A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems
Abstract
Central in entanglement theory is the characterization of local transformations among pure multipartite states. As a first step towards such a characterization, one needs to identify those states which can be transformed into each other via local operations with a nonvanishing probability. The classes obtained in this way are called SLOCC classes. They can be categorized into three disjoint types: the nullcone, the polystable states and strictly semistable states. Whereas the former two are well characterized, not much is known about strictly semistable states. We derive a criterion for the existence of the latter. In particular, we show that there exists a strictly semistable state if and only if there exist two polystable states whose orbits have different dimensions. We illustrate the usefulness of this criterion by applying it to tripartite states where one of the systems is a qubit. Moreover, we scrutinize all SLOCC classes of these systems and derive a complete characterization of the corresponding orbit types. We present representatives of strictly semistable classes and show to which polystable state they converge via local regular operators.
1 Introduction
Due to the relevance of multipartite entanglement in many areas of physics, entanglement theory has developed into an important research topic over the last decades [39, 22]. Multipartite entanglement does not only play an essential role in quantum communication [21, 16], in quantum computation, e.g., measurementbased quantum computation [45], or quantum metrology (see e.g. [15]), but it does also play a role in condensed matter physics. There, a connection between entanglement present in the wave function and quantum phase transitions can be drawn (for a review see e.g. [1] and references therein). Moreover, tensor network methods such as matrix product states [13, 44, 60] or projected entangled pair states [55, 57] embody powerful numerical tools in condensed matter physics (see e.g. [41] and references therein).
Entanglement theory is a resource theory, which becomes apparent in the scenario of spatially separated parties sharing a joint quantum state, an often considered scenario in quantum information theory. There, it is natural to restrict the allowed operations to Local quantum Operations assisted by Classical Communication (LOCC). Then, entanglement arises as a resource allowing to achieve certain tasks that are not possible by LOCC alone. Moreover, it is apparent that any state , which can be deterministically converted to another state via LOCC, must be at least as entangled as according to any reasonable measure of entanglement. The reason for that is that any task starting out with the state could as well be implemented starting out with the state instead (transforming it to as a first step). The partial order that LOCC imposes on the state space must thus be taken into account by any entanglement measure, which is also known as the LOCCmonotonicity condition for entanglement measures [22].
Studying multipartite entanglement is hence intimately connected to understanding which state transformations are possible via LOCC. A first step in studying whether a state transformation is possible via LOCC deterministically, is studying whether the state transformation is possible via stochastic LOCC (SLOCC), i.e., with a nonvanishing probability of success. Clearly, if a state transformation is not possible via SLOCC, it is not possible via LOCC. It has been shown that if one considers fully entangled states, i.e., states for which all singleparticle reduced density matrices have full rank (here, denotes the number of parties), then SLOCC is an equivalence relation [11]. Mathematically stated, two partite states and are SLOCCequivalent if and only if there exist regular operators such that [11]. The state space thus partitions into socalled SLOCC classes. Another often considered class of operations are Local Unitaries (LUs) (for pure states see e.g. [28, 29]). Such operations are reversible and, hence, do not alter the entanglement of a given state.
For bipartite pure states, necessary and sufficient conditions for when LOCC transformations are possible have been derived [40]. Moreover, for such states, the concept of SLOCC classes is trivial in the sense that there exists exactly one SLOCC class containing all fully entangled states, i.e., states that have full Schmidt rank. All this is no longer true in the multipartite scenario. Deciding which state transformations are possible via LOCC becomes a notoriously difficult problem due to the intricate structure of multipartite LOCC protocols [6, 7, 5]. Moreover, considering the simplest multipartite quantum system, three qubits, there exist two distinct SLOCC classes (considering fully entangled states), the wellknown GHZ and Wclass [11]. Furthermore, already for systems as simple as four qubits, the Hilbert space partitions into an infinite number of SLOCC classes [56]. Besides small system sizes, SLOCC classes have been studied for special types of states, such as permutationsymmetric states [34, 2, 35]. The notion of SLinvariant polynomials has been used to distinguish certain SLOCC classes (see [31, 61, 12] and references therein). These are polynomials in the coefficients of a state , which fulfill that for any ; a complete set of such polynomials can be constructed ^{1}^{1}1For first complete sets of SLinvariant polynomials see [11] and [31].[17, 42].
Recently, a notable step in the characterization of state transformations has been taken for homogeneous systems (systems for which all local dimensions equal). In particular, it has been shown that in such systems, generically, no LOCC transformations are possible [19, 48]. Important in the proof was the fact that within such systems, states for which all singleparticle reduced density matrices are proportional to the identity, socalled critical states, exist. A simple criterion that allows to decide whether a critical state exists for given local dimensions has been derived in [4].
It has been realized that there is an intrinsic connection between the existence of a critical state within an SLOCC class and the geometry of that SLOCC class. Due to the KempfNess theorem, an SLOCC class contains a critical state if and only if it is closed (wrt. standard complex topology) [26]. We call (states within) SLOCC classes containing a critical state polystable ^{2}^{2}2Such states are also known as balanced states [43] and critical states are also known as stochastic states [56].. It moreover holds that, within an SLOCC class, critical states are unique (up to LUs). Moreover, within polystable classes, the norm , where , does attain its minimum for being a critical state [26]. Polystable SLOCC classes have been a subject of interest in previous works [27, 43, 19, 48, 18, 17, 47]. It has been shown that such SLOCC classes can be distinguished by ratios of SLinvariant polynomials [17]. SLOCC classes (states) for which all nonconstant homogeneous SLinvariant polynomials vanish, form the socalled nullcone, such classes have been studied e.g. in [38, 63, 51, 32, 33, 25]. These SLOCC classes contain the vector in their closure and, obviously, cannot be distinguished with SLinvariant polynomials. SLOCC classes (states) that are not in the nullcone are called semistable. Thus, the polystable classes are those semistable classes that do contain a critical state. We call (states within) the remaining classes strictly semistable. These SLOCC classes do not contain a critical state, however, they do contain a critical state in their closure [26].
Numerical tools which allow distinguishing between certain of the aforementioned types of SLOCC classes have been developed. In [59], an algorithm which transforms a state into its socalled normal form has been presented. In case is in the nullcone, this normal form is . Otherwise, it coincides with the critical state within the SLOCC class of (closure of the SLOCC class of ), in case is polystable (strictly semistable), respectively. In [63, 51], it has been shown that a numerical algorithm, which follows the gradient of the sum of the linear entropies of the eigenvalues of , also allows to determine the critical state within a polystable SLOCC class, or the critical state within the closure of a strictly semistable SLOCC class. Moreover, the latter method also allows to distinguish certain SLOCC classes within the nullcone [38, 51, 50, 32, 33, 49].
Here, we study the existence of semistable states, i.e., states that are neither in the nullcone, nor SLOCCequivalent to a critical state using tools from Geometric Invariant Theory (GIT). One of the main results of this article is a criterion for the existence of such states in terms of the dimensions of polystable SLOCC classes. Moreover, we study threepartite systems with local dimensions , , and in more details. SLOCC classes in such systems have been characterized in [8] (see also [20]) with the help of the theory of matrix pencils [30]. We apply the derived criterion for the existence of strictly semistable states in the context of such systems. Moreover, we not only answer the question of whether strictly semistable states exist, but also derive a full characterization of the orbit types of all SLOCC classes within such systems.
The outline of the remainder of the paper is the following. First we introduce our notation and some preliminary results on SLOCC classes and the normal form of multipartite states [59]. We will also recall how SLOCC classes of polystable states can be distinguished via SLinvariant polynomials [17]. In Sec. 3 we will focus on semistable states. Using tools from GIT, we will show that strictly semistable states exist if and only if there exist two polystable states whose orbits have different dimensions. In Sec. 4 we illustrate the usefulness of this criterion by characterizing all SLOCC classes of tripartite states in . Moreover, we will identify the representatives of strictly semistable states and will show to which polystable states they converge via SLOCC.
2 Notation and Preliminaries
In this section we first introduce our notation and then review some important concepts utilized in the characterization of SLOCC classes.
We consider the Hilbert space, with arbitrary local dimensions . We are mainly interested in pure normalized states in . Hence, we consider the complex projective space , which is obtained from by identifying any two vectors which are proportional to each other via a nonzero complex number. For any vector the corresponding quantum state is denoted by . Throughout the paper we will consider actions of appropriate Lie groups on both and ). The action of a Lie group on will be denoted by and the action on by , respectively. Given a Lie group , the –orbit of a vector is defined as
(1) 
The stabilizer of a vector with respect to the group is defined as
(2) 
Any orbit is an embedded submanifold of which is isomorphic to the left coset of in , namely . Hence, the larger the dimension of the stabilizer of the state, which is equal to the number of linearly independent generators of the stabilizing Lie subgroup, the smaller the dimension of the orbit. More precisely, the dimensions satisfy the formula
(3) 
The action of a Lie group on is induced by the action via
(4) 
Hence, the orbit through any state is given by
(5) 
i.e. the –orbit of any in is the projection of the orbit . Thus, in contrast to , the –orbit of , , might contain unnormalized states. The stabilizer of a state with respect to the group is defined as
(6) 
We also have that and
(7) 
The stabilizer is the set of symmetries of in . Note that for any and any Lie group . The two groups we will consider are

Special local unitary operators, i.e. elements of

SLOCC operators, i.e. elements of .
Both groups act in a natural way on and , respectively. The action of local unitary operators does not alter entanglement and it defines an equivalence relation. Two states are in the same LUequivalence class if they belong to the same –orbit in . Hence, instead of considering all states in , we will consider representatives of LU–equivalence classes. Similarly the –orbit of a state, , is the SLOCC–class which contains as an element. Two states, are in the same SLOCC–class, i.e. they are SLOCC–equivalent, if there exist such that .
There is a qualitative difference between the actions of and . The group is compact and all –orbits are closed. On the other hand the group is not compact and –orbits may not be closed. The closure of an orbit in the standard complex topology in can obviously be obtained by adding to the limits of all sequences of vectors in . It is well known (see e.g. [62]), that the closure in standard topology coincides with the closure in Zariski topology. We refer the reader to [62] for more details on the Zariski topology and the relations between those two topologies. Let us just mention here that a set is Zariski closed if there exists a set of polynomials, such that , where denotes the polynomial ring. A Zariski closed set is called an algebraic variety. As mentioned above, the closures of in coincide in both topologies. It will become clear later on, which orbits are considered to be closed.
States which play a particularly important role in the classification of multipartite states are so–called critical states. A vector is called critical if all singleparticle reduced states are completely mixed, i.e. for all . In the following we will denote the set of critical vectors by and the set of critical states by .
2.1 Normal form of multipartite states
As mentioned above, orbits in may not be closed. In fact, the KempfNess theorem [26] states that is closed if and only if it contains a unique (up to LUs) critical vector ^{3}^{3}3 Note that a particular example of a closed orbit is the orbit containing the vector.. Moreover, for any .
Let us note that orbit is open in its closure , therefore the boundary of can be defined as ^{4}^{4}4See e.g. [3] p. 4, Proposition 1.11..
The following lemma describes the structure of the boundary of an orbit and follows from wellknown facts  about the dimensions of orbits in the boundary of orbit ^{5}^{5}5See e.g. [3] p. 4, Proposition 1.11. and basic facts in GIT theory ^{6}^{6}6See e.g. [36] p. 161, Theorem 5.3. and p. 162, Corollary 5.5..
Lemma 1.
For each , the boundary of its orbit, , is a union of –orbits of strictly smaller dimension. Each orbit closure contains a unique closed –orbit. This closed orbit is the only orbit in the closure which has minimal dimension, i.e. the closed orbit has the smallest dimension of all orbits in the closure.
Starting from a vector one can look for
(8) 
According to the above, one of the following must be the case.

The infimum is .

The infimum is not , it is attained at some critical vector within the orbit , and the orbit is closed.

The infimum is not and it is attained at some critical vector , which is not in the orbit , but within its closure, .
In case (i) the vectors are said to be unstable (they form the socalled nullcone, which will be denoted by ). For such vectors, there exist sequences such that . Vectors that are not unstable are called semistable and they form a set . In case (ii) the vectors are called polystable; they are, hence, elements of an SLOCC class which contains a critical vector. In case (ii) one can distinguish two subcases: (ii a)  when the orbit of a polystable vector is of maximal dimension; (ii b)  when the orbit of a polystable state is not of maximal dimension. Note that maximal dimension refers here and in the following to the maximal dimension among all polystable orbits. In case (ii a), we call a vector stable ^{7}^{7}7Notice that in the literature concerning GIT (see e.g. [9] Chapter 8), the usual definition of stable vector is different  a vector is stable if its orbit is closed (i.e. the state is polystable) and has a finite stabilizer. Thus, our definition is much weaker since we only require the dimension of the stabilizer to be minimal (among all polystable orbits), not necessarily zero. and in case (ii b), we call a vector strictly polystable. In case (iii), vectors are semistable but not polystable  we call such vectors strictly semistable. For such states, an infinite sequence is required in order to reach a critical state, i.e., . It has been shown that any strictly semistable vector is a superposition of a polystable vector and a vector from the nullcone (see the Supplemental Material for [17] p. 1, Proposition 2). Finally, a state in the projective space is called unstable/semistable/polystable etc. if it is the projection of some unstable/semistable/polystable vector , respectively, i.e. . We denote the set of semistable states by . In case of a system, the W–state, is a prominent example of a state in the nullcone (case (i)), whereas the critical GHZ–state is a stable state (case (ii a)). In order to give an example of strictly semi and polystable states we need to consider systems of larger dimension. In case of a system, a state 4 and Appendix F.7). We illustrate the relationship between the types of states in Figure 1. is strictly semistable (case (iii); cf. Section is strictly polystable (case (ii b) and a state
Note that there is a numeric algorithm, which for any state , determines a sequence of SLoperators, which converges, if applied to , to the normal form as discussed above [59]. If is in the nullcone, the normal form vanishes. If is polystable, then the normal form coincides with the critical state within the SLOCC class of . Finally, if is strictly semistable, the algorithm converges to, but does not reach, a critical state in the closure of the SLOCC class of ^{8}^{8}8Let us remark here that for system sizes we are considering here, numerical errors become relevant. Keeping track of the state by storing its computational basis coefficients may result in jumping to another SLOCC class due to precision errors. States that are in the nullcone may be erroneously identified as semistable, e.g., and in Appendix F.9. A more robust implementation can be achieved by keeping track of the accumulated local operators, instead..
The KempfNess theorem [26] is a powerful tool characterizing polystable states. Any SLOCC–class of a polystable state contains a unique critical state (up to LUs). It is known that , is open, dense, and of full measure in [18, 62], provided that is nonempty. We will use this fact in the proof of our main theorem which gives a criterion for the existence of strictly semistable states.
2.2 invariant polynomials
–invariant polynomials constitute an important tool in the study of SLOCC–classes. A polynomial, is called –invariant if for any and any . The vector space (over ) of all –invariant polynomials is spanned by homogeneous polynomials of degree . The set of –invariant polynomials forms a ring which is finitely generated [36].
As polynomials are continuous functions, any –invariant polynomial will have the same value on any two orbits, and in , whose closures intersect. Hence, as strictly semistable vectors converge to critical vectors, the SLOCC–class of a strictly semistable state cannot be differentiated from the one of the critical state, which is in its closure using invariant polynomials. Moreover, for any vector in the nullcone any of these polynomials vanish and vectors which are in the different SLOCC–classes within the nullcone cannot be distinguished via invariant polynomials ^{9}^{9}9Note however, that some orbits in the nullcone can be distinguished by studying the gradient flow of the function , where is the th reduced state of [38, 51, 50, 32, 33].. However, in [17] it has been shown that SLOCC–classes of polystable states can be distinguished using ratios of –invariant polynomials, as we recall in the following.
Two polystable states are SLOCC equivalent if and only if [17]
(9) 
for any –invariant homogeneous polynomials with of degree for any . As mentioned before, polystable states constitute a full measure set of states in case at least one critical state exists. Hence, if , the criterion above solves the SLOCC–equivalence problem for generic states.
The necessary and sufficient condition for the existence of a critical state has been presented in [4]. For arbitrary dimensional Hilbert spaces, a critical state exists if and only if
(10) 
We will elaborate on that for the Hilbert spaces , which we denote by , in Section 4.
In summary, there are many important facts known about polystable states and also the nullcone has been investigated previously [38, 24, 25, 51, 33]. However, little is known about strictly semistable states. In the subsequent section we will derive a simple necessary and sufficient condition for their existence. We will use this criterion for the simplest examples of multipartite systems, namely the Hilbert spaces . There, we will not only identify the dimensions for which strictly semistable states exist, but we will scrutinize all SLOCC classes and present a complete characterization of them.
3 The existence of strictly semistable states
In this section we will show that there exists a strictly semistable state if and only if there exist two critical states whose –orbits do not have the same dimensions (see Theorem 1). Note that the latter means that there exist a strictly polystable state (i.e. not all polystable states are stable). To this end, we will first restrict ourselves to the set of semistable states. We will then introduce the definition of –equivalence, which leads to a coarser classification than SLOCC–equivalence. Next, using tools from GIT we will prove the main theorem of the paper.
3.1 The set of semistable states
Our aim is to derive a criterion for the existence of a strictly semistable state. Hence, we can restrict our attention to the Hilbert space excluding the nullcone, i.e. , or, more precisely, . However, we will first demonstrate that without excluding the nullcone, taking the closure in the projective space might become problematic. To give a simple example, let us consider the 3–qubit GHZ–state, and the regular matrix
Observation 1.
For any vector , we have
(11) 
Hence, considering as the full (topological) space we have that the orbit is closed (in ) if and only if is closed in . Moreover, for any pair of states it holds that if and only if . Thus if we treat and as full topological spaces, we have a well defined correspondence between closures in and . In the following we will always consider and as full topological spaces.
Due to the results summarized in Sec. 2 we have that the closure of a –orbit in always contains a unique critical state (up to LUs). It might also contain strictly semistable states of various SLOCC classes, which converge to this critical state. In order to derive a criterion for the existence of these strictly semistable states, we adapt now Lemma 1 for states in . To this end, we use the following observation, which follows from Corollary 10 of [18].
Observation 2.
For any it holds that
That is, the dimension of the –orbit in the projective space coincides with the dimension of the –orbit in the Hilbert space for any semistable state.
Proof.
As we consider here, in contrast to [18], the projective space, we adapt the proof of [18] to our setting. Of course . Assume next that there is such that . When this implies that and when it means . In both cases this is contradiction as is semistable. Thus the only possibility is . We will show next that belongs to a finite set. Let be a set of homogeneous generators of invariant polynomials, . Then
(12) 
Thus . This means that is a finite set and hence ^{10}^{10}10Notice that in fact it suffices to consider any chosen –invariant homogeneous polynomial.. ∎
Due to the relation given in Eqs. (3) and (7) the above statement is equivalent to the statement that the dimensions of the stabilizers coincide. That is, the dimension of the set of operators, which leave the vector invariant, , coincides with the dimension of the set of operators, which leave the vector up to a proportionality factor invariant, .
Combining now Lemma 1 with the Observations 1 and 2 leads to the following adaptation of Lemma 1 for states in .
Lemma 2.
For each , the boundary of an orbit in is a union of –orbits in of strictly smaller dimension. Each orbit closure contains a unique polystable –orbit. This polystable orbit is the only orbit in the closure which has minimal dimension.
3.2 c–equivalence
The number of SLOCCclasses, i.e. –orbits, is in general infinite [11]. Furthermore, as mentioned before, orbits may not be closed. This motivates the introduction of closureequivalence (equivalence) which is used in GIT. By the Theorem of Nagata and Mumford it is known that –equivalence can be defined using three equivalent definitions (see [36]). As we are concerned with semistable states, it will suffice to consider the following definition (see also Figure 2).
Definition 1.
Two states are called –equivalent if
(13) 
We note that –equivalence is indeed an equivalence relation. As the closure of a semistable state contains a critical state, which does not belong to its –orbit, we have that –equivalence induces a coarser classification than SLOCCequivalence. As explained in Sec. 2.2, –classes can be distinguished using the ratios of SL–invariant polynomials. Each –class contains exactly one closed –orbit. Due to Lemma 2 this –orbit is the only orbit of minimal dimension (in the closure of the –orbit). This closed orbit lies in the closure of every –orbit from the –class [36] and is the one mentioned in Lemma 2. Moreover, due to the KempfNess theorem [26], each –class is uniquely characterized by the unique (up to LU) critical state. Hence, –classes are in 11 correspondence with critical states (up to LU) (see Figure 2).
In order to state these facts mathematically rigorously, several definitions would be in order. As these definitions are cumbersome, we do not state them here, but refer the reader to Appendix C. We consider the set of classes given by the –equivalence relation in . It turns out that the set can be equipped with the structure of a projective algebraic variety denoted by . The GIT construction gives a regular (see Definition 6), hence continuous mapping called the GIT quotient [3]. Thus, is a point in a quotient variety corresponding to the –class of and if and only if and are –equivalent. The KempfNess theorem gives us the bijection . Hence, the GIT quotient maps a semistable state to the unique (up to LUs) critical state in .
In what follows we call a –class trivial if it is a singleton, i.e. if it consists of exactly one (thus closed/polystable) –orbit. We will say that the –equivalence in is trivial if each –class (if there is any) is trivial, i.e. the GIT quotient does not identify any two distinct –orbits or there are no critical states (so no –classes). Otherwise the –equivalence is called nontrivial (see Figure 2). For example, one may check that in case of system, the left hand side of Formula 10 is negative, what indicates that there are no critical states in this system (cf. Example F.6 from Appendix F). Hence, –equivalence in system is trivial. Stated differently, the –equivalence in is trivial if each –class is also a SLOCC–class, i.e. –equivalence is not a coarser classification than SLOCC. Note that a –class is nontrivial if and only if there exists a strictly semistable state whose –orbit belongs to this class. Similarly the –equivalence is nontrivial if and only if there exists a strictly semistable state (given by some nontrivial –class).
3.3 Stability of the action of a group
We briefly discuss here the fact that, in case critical states exist, the action of on is stable, which is required in the proof of the main theorem of this paper. Let us first introduce the definition of stability of an action of a group.
Definition 2.
An action is stable if there exists an open dense set that is a union of polystable orbits.
We show in Appendix B that the existence of an open dense set that is a union of polystable orbits implies the existence of an open dense set that is a union of polystable orbits of maximal dimension (among all orbits). In particular, is a union of stable orbits so in case the action is stable, the maximal dimension among polystable orbits (i.e. the dimension of stable orbits) coincides with the maximal dimension among all orbits, which we will denote in the following by .
The action is stable if and only if critical states exist, as the union of the SLOCC–orbits of critical states (i.e. polystable orbits) is an open dense subset , whenever [18]. In particular, we have ^{11}^{11}11The reason for that is that the dimension of an open and dense set is the same as the dimension of the whole space.
(14) 
Let us note here that having maximal dimension does not necessarily mean that is finite, i.e. that the dimension of the local stabilizer is zero. An example would be the –orbit of the 3–qubit GHZ state, which is of maximal dimension, as it is of full measure. However, its dimension does not coincide with the dimension of since it admits continuous symmetries and its stabilizer has a positive dimension [4] ^{12}^{12}12More precisely, we have that and [58].. However, we have that , as the –state is the only critical state for three qubits.
3.4 Existence of strictly semistable states
In this subsection, we prove the main theorem (Theorem 1), which provides a necessary and sufficient condition for the existence of strictly semistable states and discuss its consequences.
In order to do so, we will restate a lemma from [37], which we will apply later in the context of the GIT quotient and which is a direct consequence of Corollary 15.5.4 from [54].
Lemma 3 (On the fibres of the mapping [37]).
Let and be quasiprojective irreducible varieties and a regular surjective map. Then, there exists an open and dense set , such that for any points and , we have
(15) 
and .
We defer a more detailed discussion of the assumptions of the lemma as well as proper definitions of quasiprojective varieties, irreducible varieties, and regular maps to Appendix C. However, for our purposes here, note that Lemma 3 is applicable to the GIT quotient . Let us briefly outline the reason for that. First, and are indeed irreducible quasiprojective varieties (see Appendix C). Moreover, is a regular map, which follows from the fact that a finite number of rational polynomials can be used to decide the class of any given semistable state , as explained in Section 3.2.
The sets in Lemma 3 are called fibres and the dimension is called the minimal fibre dimension. Notice that is an open and dense subset of and generic fibres have minimal fibre dimension . We are now in the position to prove the main theorem, which presents a criterion for when a –equivalence is trivial (i.e. all GIT quotient fibres are single –orbits).
Theorem 1 ( Main Theorem ).
There exist no strictly semistable state if and only if the dimensions of all polystable –orbits are the same (i.e. they are all stable).
Proof.
First, note that if there exists no critical state, then there also does not exist a strictly semistable state and thus the statement holds trivially.
Let us now assume that there does exist a critical state. As mentioned before, we denote by the maximal dimension among all –orbits. As discussed in Section 3.3, coincides with the maximal orbit dimension among all polystable –orbits, i.e., the dimension of stable orbits. We prove both directions by contradiction.
If: If there exists a strictly semistable state, , we have that contains a polystable –orbit of strictly smaller dimension than (Lemma 2). Therefore, there has to exist a polystable state whose orbit is of strictly smaller dimension than , i.e., there has to exist a strictly polystable state.
Only if: As explained earlier in this subsection, fulfills the conditions of Lemma 3. Let us first show that the minimal fibre dimension of , , equals the maximal orbit dimension, . To this end, let be the union of all stable orbits. Suppose that . Obviously . Next, by Lemma 2 there exists no state such that as this would imply that , which contradicts the fact that is the maximal dimension. This means that the –class of any is given by , i.e. for each we have . In particular, for each we have . Recall that is open and dense. Let be the open and dense set from Lemma 3. From continuity of the GIT quotient we have that is an open and dense set so , as two open and dense sets need to intersect. Thus, we have that and therefore [recall that is the minimal fibre dimension from Eq (15)]. Hence, the dimension of any –class must be greater than or equal to .
We are now in the position to prove that if there exists a polystable –orbit of smaller than maximal dimension (i.e. a strictly polystable orbit), then there exists a strictly semistable state. Suppose that there exists a such that . From the fact that the minimal fibre dimension is we know that . Hence, the fibre must contain another –orbit (which cannot be polystable due to the KempfNess theorem), i.e. there must exist a strictly semistable state. ∎
As the dimension of the orbits is directly related to the dimension of the stabilizer (see Eq. (7)) and as the –equivalence is trivial iff there exists no strictly semistable state, the following statements are equivalent to Theorem 1.

The equivalence is nontrivial if and only if there exists a critical state whose stabilizer has strictly larger dimension than the stabilizer of a generic state.

There exists a strictly semistable state if and only if there exists a polystable state whose –orbit has smaller than the maximal dimension .

The existence of strictly semistable states is equivalent to the existence of strictly polystable states.
Moreover, note that the proof of Theorem 1 can also be applied to individual –classes, as stated in the following
Observation 3.
The –class of a given polystable orbit is nontrivial if and only if is smaller than the dimension of a generic orbit (i.e. has more continuous symmetries). If this is the case then the dimension of the –orbit of every strictly semistable state in such a –class is larger than the dimension of (see also Figure 2).
Let us remark here that Theorem 1 can be extended to other groups whose action is stable and is the complexification of , i.e. . In the subsequent section we demonstrate the usefulness of this criterion for Hilbert spaces of dimension .
4 The categorization of SLOCC classes of states in
In this section, we apply the derived criterion (Theorem 1) for the existence of strictly semistable states to three partite systems in a Hilbert space (). We show that strictly semistable states exist (and thus, cequivalence is nontrivial) if and only if and . Then, we go beyond that and provide a full characterization of orbit types of all SLOCC classes, i.e., orbits, for systems.
The outline of this section is as follows. First, we will briefly recall the characterization of SLOCC classes in systems using the theory of matrix pencils [8] in Subsection 4.1. Then, in Subsection 4.2, we will recall a characterization of critical states in systems obtained in [4]. In Subsection 4.3, we apply Theorem 1 to systems. In Subsection 4.4, we perform a complete characterization of the orbit types of SLOCC classes in systems, which we summarize in form of a flowchart in Subsection 4.5. Finally, we provide tables of SLOCC classes in systems equipped with orbit types for small system sizes (up to ) in Appendix F.
4.1 SLOCCclasses of states in
The SLOCC classification in systems was obtained via the Kronecker Canonical Form (KCF) of linear matrix pencils [8]. As we will use this characterization, we recall here briefly the idea. The reader is referred to [14, 30, 8, 20] for the definitions used in the categorization of matrix pencils. A linear matrix pencil is a homogeneous matrix polynomial of degree 1 in variables and ,
(16) 
where and are matrices. The matrix pencil associated to an arbitrary state in , with
(17) 
where , is given by Eq. (16). Let us now recall how the matrix pencil corresponding to a state is transformed under SLOCC operators . Let
(18) 
then the matrix pencil transforms as
(19)  
(20) 
A normal form of matrix pencils, the KCF, under regular operators and as in Eq. (20) has been derived in [30]. It is always possible to find operators and which transform a matrix pencil into its KCF. In the following, we only consider fully entangled states, i.e., states for which all singleparticle reduced density matrices have full rank. Restricting to matrix pencils that correspond to fully entangled states, the KCF of a matrix pencil is the (generalized) blockdiagonal form
(21) 
where
(22) 
has size ^{13}^{13}13Here and in the following, matrix elements that are not displayed are . and
(23) 
where the pairwise different are called the finite eigenvalues of a matrix pencil, are called size signatures of the eigenvalue , and denotes the number of (distinct) eigenvalues. Moreover, an eigenvalue may occur several times in Eq. (23) and is called the multiplicity of the eigenvalue . blocks correspond to an infinite eigenvalue (which we denote by ). The blocks take the form
(24) 
The sizes of these matrices are and , respectively. The KCF of a matrix pencil is invariant under the operators , , which makes it a useful tool for studying SLOCC classes. However, the operator [given in Eq. (18)] changes the finite eigenvalues according to
(25) 
and the infinite eigenvalue according to
(26) 
Other than that, does not change the KCF of a matrix pencil ^{14}^{14}14In particular, the occurrence of  and blocks as well as their sizes are unchanged, as are the size signatures of the eigenvalues.. Note that with an operator it is always possible to set three of the distinct eigenvalues to arbitrary new (distinct) values. Moreover, it is always possible to bring a matrix pencil into a form without infinite eigenvalues [8]. Therefore, in the following, we only consider finite eigenvalues for convenience. Degeneracies of eigenvalues cannot be changed with any finite . However, we will see later on, it is indeed possible to create degeneracy when considering the limits of sequences of operators.
All SLOCC classes of states in can be characterized by the KCF of the corresponding matrix pencils [8]. Moreover, the problem of listing all SLOCC classes becomes a combinatorial problem of listing all possible KCFs. Let us restate here the according theorem from [8].
Theorem 2 ([8]).
Two dimensional pure states and for which their corresponding matrix pencils have only finite eigenvalues and , respectively, are SLOCC equivalent if and only if the KCFs of the matrix pencils agree up to a linear fractional transformation of the eigenvalues,
(27) 
for some , where .
4.2 Critical states
The question of the existence of a critical state has been answered for general multipartite quantum systems in [4]. For systems, at least one SLOCC class containing a critical state exists if and only if or divides [4]. Moreover, in systems, there is a unique (up to LUs) critical state [4], which is given by
(28) 
More generally, in all systems for which divides , there is a unique (up to LUs) critical state which is LUequivalent to [4]
(29) 
where the system held by the second (third) party is composed of the subsystems and ( and ), respectively. The corresponding matrix pencil is .
Moreover, for systems, critical states are (up to LUs) of the form [4]
(30) 
where and for such that
(31) 
The critical state is unique in cases and [4]. Note that it is possible to associate a geometrical meaning to the conditions in Eq. (31). The pairs can be interpreted as spherical coordinates of unit vectors
(32) 
and the state in Eq. (4.2) is critical if and only if the vectors sum to [4].
Note that matrix pencils corresponding to the states given in Eq. (4.2) are diagonal, i.e. direct sums of blocks and the eigenvalues of the matrix pencil, , are related to the unit vectors by [4]
(33) 
Recall, however, that the operator on the first system () may change the eigenvalues of the matrix pencil ^{15}^{15}15Operators and do not change the eigenvalues, though, as the KCF of a matrix pencil and thus the eigenvalues are invariant under and .. It will become important later on to understand how the vectors are transformed under such an operator . Let us consider the singular value decomposition, , where . Writing in Euler decomposition and absorbing the third () rotation into (as the rotation commutes with the diagonal matrix), we have . Note that we will disregard , as it is only a LU transformation of the corresponding states. . We are now interested in how this transformation changes the eigenvalues and thus the vectors
Let us first consider the application of in case , i.e., we only have the transformation , which acts on all vectors simultaneously as
(34) 
Note that—under this nonlinear transformation — increases strictly with for all , unless or . In the latter cases is left invariant. The transformation can be pictorially understood as hinging all (except those pointing in direction or ) down towards the direction without changing their azimuthal angle, .
Let us now consider the full operator also taking the unitary into account. It can be easily seen that unitary transformations by correspond to rotations of the [4]. Taking the operation stemming from into account, the vectors are simply rotated before hinging them down by according to Eq. (4.2) along the axis. Equivalently, the operation stemming from can also be understood as choosing an axis along which the “hinging”transformation is performed (on static vectors).
4.3 cequivalence and existence of strictly semistable states
In this subsection, we will apply Theorem 1 to systems to characterize those , for which strictly semistable states exist.
Recall that in the cases , or [4] there exists either no critical state, or a unique critical state (up to LUs). Combining this fact with Theorem 1 we obtain the following corollary.
Corollary 2.1.
In systems, there exists no strictly semistable state if , or .
In the following we will show that in all other cases, i.e. for , strictly semistable states exist and thus cequivalence is nontrivial. To this end, it suffices to identify two critical states whose dimensions of the stabilizer group differ (see Theorem 1).
It can be easily verified that the (complex) dimension of the stabilizer of states of the form given in Eq. (4.2) is given in terms of the degeneracies of the eigenvalues of the corresponding (diagonal) matrix pencil and reads
(35) 
where denotes the number of distinct eigenvalues and their degeneracies. To this see this, note that symmetries of the form for diagonal matrix pencils must satisfy as it must hold that . Moreover, must commute with . Hence, the number of free parameters is , which gives rise to the second summand on the righthand side of Eq. (35). Moreover, any operator that gives rise to a symmetry must act in such a way that all sets of eigenvalues with coinciding multiplicity are mapped into themselfs. This gives rise to some finite freedom in choosing a permutation of the eigenvalues in an appropriate way. Then, the image of three distinct complex numbers uniquely determines an operator , while any number less then three allows free complex parameters in as in the first summand on the righthand side of Eq. (35).
It is clear that is always minimized by , i.e., by having all distinct eigenvalues, as summarized in the following observation.
Observation 4.
The dimension of the SLOCC orbit of a state corresponding to a diagonal matrix pencil with all distinct eigenvalues is strictly larger than the orbit dimension of a state corresponding to a diagonal matrix pencil with any degeneracy in the eigenvalues.
Using this observation, it is easy to prove the following theorem.
Theorem 3.
In systems, there exist striclty semistable states if and only if .
In other words, cequivalence is nontrivial if and only if .
Proof.
For , there exist critical states corresponding to diagonal matrix pencils that do and those which do not have degenerate eigenvalues. This guarantees the existence of strictly semistable states according to Theorem 1. This can be easily seen in the geometrical picture reviewed in Section 4.2. For , it is always possible to find distinct unit vectors in with multiplicity , which sum to . Moreover, for , it is always possible to construct one vector with as well as up to additional unit vectors with multiplicities summing up to , such that the total weighted sum of vectors yields . ∎
4.4 Characterization of orbit types
In this subsection, we go beyond Theorem 1 and derive a full characterization of the orbit types of SLOCC classes in systems with the help of matrix pencils. More precisely, we provide a systematic way of checking properties of a matrix pencil (corresponding to a state) in order to conclude the orbit type of the corresponding SLOCC class. Let us remark here that for some choices of and , the considerations above as well as Theorem 3 (see also [4]) already give the full characterization, e.g., cases in which , or . However, in case , the considerations above guarantee the existence of semistable states, but do not yet give a full characterization of the orbit type of all SLOCC classes for the considered system sizes. In the following, we will thus employ two observations and a theorem which will allow extending Theorem 3 to a full characterization of the orbit types. Remarkably, we will show that the orbit type does not depend on the eigenvalues of the matrix pencil, but only on their multiplicities. If this had not been the case, it would have seemed unlikely that a selfcontained characterization of orbit types is possible. In the remainder of the section, we will wlog always consider matrix pencils in KCF and states such that the corresponding matrix pencil is in KCF. As mentioned before, this is possible as operators and which transform the matrix pencil to KCF always exist.
Let us start by considering the case . In this case, Theorem 3 states that there do not exist strictly semistable states. Moreover, as discussed in Section 4.2, critical states do only exist if divides [4]. Furthermore, if does divide , there exists a unique stable SLOCC class corresponding to the matrix pencil . Thus, for , there is either exactly one stable SLOCC class or no stable SLOCC class, which implies that there neither exist any strictly polystable, nor strictly semistable classes. Thus, all SLOCC classes except the aforementioned stable class are in the nullcone. Hence, the characterization of orbit types is already complete for the case .
In order to tackle the open cases (within systems such that