimpkrV s G q ď dimpkrV sq " dimpV q for any G ñ V .

SLOCC invariants for any number of qubits
We now wish to relate the invariants of SL 2 :"ˆn i"1 SLpC 2 q by left multiplication on V 'm , where V " pC 2 q bn , to the invariants of SL 2 by conjugation on EndpV q 'm . The relevant property we use is that the action of SL 2 on V 'm is self-dual. This means that the standard action of SL 2 on C 2 is isomorphic to the representation of SL 2 on pC 2 q˚given by g.ϕ " ϕpg´1q. To state this more formally: where ρ˚is the induced contragradient representation on V˚.
The action of SLpC 2 q on C 2 by left multiplication is self-dual. Let T "ˆ0 1 1 0˙.
Then for any g P SLpC 2 q, T gT´1 " pg´1q T . We consider the map φ : C 2 Ñ pC 2 q˚given by φpvq " pT vq T . Then φpgvq " pT gvq T "`T gT´1T v˘T " pT vq T g´1.
This gives an equivariant isomorphism between the standard action of SLpC 2 q and its induced contragradient representation.
Lemma 5.2. The action of ρ : SL 2 Ñ GLpV 'm q by left multiplication is self-dual.
Proof . Let φ : V 'm Ñ pV˚q 'm be the linear map given by φp' m i"1 v i q " Let G ñ V be a self-dual representation, given by ρ. Then there is an isomorphism φ : ρ Ñ ρ˚. Since it is a linear map, there is a matrix S such that φpvq " pSvq T . Then φpρpgqvq " pSρpgqvq T " pSvq T`S ρpgqS´1˘T " pSvq T g´1.
Thus we have that a representation ρ is self-dual if and only if there exists a matrix S such that SρpgqS´1 " ρpg´1q T for all g P G.
Suppose the representation ρ : G Ñ GLpV q on V is self-dual. Let φ : ρ Ñ ρ˚be the equivariant isomorphism. This induces an action on V 'm , which is clearly self-dual. Then there is an equivariant inclusion of ψ : V 'm ãÑ pV ' V˚q 'm given by So let us consider the invariants on pV ' V˚q 'm with the above action. We first look at the multilinear invariants; from these we can construct all invariants. Let I be the ideal defining the image of V ' V˚inside of EndpV q under the Segre embedding. Recall that the Segre embedding of V ' W is the map pv, wq Þ Ñ v b w. Also recall that the ideal defining a variety is the set of polynomials that vanish identically on the variety. The image of the Segre embedding is G-stable and so its ideal is also G-stable. Proposition 5.3 ([37]). Let G act on a subvariety X Ď V . If G is reductive, and its ideal, I Ď krV s, is a G-stable ideal, then krV s G {pI X krV s G q -pkrV s{Iq G .
Proof . The multilinear invariants are elements of EndpV q 'm of degree d are elements of the space pEndpV q bd q˚by the universal property of tensor product.The multilinear invariants of pV 'V˚q of degree d, are also elements of pEndpV q bd q˚, lying in the image of the Segre embedding V ' V˚ãÑ EndpV q. Furthermore, notice that the action of G on pV ' V˚q 'd and on EndpV q 'd both turn into the action on EndpV q bm given by g.
So the multilinear invariants are the same and by Proposition 2.3, the restitutions are the same. Proposition 5.3 finishes the proof.
Of course, we are not interested in the entire space pV ' V˚q 'm but rather the subset defined by the image of φ : V 'm ãÑ pV ' V˚q 'm . This is also a G-invariant variety.
Letφ : V 'm Ñ EndpV q 'm be the map given by For the case that m " 1, the image of V P EndpV q is matrices of the form v b pv T S T q, which is isomorphic to the Veronese variety of matrices of the form v b v T . Thus the image of V 'm P EndpV q 'm is isomorphic to a direct sum of these Veronese varieties. Now consider its ideal I Ă CrEndpV q 'm s. The action of G on EndpV q 'm induces an action on the coordinate ring. As I defines an G-invariant variety, it is clear that I is a G-stable ideal.
Theorem 5.5. Suppose ρ : G Ñ GLpV q acting on V 'm is self-dual and reductive. Let I be the ideal of Impφq. Then We know that SL 2 is self-dual by Lemma 5.2. Unfortunately, SLpC n q is self-dual only when n " 2. So this method only works for the group SL 2 . We relate this to the invariant ring CrEndpV q 'm s SL 2 , which we have already described.
This turns the polynomials Tr P σ into polynomials in CrV 'm s. These polynomials generate the ring of invariants. However, we haven't accounted for the relations introduced among them from restricting the variety defined by the image ofφ, so many of these polynomials will be redundant.
Proof . By Lemma 5.2, the action of SL 2 on V by left multiplication is self-dual and reductive. Then by Theorem 5.5, the generators of CrEndpV qs SL 2 applied to the image ofφ gives a generating set for CrV s SL 2 . The bound comes from applying Theorem 4.10. The degree of SL 2 is at most two as it is defined by determinants of 2ˆ2 matrices. dimpCrV 'm s SL 2 q ď dimpV q, A is n as we are taking a Kronecker product of n matrices, and SL 2 has dimension 3n.
While Theorem 5.6 gives a complete accounting of all the polynomial SLOCC invariants for an n qubit system, as well as a finite generating set of the ring, further work is necessary. The most obvious problem is that the degree bound is obtained by appealing to a general degree bound for reductive group actions. There is no reason to expect that it is optimal; indeed, we conjecture that a degree bound exists that is polynomial in the dimension of V . For small n, explicit generating sets are known and the following table compares these degree bounds to the ones given by Theorem 5.6, for m " 1. We see that the above degree bound is very far off. While one might be tempted to algorithmically find minimal sets of invariants by enumerating all invariants, the above bound does not give an indication of how long such a enumeration would take. The known minimal degree bounds have been found by a variety of methods. However, as the number of qubits grows, the general approach has been an analysis of the Hilbert series of the rings to determine degrees of generators along with the computations of covariants. For 5 qubits, this method is already computationally prohibitive. As such, if any progress is to made in this direction, a better theoretical understanding of these invariants is necessary rather than relying on computation.
The second issue is that the above invariants might not all be necessary. Indeed, for the case of four qubits, this turned out to be the case [47], although this case was special as there were a finite number of normal forms describing all of the orbits. A classification in terms of geometric properties was later carried out for four qubits [20]. This is not likely to be the case as the number of qubits grows. Nevertheless, there may be relations (although necessarily non-algebraic) among the invariants as a result of restricting to quantum states.