Transformations Among Pure Multipartite Entangled States Via Local Operations Are Almost Never Possible

Transformations among pure multipartite entangled states via local operations are almost never possible

David Sauerwein,1, ∗ Nolan R. Wallach,2, † Gilad Gour,3, ‡ and Barbara Kraus1, § 1Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria 2Department of Mathematics, University of California/San Diego, La Jolla, California 92093-0112 3 Department of Mathematics and Statistics, and Institute for Quantum Science and Technology (IQST), University of Calgary, AB, Canada T2N 1N4 Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure states of n > 3 d–level systems with d > 2. Combined with the analogous results for n-qubit states shown in G. Gour, B. Kraus, N. R. Wallach, J. Math. Phys. 58, 092204 (2017) this gives a characterization of almost all local transformations among multipartite pure states. We show that non-trivial LOCC transformations among generic fully entangled pure states are almost never possible. Thus, almost all multipartite states are isolated. They can neither be deterministically obtained from local unitary (LU)-inequivalent states via local operations, nor can they be deterministically transformed to pure fully entangled LU-inequivalent states. In order to derive this result we prove a more general statement, namely that generically a state possesses no non-trivial local symmetry. We show that these results also hold for certain tripartite systems. We discuss further consequences of this result for multipartite entanglement theory.

I. INTRODUCTION glement theory is a resource theory where the free operations are LOCC. In particular, if |ψi can be transformed to |φi via LOCC, then E(|ψi) ≥ E(|φi) for any entangle- Entanglement lies at the heart of quantum theory and ment measure E. Thereof it also follows that studying all is the essential resource for many striking applications possible LOCC transformation among pure states leads of quantum information science [1–6]. The entanglement to a partial order of entanglement. properties of multipartite states are moreover fundamen- In the bipartite case, simple necessary and sufficient tal to important concepts in condensed matter physics conditions for LOCC transformations among pure states [7]. This relevance of entanglement in various fields of sci- were derived [8]. This is one of the main reasons why bi- ence has motivated great research efforts to gain a better partite (pure state) entanglement is so well understood, understanding of these intriguing quantum correlations. as those conditions resulted in an elegant framework that Local operations assisted by classical communication explains how bipartite entanglement can be character- (LOCC) play an essential role in the theoretical and ex- ized, quantified and manipulated [2]. In particular, the perimental investigation of quantum correlations. Spa- optimal resource of entanglement, i.e. the maximally en- tially separated parties who share some entangled state tangled state, could be identified. It is, up to Local Uni- can utilize it to accomplish a certain task, such as tele- tary (LU) operations, which do not alter the entangle- portation. The parties are free to communicate classi- P ment, the state i |iii. This state can be transformed cally with each other and to perform any quantum op- into any other state in the Hilbert space via LOCC. eration on their share of the system. To give an exam- Many application within quantum information theory, ple, party 1 would perform a generalized measurement such as teleportation, entanglement-based cryptography, on his/her system, sends the result to all other parties. or dense coding utilize this state as a resource. Party 2 performs then, depending on the measurement In spite of considerable progress [2, 9], an analogous outcome of party 1, a generalized measurement. The out- characterization of multipartite LOCC transformations come is again sent to all parties, in particular to party remains elusive. The reasons for that are manifold. 3 who applies a quantum operation, which depends on Firstly, the study of multipartite entangled states is both previous outcomes, on his/her share of the system, difficult, and often intractable, due to the exponential etc.. Any protocol which can be realized in such a way growth of the dimension of the Hilbert spaces. Secondly, is a LOCC protocol. This physically motivated scenario multipartite LOCC is notoriously difficult to describe led to the definition of entanglement as a resource that mathematically [10]. Thirdly, there exist multipartite cannot be increased via LOCC. Stated differently, entan- entangled states, belonging to the same Hilbert space, that cannot even be interconverted via stochastic LOCC (SLOCC) [11] and thus there is no universal unit of multipartite entanglement. ∗Electronic address: [email protected] †Electronic address: [email protected] Apart from LOCC transformations, other, more ‡Electronic address: [email protected] tractable local operations were considered. Local unitary §Electronic address: [email protected] (LU) operations, which as mentioned before, do not alter 2 the entanglement, have been investigated [12]. SLOCC now become an experimental reality, or new theoretical transformations, which correspond to a single branch of a tools in condensed matter physics. LOCC protocol have been analyzed [11]. Both relations define an equivalence relation. That is, two states are Instead of investigating particular LOCC transforma- said to be in the same SLOCC class (LU class) if there tions we follow a different approach, which is based on exists a g ∈ Ge (g ∈ Ke) which maps one state to the other, the theory of Lie groups and algebraic geometry (see also respectively. Here, and in the following Ge (Ke) denote the [18]). This new viewpoint allows us to overcome many set of local invertible (unitary) operators, respectively. of the usual obstacles in multipartite entanglement the- Clearly, two fully entangled states, i.e. states whose ory described above. It enables us to characterize, rather single-subsystem reduced states have full rank, have to unexpectedly, all LOCC (and SEP) transformations, i.e. be in the same SLOCC class in case there exists a LOCC all local transformations, among pure states of a full- transformation mapping one into the other. That is, it measure subset of any (n > 3)-d–level (qudit) system must be possible to locally transform one state into the and certain tripartite qudit systems. We show that there other with a non-vanishing probability in case the trans- exists no non-trivial LOCC transformation from or to formation can be done deterministically. Apart form LU, any of the states within this full-measure set. We call a and SLOCC, where a single local operator is considered, local transformation non-trivial if it cannot be achieved transformations involving more operators, such as LOCC by applying LUs, which can of course always be applied. transformations using only finitely many rounds of clas- To be more precise, we show that a generic state |ψi sical communication [13], or separable operations (SEP) can be deterministically transformed to a fully entangled [14] have been investigated. Considering only finitely state |φi via LOCC (and even SEP) if and only if (iff) many rounds of classical communication in a LOCC pro- |φi = u1 ⊗ ... ⊗ un|ψi, where ui is unitary; that is, only tocol is practically motivated and led to a simple char- if |ψi and |φi are LU-equivalent. As LU-transformations acterization of (generic) states to which some other state are trivial LOCC transformations, all pure multi-qudit can be transformed to via such a protocol. However, it states are isolated. That is, they can neither be determin- has been shown that there exist transformations which istically obtained from other states via non-trivial LOCC, can only be accomplished with LOCC if infinitely many nor can they be deterministically transformed via non- rounds of communication are employed [15]. SEP trans- trivial LOCC to other fully entangled pure states. This formations are easier to deal with mathematically than also holds if transformations via the larger class of SEP LOCC. However, they lack a clear physical meaning as are considered. they strictly contain LOCC [10, 16]. Any separable map We derive this result by using the fact that the exis- P † ΛSEP can be written as ΛSEP (·) = k Mk(·)Mk , where tence of local symmetries of a state are essential for it to (1) (n) be transformable via LOCC or SEP (see [18, 19] and Sec. the Kraus operators Mk = M ⊗...⊗M are local and k k II). We prove that for the aforementioned Hilbert spaces fulfill the completeness relation P M †M = 1l. In [19] k k k there exists a full-measure set of states which possess no necessary and sufficient conditions for the existence of a non-trivial symmetry. These results are a generalization separable map transforming one pure state into another of those presented in [18], where (n > 4)-qubit (d = 2) have been presented. Clearly, any LOCC protocol as ex- states are considered. However, a straight forward gen- plained above, corresponds to a separable map. However, eralization beyond qubit states was impossible, as in [18] not any separable map can be realized with local oper- special properties of the qubit case, for instance the exis- ations and classical communication [16] and there exist tence of homogeneous SL–invariant polynomials (SLIPs) even multipartite pure state transformations that can be of low degree, were utilized. Note that, due to these achieved via SEP, but not via LOCC [17]. special properties of qubit-states, it was unclear whether Thus, despite all these efforts and the challenges indeed a similar result holds for arbitrary dimensions. involved in characterizing and studying LOCC, the As the statement is not true for less than five qubits, it fundamental relevance of LOCC within entanglement could furthermore have been, that the number of par- theory makes its investigation inevitable in order to ties for which almost all states have a trivial stabilizer reach a deeper understanding of multipartite entangle- depends on the local dimension, i.e. that n depends on ment. Already the identification of the analog of the d. We show here, however, that this is not the case by maximally entangled bipartite state, the Maximally employing new tools form algebraic geometry. Clearly, Entangled Set (MES) requires the knowledge of possible the investigation of higher local dimensions is central in LOCC transformation. This set of states, which was quantum information processing, where for e.g. in quan- characterized for small system sizes [17, 20, 21], is tum networks the parties have access to more than just the minimal set of states from which any other fully a single qubit. Moreover, in tensor network states, which entangled state (within the same Hilbert space) can be are employed for the investigation of condensed matter obtained via LOCC. The investigation of LOCC trans- systems, the local dimension is often larger than two. formations, in particular for arbitrary local dimensions, A direct consequence of this result is that the maxi- might also lead to new applications in many fields of mally entangled set (MES) [20] is of full measure in sys- science, e.g. new ways to use quantum networks, which tems of n > 3 qudits (and certain tripartite systems). 3

The intersection of states which are in the MES and n > 4 qubits and n > 3 qudits is trivial. This result also are convertible, i.e. which can be transformed into some holds for three qudits with local dimension d = 4, 5, 6. other (LU–inequivalent) state are of measure zero. These In Sec. IV we illustrate and discuss the picture of mul- states are the most relevant ones regarding pure state tipartite pure state transformations that emerges if we transformations. Prominent examples of these states are combine this work with previous findings on bipartite the GHZ-state [24] or more generally stabilizer states [4]. [8], 3-qutrit [17] and qubit systems [18, 20]. In Sec. V Hence, the results presented here do not only identify we present our conclusions. the full–measure set of states which are isolated, but also indicate which states can be transformed. As generic LOCC transformations are impossible, it is II. MAIN RESULTS AND IMPLICATIONS crucial to determine the optimal probabilistic protocol to achieve these transformations. Given the result presented Let us state here the main results of this article and here, the simple expression for the corresponding opti- elaborate one its consequences in the context of entan- mal success probability presented in [18] also holds for a glement theory. generic state with arbitrary local dimensions. Moreover, We consider pure states belonging to the Hilbert space n d we will show that the fact that almost no state possesses a Hn,d ≡ ⊗ C , i.e. the Hilbert space of n qudits. When- non-trivial local symmetry can be used to derive simple ever we do not need to be specific about the local dimen- conditions for two SLOCC-equivalent states to be LU– sions, we simply write Hn instead of Hn,d. As before, Ge equivalent. Furthermore, we also show that our result denotes the set of local invertible operators on Hn. Our leads to some new insight for the situations where more main result concerns the group of local symmetries of a than one copy of a state is considered. In particular, a multipartite state |ψi, also referred to as its stabilizer in lower bound on the rate, with which n copies of a state Ge, which is defined as |ψi can be transformed into m copies of a state |φi can n o be derived. This bound holds for arbitrary states, i.e. Geψ ≡ g ∈ Ge g|ψi = |ψi ⊂ G.e (1) even those which are not generic, and arbitrary numbers of copies, n, m. We prove that for almost all multi-qudit Hilbert spaces, The rest of this paper is organized as follows. In Sec. Hn,d, there exists a full-measured set of states whose sta- II we present the main result of the paper and emphasis bilizer is trivial. Recall that a subset of Hn is said to be the physical consequences thereof. In particular, we will of full measure, if its complement in Hn is of lower di- first state that there exists a full–measure set of states (of mension. Stated differently, almost all states are in the almost all Hilbert spaces with constant local dimension), full-measured set and its complement is a zero measure with the property that the local stabilizer of any state in set. this set is trivial (Theorem 1). We then recap why local The main result presented here is given by the follow- symmetries play such an important role in state trans- ing theorem. formations and that Theorem 1 implies that generically there is no state transformation possible via LOCC. We Theorem 1. For any number of subsystems n > 3 and then derive simple necessary and sufficient conditions for any local dimension d > 2 there exists a set of states LU–equivalence of two states, which belong to the same whose stabilizer in Ge is trivial which is open, dense and SLOCC class of a generic state. Regarding probabilistic of full measure in Hn,d. Such a set of states also exists transformations, the result presented here, allows to ex- for n = 3 and d = 4, 5, 6. tend the simple formula for the optimal success probabil- That is, almost all multi-qudit states |ψi have only ity of converting a generic state into another, to arbitrary the trivial local symmetry, i.e. G = {1l}. This result local dimensions BK: We will discuss the multi-copy eψ has deep implications for entanglement theory. In or- case.. der to explain them, we briefly review the connection In Sec. III A we introduce our notation and briefly between the local symmetries of multipartite states and recap the results presented in [18], where qubit systems their transformation properties under LOCC and SEP. were considered. In Sec. III B we develop these meth- As mentioned in the introduction, we say that |ψi ods further and employ new tools from the theory of Lie can be transformed via SEP into |φi if there exists groups and algebraic geometry to show that whenever † a separable map Λ (·) = P M (·)M such that there exists a critical state whose unitary stabilizer is SEP k k k ΛSEP (|ψihψ|) = |φihφ|, where the Kraus operators Mk = trivial then the stabilizer of a generic multipartite state (1) (n) is trivial (Theorem 12). In Sec. III C we then present ex- Mk ⊗ ... ⊗ Mk are local and fulfill the completeness P † amples of n–qudit systems for all local dimensions (d > 2) relation k Mk Mk = 1l. The transformation is possi- and any number of subsystems (n > 3) of states which ble via LOCC if ΛSEP can be implemented locally. It is have these properties. In particular, we prove there that clear that a fully entangled state |ψi can only be trans- the stabilizer of these states is trivial. Combined with the formed into an other fully entangled state |φi if these results presented in [18] this shows that the stabilizer of states are SLOCC equivalent, i.e. |φi = h|ψi for some a generic state, i.e. of a full measure subset of states, of h ∈ Ge. In [19] it was shown that a fully entangled 4 state |ψi can be transformed via SEP to |φi = h|ψi iff A. Nontrivial deterministic local transformations m there exists a m ∈ N and a set of probabilities {pk}k=1 are almost never possible Pm m ˜ (pk ≥ 0, k=1 pk = 1) and {Sk}k=1 ⊂ Gψ such that In [18] it was shown that states with trivial stabilizer are isolated. That is, a state with trivial stabilizer can X † pkSkHSk = r1l. (2) neither be obtained from LU-inequivalent states via SEP, k nor can it be transformed to LU-inequivalent fully entangled states via SEP. The same holds for transformations † N † N via LOCC. Indeed, for such a state the only solution to Here, H = h h ≡ Hi, and G = g g ≡ Gi are local Eq. (2) is H = 1l, which means that |ψi is LU equiv- |||φi||2 operators and r = |||ψi||2 . This criterion for the exis- alent to |φi. It was then shown in [18] that this holds tence of a SEP transformation can be understood as fol- for almost all states of n > 4 qubits. Theorem 1 ensures lows. Let Mk denote the local operator which maps |ψi to that the same holds true for almost all multi-qudit states, |φi = h|ψi, i.e. Mk|ψi = ckh|ψi for some ck 6= 0. Hence, which is stated in the following theorem. −1 h Mk must be proportional to a local symmetry of |ψi. P † Theorem 2. [18] Let Hn,d be one of the multipartite Using then the completeness relation M Mk = 1l k k qudit Hilbert spaces in Theorem 1 and let |ψi ∈ Hn,d leads to the necessary and sufficient conditions in Eq. be a fully entangled n-partite state with trivial stabilizer, (2) for the existence of a separable map transforming i.e. G = {1l}. Then, |ψi can be deterministically ob- one fully entangled state into the other. eψ tained from or transformed to a fully entangled |φi via As LOCC is contained in SEP, it is evident from this re- LOCC or SEP iff |ψi and |φi are related by local uni- sult that the local symmetries of a state play also a major tary operations; that is, iff there exists a u ∈ K˜ such that role in the study of LOCC transformations. However, in |ψi = u|φi. order to characterize LOCC transformation among fully- entangled states using Eq. (2) one has to determine their Recall that K˜ denotes the group of local unitary op- local symmetries, find all solutions of Eq. (2) and check erators. On the one hand, this result shows that, rather if the corresponding separable measurement can be im- unexpectedly, a characterization of LOCC transforma- plemented locally. Each one of these steps can be highly tions of almost all multi-qudit states is possible. On nontrivial. Our main result (see Theorem 1) allows to ac- the other hand, it proves that these transformations are complish all these steps for almost all multipartite qudit generically extremely restricted and non-trivial transfor- states and thereby provides a characterization of deter- mations are generically impossible. That is, the parties ministic SEP and LOCC transformations for almost all who share a generic state cannot transform it via LOCC qudit states. This is one of the reasons why Theorem 1 deterministically into any other (LU–inequivalent) state. has such deep implications in entanglement theory, as we Moreover, this even holds for transformations via the explain below. more powerful separable operations. This result might also be the reason why there has been so little progress In [18] some of us proved a similar result as stated in on multipartite state (or entanglement) transformations Theorem 1 for qubit states. There, so-called SL-invariant via local operations. polynomials (SLIPS) [32] were used to identify a full- measure subset of all (n > 4)-qubit states that have triv- As the MES is defined as the minimal set of states ial stabilizer. As the special characteristics of the qubit which can be transformed into any other state in the case, for instance the existence of SLIPS of low degree, Hilbert space [20], Theorem 2 implies that the MES cannot be utilized for higher dimensions, this proof does of (n > 3)-qudits is of full measure. Note that this not hold beyond qubit states. However, due to the char- is in strong contrast to the bipartite case, where a acteristics of qubit states, it was unclear whether indeed single state, namely the maximally entangled state a similar result holds for arbitrary dimensions. Moreover, + P |Φ i = i |iii can be transformed into any other as the analog of Theorem 1 is not true for less than five state in the Hilbert space with LOCC. In Sec. IV we qubits, it could have been, that the number of parties for discuss in detail the picture of multipartite pure state which almost all states have a trivial stabilizer depends transformations that emerges if we combine our find- on the local dimension, i.e. that n depends on d. Theo- ings with previous results on the subject (see also Fig. 1). rem 1 shows that this is not the case. In order to tackle the case of arbitrary local dimensions, we employ in this Theorem 2 also has implications for the construction work new tools from the theory of Lie groups and ge- of entanglement measures. Recall that an entanglement ometric invariant theory without explicitly using SLIPs measure for pure states is a function E : H → R≥0 such (see subsequent sections). We also show in Sec.III that that E(ψ) ≥ E(φ) holds whenever the transformation the new results encompass the qubit case. from |ψi to |φi can be performed deterministically Let us now discuss the consequences of Theorem 1 in via LOCC. Since generic multi-qudit states cannot be the context of entanglement theory. reached via non-trival deterministic LOCC, one only has to verify if E is invariant under LU-transformations 5 and nonincreasing under LOCC transformations to and h = h1 ⊗ ... ⊗ hn ∈ Ge, is implemented as follows. The within the zero-measure subset of states with nontrivial first party applies a local generalized measurement that stabilizer, e.g. to states that are not fully entangled. contains an element proportional to h1. Similarly, party 2 applies a local generalized measurement that contains an element proportional to h2 etc. The successful branch is the one where all parties managed to apply the op- B. A characterization of optimal probabilistic local erator hi. Due to the fact that the local measurements transformations for almost all multi-qudit states have to obey the completeness relations one can show that the maximal success probability is given as in The- Given the fact that it is not possible to transform orem 3. Note that this protocol can of course also be generic states via local transformations into any other performed if the corresponding state has nontrivial sym- state, it is crucial to determine the optimal probability metries. That is, the success probability given in Eq. (3) to achieve these conversions. Note that if both, the ini- is always a lower bound on the success probability. tial and final states, are fully entangled, this probability Due to Theorem 2, the optimal success probability can is only nonzero if they are elements of the same stochas- only be one if the states are LU–equivalent. Let us ver- tic LOCC (SLOCC) class [11]. In [18] some of us found ify that this is indeed the case. Given the premises of an explicit formula for this probability for qubit–states. Theorem 3 we make the following observation. Theorem 3 shows that this formula indeed holds for arbitrary local dimensions. Observation 4. The optimal success probability as given in Theorem 3 is equal to one iff H ≡ h†h = 1l. Theorem 3. [18] Let Hn,d be one of the multipartite qu- This can be easily seen as follows. As |ψi and |φi = dit Hilbert spaces in Theorem 1, let |ψi ∈ Hn,d be a normalized fully entangled n-partite state with trivial stabi- h|ψi are both normalized we have that lizer, i.e. Geψ = {1l}, and let |φi = h|ψi be a normalized hχ|H|χi hψ|H|ψi state in the SLOCC class of |ψi. Then the maximum λmax(H) = maxχ ≥ = 1. (4) hχ|χi hψ|ψi probability to convert |ψi to |φi via LOCC or SEP is given by Due to Eq. (3) the success probability is one iff the max- 1 imal eigenvalue of H is one. We hence obtain that |ψi is pmax(|ψi → |φi) = † , (3) an eigenstate of H, i.e. H|ψi = λmax(H)|ψi. However, λmax(h h) as H is in Ge and as |ψi does not have any non-trivial where λmax(X) denotes the maximal eigenvalue of X. local symmetry it must hold that H = 1l. Due to Theorem 1 this theorem gives a simple expression for the optimal probability pmax(|ψi → |φi) to C. A simple method to decide LU-equivalence of locally transform a generic (n > 3)-qudit state |ψi into generic multi-qudit states another fully entangled state |φi. These results also hold for tripartite d-level systems with d = 4, 5, 6. It should Since local unitary transformations are the only triv- be noted here that the optimal success probability was ial LOCC transformations of pure states, i.e. the only only known for very restricted transformations prior to transformations that do not change the entanglement of these results (see e.g. [19, 38] and references therein). a state, it is important to know when two states are LU- Theorem 2 and Theorem 3 now provide a characteriza- equivalent. That is, given two states |ψi, |φi one would tion of all deterministic and optimal probabilisitc local transformations for almost all multi-qudit states. like to know whether there exists a local unitary u ∈ Ke such that |ψi = u|φi. In general this is a highly non- Note that the optimal success probability given in Eq. trivial problem (see e.g. [12]). However, we show now (3) is optimal for transformations via LOCC and via that the results in this article also allow us to solve the SEP. This shows that, despite the fact that there are LU-equivalence problem for generic multi-qudit states, as pure state transformations that can be achieved via SEP stated in the following theorem. but not via LOCC [17], the two classes of operations Theorem 5. Let |ψi, |φi ∈ Hn be both states in the are equally powerful for transformations among generic SLOCC class of a state, |ψsi, with trivial stabilizer, i.e. (n > 3)-qudit states. The reason for this is that the op- Geψs = {1l}. That is, |ψi = g|ψsi and |φi = h|ψsi. Then timal SEP protocol is a so-called one-successful-branch |ψi is LU–equivalent |φi iff g†g = h†h. protocol (OSBP), which can always be implemented via LOCC in one round of classical communication. As sug- Proof. As before we use the notation G = g†g and H = gested by the name, a OSBP is a simple protocol for h†h. First, note that G = H holds iff g = uh for some which only one measurement branch leads to the final local unitary u ∈ K. Hence, |ψi = g|ψsi = uh|ψsi = state, while all other branches lead to states that are u|φi and therefore the states are LU–equivalent. The no longer fully entangled (see [18]). This optimal proto- other direction of the proof can be seen as follows. If −1 col to transform |ψi into |φi = h|ψi via LOCC, where |ψi = g|ψsi = uh|ψsi, then g uh = 1l must hold, as |ψsi 6 does not possess any non-trivial local symmetry. Thus, Proof. If: Let G = u†Hu. Then there exists a unitary we have that G = H. v such that g = vhu. As all operators, u, h, and g are local and invertible, v is a local unitary operator. This strong implication follows only from the fact that 0 Hence, |ψi = g|ψsi = vhu|ψsi = vh|ψsi. Only if: If |ψsi has trivial stabilizer, which implies that the standard there exists a local unitary, v transforming |φi into |ψi form g|ψsi with which a state in the SLOCC class of |ψsi we have |ψi = g|ψ i = v|φi = vh|ψ0 i = vhu|ψ i. The 0 s s s can be represented, is unique. That is, the only g such last equality follows from the uniqueness of the critical 0 0 0 −1 that g|ψsi = g |ψsi is g = g , as otherwise (g ) g would states in a SLOCC orbit. As |ψsi does not possess any be a symmetry of the state |ψsi. Due to the results (see non–trivial local symmetry it must hold that g = vhu. Theorem 1) presented here, Theorem 5 applies to almost Therefore, we have G = u†Hu. all multi-qudit states. Let us now generalize this result to the situation where it is known that the two states are in the same SLOCC class, but the local invertible operator transforming one −1 into the other (for the states above the operator hg ) is D. Multi-copy transformations and asymptotic unknown. To this end, we introduce now the notion of conversion rates critical states. A state is called critical if all of its single subsystem reduced states are proportional to the completely mixed state [19]. Prominent examples of critical Let us briefly discuss which consequences the results states are Bell states, GHZ states [24], cluster states [3], presented here have in the case where transformations of graph states [25], code states [1], and absolutely maxi- many copies of a state are considererd. First of all, note mally entangled states [26]. The set of all critical states that the fact that |ψi has only trivial local symmetries does not imply that the same holds for multiple copies in Hn,d, denoted Crit(Hn), plays an important role in entanglement theory as the union of all SLOCC classes of this state. In fact, any k copies of a state |ψi, i.e. |ψi⊗k, do have local symmetries, namely a local permu- of critical states is of full measure in Hn,d [27]. For more details and properties of critical states we refer the reader tation operator (SWAP) applied to all parties. Hence, to the subsequent section. our proof that almost all multi-qudit states are isolated does not generalize to the multicopy case as it was based Let us note that the standard form, |ψi = g|ψsi, of a generic state, corresponds to the normal form introduced on the fact that these states have trivial local symme- in [28]. The numerical algorithm presented in [28] can be tries. Indeed, it has recently been shown in [29] that used to find the normal form a generic state, i.e. a local there are cases where two copies of a state can be transformed to states which are not reachable in the case of a invertible g ∈ Ge and a critical state |ψi such that |ψi = single copy. Hence, the MES can be made smaller even g|ψsi. Due to the Kempf-Ness theorem (see Appendix B), there exists, up to local unitaries, only one critical if only two copies of the state are considered. state in a SLOCC class. Hence, computing the normal However, since we know the optimal probability to lo- form for two states in the same SLOCC class leads to cally transform a single copy of a generic state |ψi into 0 0 a fully entangled state |φi, it is straighforward to obtain |ψi = g|ψsi and |φi = h|ψsi, where |ψsi = u|ψsi, with u a local unitary. The question we address next is when a lower bound on the optimal probability to transform k are these two states LU–equivalent. The necessary and copies of |ψi into m ≤ k copies of a fully entangled state sufficient condition is given by the following lemma. |φi = g|ψi via LOCC, namely

⊗k ⊗m Lemma 6. Let |ψi, |φi ∈ Hn be both states in the pmax(|ψi → |φi ) ≥ SLOCC class of a critical state, |ψ i with trivial stabi- s k lizer. Let further |ψi = g|ψ i and |φi = h|ψ0 i be the X k s s p (|ψi → |φi)⊗j(1 − p (|ψi → |φi))⊗k−j normal forms of the states derived with the algorithm j max max j=m presented in [28]. Then |ψi is LU–equivalent to |φi iff 0 (5) the local unitary u which transforms |ψsi into |ψsi, i.e. 0 |ψsi = u|ψsi (which must exist and is unique) fulfills G = u†Hu. Although this bound follows trivially from our results on single copy transformations it can provide new insights Due to Theorem 1 this theorem again applies to al- into the many copy case. This is examplified if one con- most all multi-qudit states. It provides an easy way to siders the asymptotic limit of k → ∞, where one is inter- solve the, a priori highly nontrivial, problem of deciding ested in the optimal rate R(|ψi → |φi) at which asymp- LU equivalence of two generic states that are SLOCC totically many copies of a state |ψi can be transformed equivalent. into copies of a state |φi, which is defined as 7

Here, the inﬁmum√ is taken over all LOCC maps and Furthermore, we summarize some of the results which † kXk1 = Tr( X X) denotes the trace norm of X. It was were presented in [18]. In Sec. III B we ﬁrst give a con- recently shown in [30] that for tripartite states |ψi, |φi it cise outline of the proof of Theorem 1. We then continue holds that with a presentation of the detailed proof. In Sec. III C we give examples of states with trivial stabilizer, which ( (1) (2) (3) ) S(ρ ) S(ρ ) S(ρ ) are required to complete the proof. R(|ψi → |φi) ≥ min ψ , ψ , ψ , (2) (3) (2) (3) S(ρψ ) + S(ρψ ) S(ρφ ) S(ρφ ) (7) A. Notations and preliminaries

(i) where ρψ = Trl6=i(|ψihψ|) (and similar for |φi). Little Throughout the remainder of this paper we use the is known on lower bounds on R(|ψi → |φi) for states of following notation. We consider the following 4 different more than three parties. However, we prove the following groups all acting on Hn: lemma. G ≡ SL(d, C) ⊗ · · · ⊗ SL(d, C) ⊂ SL(Hn) (10) Lemma 7. Let |ψi, |φi ∈ Hn be two multipartite entan- K ≡ SU(d) ⊗ · · · ⊗ SU(d) ⊂ SU(Hn) (11) gled states and let pmax(|ψi → |φi) denote the optimal ˜ success probability to transform |ψi into |φi via LOCC. G ≡ GL(d, C) ⊗ · · · ⊗ GL(d, C) ⊂ GL(Hn) (12) Then the asymptotic LOCC conversion rate from |ψi to K˜ ≡ U(d) ⊗ · · · ⊗ U(d) ⊂ U(H ) (13) |φi fulfills, n Note that G = ×G, where × = \{0} and that K = R(|ψi → |φi) ≥ pmax(|ψi → |φi). (8) e C C C e Ge ∩ U(n). That is Ke = {zu|u ∈ K, |z| = 1}. We provide the proof of this lemma in Appendix A. Given a subgroup H ⊂ GL(Hn), the stabilizer sub- For a normalized generic multi-qudit state |ψi (i.e. with group of a state |ψi ∈ Hn with respect to this group is trivial stabilizer) and a normalized state |φi = h|ψi we defined as can insert the expression of Eq. (3) for pmax(|ψi → |φi) n o into Eq. (8) and we obtain the following bound, Hψ ≡ h ∈ H h|ψi = |ψi ⊂ H. (14) 1 R(|ψi → |φi) ≥ . (9) If we refer to the stabilizer of a state |ψi without explic- λ (H) max itly mentioning the corresponding group, we mean Geψ. Moreover, the orbit of a state |ψi under the action of H Note that, even in the tripartite case (e.g. for three is defined as 4-level systems), one can easily construct examples n o where the bound in Eq. (9) is better than the bound in H|ψi ≡ h|ψi h ∈ H . (15) Eq. (7), while there are also tripartite states for which the opposite holds. Note that the orbit contains states that are not necessarily normalized, and any orbit H|ψi is an embedded Let us finally mention that the results presented here submanifold of Hn. Hence, any orbit H|ψi has a dimen- guide us very naturally towards several research direc- sion, which we denote by dim(H|ψi). tions, which are discussed in the conclusion. In Sec. II we briefly mentioned the set of critical states, Crit(Hn), in Hn which contains all states whose single- subsystem reduced states are proportional to the com- III. MATHEMATICAL CONCEPTS AND PROOF OF THE MAIN RESULT pleteley mixed stated. Denoting by Lie(G) the Lie alge- bra of G, this set can also be expressed as

In this section we present the proof of our main result, Crit(Hn) ≡ {|φi ∈ Hn| hφ|X|φi = 0, ∀X ∈ Lie(G)}.(16) Theorem 1. In fact, we prove Theorem 1 by deriving results that are stronger than actually required. How- As mention in Sec. II the union of all orbits (in G) con- ever, we believe that these tools are also useful in other taining a critical state, i.e. G · Crit(Hn), is open, dense, contexts and should therefore be presented in the main and of full measure in Hn [19, 23]. Moreover, the sta- text of this article. We first introduce in Sec. III A our bilizer of any critical state is a symmetric subgroup of notation and the main mathematical tools that we use. GL(Hn), i.e. it is Zariski-closed (Z-closed) (see e.g. [23] 8 for the definition of the Zariski topology) and invariant of lower dimension in Hn, we used for the qubit case ho- under the adjoint [18]. The latter means that, if g ∈ Geψ, mogeneous SL–invariant polynomials (SLIPs) [32]. With † these SLIPs we were able to identify a full measure sub- for |ψi ∈ Crit(Hn), then g ∈ Geψ. 0 Let us now briefly recall how some of us proved in [18] set A ⊂ GC with the desired property that for any state ˜ that there exists an open and full measure set of states |φi ∈ A, Gφ = {1l}. in the Hilbert space corresponding to n-qubit states with As mentioned before, Lemma 9 holds for arbitrary n ≥ 5, which contains only states with trivial stabilizer qudit-systems. Note, however, that in order to obtain the strong implications in entanglement theory (see Sec. in Ge. In order to do so, we define the following subset of critical states, II and IV) it is required to prove that the stabilizer in G˜ (and not only in G ) is trivial. Hence, the last step of n o the proof, as outlined above, is essential here. However, C ≡ |ψi ∈ Crit(Hn) dim(G|ψi) = dim(G) . (17) it is precisely this step, which cannot be easily general- That is, C consists of all critical states whose orbits (un- ized to arbitrary local dimensions. Hence, we will employ der G) have maximal dimension (i.e. the dimension of new proof methods in the subsequent section to prove di- ∼ rectly the existence of a set A, which contains only states G). Due to the identity G|ψi = G/Gψ it follows that whose stabilizer in G is trivial, and which is open and of |ψi ∈ C if and only if (iff) |ψi is critical and Gψ is a finite e full measure in H . group (or equivalently dim(Gψ) = 0). Using algebraic n geometry and the theory of Lie groups we showed in [18] the following important properties of this subset. Lemma 8. [18] The set C defined in Eq.(17) has the B. Genericity of states with trivial stabilizer following properties: Using Lemma 8, we prove now one of the main results (i) Gψ = Kψ for all |ψi ∈ C. of this paper. We have already presented this theorem (ii) The set GC ≡ {g|ψi | g ∈ G ; |ψi ∈ C} is open with and its profound implications in entanglement theory in Sec. II. Let us restate it here to increase readability. complement of lower dimension in Hn.

(iii) C is a connected smooth submanifold of Hn, and K Theorem 1. For any number of subsystems n > 3 and acts differentiably on C. any local dimension d > 2 there exists a set of states whose stabilizer in G is trivial which is open, dense and The principal orbit type theorem [31] (see also Ap- e of full measure in H . Such a set of states also exists pendix B) was then central to the proof that the set of n,d for n = 3 and d = 4, 5, 6. states whose stabilizer in G is trivial is open and of full measure. Defining the set We show in the following that, in order to prove 0 Theorem 1 for given values of n and d, it will eventually C = |ψi ∈ C Gψ = {1l} (18) be enough to find only one critical state |ψi ∈ Hn,d with we proved that if C0 is not empty, then C0 is open, dense, trivial unitary stabilizer. Due to that we believe that the and of full measure in C [18]. Moreover, in this case, techniques used here are also useful in other contexts. 0 0 the set GC = g|ψi |ψi ∈ C , g ∈ G is open, dense, Hence, despite the fact that the proof is technical we and of full measure in GC. Clearly, any state |φi in GC0 present it here. has a trivial stabilizer in G. Using now that GC is open and of full measure in Hn (see property 2 in Lemma 8), Let us first give an outline of the proof of Theorem 1. 0 we also have that GC , which contains only states with First, we consider the set of critical states, Crit(Hn) Gψ = {1l}, is open and of full measure in Hn. As can be (see also Eq. (16)). We show that if a critical state has seen from the proofs in [18] this result holds for arbitrary only finitely many local unitary symmetries then there multipartite quantum systems (as long as it can be shown exists no further local (non-unitary) symmetry of this that C0 is not empty). In particular, we have [46] state (see Lemma 11). We then use this together with the results from [18] and tools from geometric invariant 0 Lemma 9. If there exists a state |ψi ∈ C , then the set theory to prove the following statement (see Theorem 12). If there exists one critical state |ψi ∈ Hn with |φi ∈ Hn Gφ 6= {1l} (19) trivial unitary stabilizer, then there exists a set A ⊂ Hn of states with trivial stabilizer in G˜ that is open and of is of measure zero in Hn. full measure in Hn. Due to this theorem it is sufficient For n ≥ 5 we presented in [18] a n–qubit state |ψi, to find one criticial state with trivial stabilizer in Hn,d to which is contained in C0. Hence, for n ≥ 5 a generic n– proof Theorem 1 for these values of n and d. Finally, we qubit state has only a trivial stabilizer (in G). In order to explicitly construct such states and therefore complete define the set A containing states with trivial stabilizer in the proof of Theorem 1 (see Secs. III C, App. C). G˜ (not only G) which is also open and with complement 9

Let us now present the details of the proof of Theo- With Lemma 11 it is now easy to see that the following rem 1. We first show that the set of critical states with sets all coincide: finite stabilizer in Ke coincides with the set of critical (i) states with finite stabilizer in Ge, as stated in the following lemma. 0 n o C = |ψi ∈ C Keψ = {1l} . (23) Lemma 10. The following subset of critical states, n o ˜ (ii) C = |ψi ∈ Crit(Hn) dim(Ke|ψi) = dim(Ke) (20) n o coincides with the set |ψi ∈ Crit(Hn) Keψ = {1l} . (24) n o

|ψi ∈ Crit(Hn) dim(Ge|ψi) = dim(Ge) . (21) (iii) Proof. This lemma is a direct consequence of a much n o |ψi ∈ Crit(Hn) Geψ = {1l} . (25) stronger theorem (Theorem 2.12) proven in [23]. This theorem states that if H is a symmetric subgroup of GL(Hn) and the so–called maximal compact subgroup We now use these results to prove the following theo- 0 0 0 of H is K = H ∩ U(Hn), then Lie(H) = Lie(K ) + rem, which states that, if C is non-empty then our main 0 k2 0 iLie(K ). That is, H = K1e , where K1 ∈ K and theorem, Theorem 1, is implied. 0 k2 ∈ Lie(K ). In [18] it was shown that Geψ is a sym- Theorem 12. If there exists a state |ψi ∈ C0, i.e. if there metric subgroup of GL(Hn). As shown in [23], K is a e exists a critical state |ψi such that Keψ = {1l}, then there maximal compact subgroup of Ge and so is Keψ of Geψ exists an open and full–measure (in Hn) set of states [47]. Thus, we have that Lie(Geψ) = Lie(Keψ)+iLie(Keψ). whose stabilizer in Ge is trivial. More precisely, if C0 6= ∅, Hence, if Keψ is finite, then also Geψ is finite. Using now then the set of states ∼ ˜ that H|ψi = H/Hψ, for H = G,e Ke, we obtain that C 0 0 coincides with the set of critical states whose stabilizer is A = GC = {g|ψi | |ψi ∈ C , g ∈ G} , (26) finite in Ge, which proves the assertion. which contains only states with trivial stabilizer in Ge, is Using the lemma above we are now in the position to open and of full measure in Hn. prove that if a critical state has a finite stabilizer in Ke (or Proof. First of all, note that K is a compact Lie group, equivalently in G), then all symmetries in G are unitary. e e e which acts differentiably on the connected smooth sub- That is, we prove now the following lemma. manifold C of Hn (see (iii) of Lemma 8). Hence, the Lemma 11. For any state |ψi ∈ C˜, with C˜ given in Eq. principal orbit type theorem (see Appendix B) can be (20), it holds that applied. This theorem implies that the set n o n o 0 Keψ = Geψ. (22) C = |ψi ∈ C Keψ = {1l} = |ψi ∈ C Geψ = {1l} (27) Proof. Due to Lemma 10 we have that C˜ is a subset of C. Hence, Lemma 8 (i) implies that for any state |ψi ∈ is, if it is non-empty, open and of full measure in C. Note ˜ C Gψ = Kψ. To prove now that this equivalence also that, in the last equality in Eq. (27) we used Lemma 11. ˜ holds for Keψ and Geψ we consider |ψi ∈ C and g ∈ Geψ. According to (ii) of Lemma 8 we have that GC is open We show that g must be unitary. The Hilbert-Mumford and of full measure in Hn. Therefore, for any open and theorem (see e.g. [23, 33]) implies that for any critical full measure set of C the union of the orbits of all states state, |ψi, there exists a homogeneous SLIP f of some in this set is also open and full measure in Hn. Using 0 degree m such that f(|ψi) 6= 0. Now, if g ∈ G˜ψ we now that for any |φi ∈ A there exist g ∈ G and |ψi ∈ C 0 0 −1 can write it as g = zg , where 0 6= z ∈ C and g ∈ G. such that Geφ = gGeψg , we have that for any |φi ∈ A it m 0 m Hence, f(|ψi) = f(g|ψi) = z f(g |ψi) = z f(|ψi). As holds that Geφ = {1l}, which completes the proof. f(|ψi) 6= 0 this implies that zm = 1. Using now the polar decomposition of g, i.e. g = upg†g, with u ∈ K, the In the subsequent section we explicitly present states e 0 Kempf–Ness theorem (see Appendix B) implies, as g†g is in C for the Hilbert spaces specified in Theorem 1, which completes the proof of this theorem. In Sec. II the im- positive, that pg†g|ψi = |ψi. Hence, also u has to be a plications of this result in the context of entanglement stabilizer of |ψi. In particular, u ∈ Keψ. Moreover, as z theory are discussed. Let us stress here that our results p † p 0 † 0 is only a phase, we have that g g = (g ) (g ) ∈ Gψ. also encompass the results of [18], where it was shown p † Using now that Gψ = Kψ we have that g = u g g ∈ Keψ, that almost all (n > 4)-qubit states have trivial stabi- which proves the statement. lizer. While no example of a state with trivial stabilizer 10 was given in [18], our results allow us to construct states there is no k with the properties described after Eq. (29). with this property, as we show in the following section. Let us further remark here that Theorem 12 holds for ar- Note that for (n > 4)-qubits the existance of states bitrary multipartite quantum systems. However, in this with trivial stabilizer was shown in [18]. However, no work we only use it for homogeneous systems, i.e. sys- examples of states with this property were given. The tems composed of subsystems with equal dimension. mathematical methods developed in this article allow us to explicitly construct such states, namely the states {|Ψn,2i}n=5,n>6 [48]. This shows that our work also in- C. Critical states with trivial stabilizer in Ge cludes, and in fact extends, the results for qubits obtained in [18]. To explicitly give an example of a qubit state with In this section we present critical states with trivial trivial stabilizer, consider the 5-qubit state, stabilizer. First, we introduce a critical state which is √ defined for n = 5, n > 6, and any d ≥ 2 and give an |Ψ5,2i = 7|00000i + |00111i + |01011i + |01101i outline of the proof that its stabilizer is trivial. The + |01110i + |10011i + |10101i + |10110i proof itself is given in Appendix C. It will become √ + |11001i + |11010i + |11100i + 5|11111i. evident from the construction of this state that the cases n = 3, 4, 6 have to be treated separately. However, also The following lemma shows that |Ψn,di has trivial sta- for these cases we construct states with trivial stabilizer bilizer for n = 5, n > 6 and d ≥ 2. It is a combination of in Appendix C. This completes the proof of Theorem 1 0 Lemma 13 and Lemma 19 in Appendix C 1 a, where we as it shows that the set C is non-empty for the systems prove this statement for n > 6 and n = 5, respectively. mentioned in this theorem. Lemma 13. For n = 5, n > 6 and d ≥ 2 the stabilizer Let us introduce the following notation before we de- of |Ψn,di is trivial, i.e. GeΨn,d = {1l}. fine the state with the desired properties for n = 5, n > 6 and d ≥ 2. Let Sn denote the symmetric group of n ele- In the following we give an outline of the proof of this ments. For a permutation σ ∈ Sn we define the operator lemma, which is divided into four main steps. Pσ via Pσ|i1i⊗...⊗|ini = |iσ−1(1)i⊗...⊗|iσ−1(n)i, for all n (i1, . . . , in) ∈ {0, . . . , d−1} . We call a state |ψi symmet- First, we note that it is sufficient to show that KeΨn,d = ric if P |ψi = |ψi for all σ ∈ S . Furthermore, we define σ n {1l} as Lemma 11 then implies that also GeΨn,d = {1l} for an arbitrary state |φi ∈ Hn,d the set of all distinct holds. In the second step, we show that any v ∈ KeΨn,d permutations of |φi as π(|φi) = {Pσ|φi | σ ∈ Sn} and ⊗n P is of the form v = u for some u ∈ U(d). The proof the symmetrization of |φi as |π(|φi)i = |χi∈π(|ψi) |χi. of this statement is presented in Appendix C 1 a. It thus Using this notation, we define for 0 ≤ k ≤ n and remains to show that the only u ∈ U(d) that fulfills the j ∈ {1, . . . , d − 1} the (unnormalized) state equation ⊗k ⊗n−k |Dk,n(j)i = |π(|ji |j − 1i )i. (28) ⊗n u |Ψn,di = |Ψn,di (30) We are now ready to introduce the critical n-qudit state also fulfills u⊗n = 1l. In the third step, we show that (n = 5, n > 6) for which we show that it has trivial Eq. (30) can only be fulfilled if u is diagonal, i.e. if stabilizer, namely P u = i ui|iihi|. We show this in Appendix C 1 a by con- d−1 d−1 sidering the 2-subsystem reduction of Eq. (30). In the X ⊗n X P |Ψn,di = cj|ji + |Dk,n(j)i, (29) fourth step, we reinsert u = i ui|iihi| into Eq. (30) and j=0 j=1 see that it is equivalent to q n n−1 ui = 1 for i ∈ {0, . . . , d − 1}, (31) where c0 = k−1 + 1, ci = 1 for 0 < i < d − 1 ukun−k = 1 for i ∈ {1, . . . , d − 1}. (32) qn−1 i i−1 and cd−1 = k + 1, with k the smallest natural number such that 3 ≤ k ≤ n − 2, n 6= 2k and Now, recall that gcd(n, k) = 1. This can be used to show m gcd(n, k) = 1. Here, gcd(n, k) denotes the greatest that the only solution of Eqs. (31-32) is u = ωn 1l, where ⊗n common divisor of n and k. The existence of k is ωn = exp(2πi/n) and m ∈ N. Hence, u = 1l holds. obvious for n = 5 and is proved for n > 6 in Appendix This completes the proof of Lemma 13. C. The condition gcd(n, k) = 1 is crucial to ensure that the state |Ψn,di has only trivial symmetries as we shall see later. Recall that |Ψn,di is critical if all of its IV. MULTIPARTITE PURE STATE single-subsystem reduced states are proportional to the TRANSFORMATIONS identity. A straightforward calculation shows that |Ψn,di indeed fulfills this property for n = 5, n > 6. Note Combining our result with previous works, the fol- further that |Ψn,di is not defined for n = 2, 3, 4, 6 since lowing picture of multipartite pure state entanglement 11 transformations emerges (see Fig. 1). For bipartite pure states (region A in Fig. 1) all deterministic and probabilisitc LOCC transformations are characterized [8, 36] and SEP = LOCC holds [37]. Entangled bipartite pure states can always be transformed via non-trivial deterministic LOCC, regardless of their local dimensions. Moreover, they can always be obtained from the maximally entangled state. Hence, this (up to LUs) single state consti- tutes the maximally entangled set of bipartite states. Three qubits (region B in Fig. 1) are the only multipartite system for which all deterministic LOCC transformations between pure states are characterized [38]. They are moreover the only tripartite system for which it is known that all fully entangled pure states can be transformed to other fully entangled states via non-trivial deterministic LOCC. Moreover, SEP = LOCC for deterministic transformations within the GHZ class, i.e. for deterministic transformations between generic states [39]. Fur- thermore, the MES, i.e. the minimal set from which all FIG. 1: Summary of results on the symmetries of n-partite other states can be deterministically obtained via LOCC, systems with local dimension d. The picture is divided into different regions (A to G) that were treated separately in the is of measure zero, albeit uncountably infinite [20]. This literature. The colors give information on the stabilizer of situation changes drastically when the local dimension is states in the corresponding system: blue (all states have non- increased by only one. compact stabilizer; regions A [8], B [20, 38]), green (generic Generic 3-qutrit states (region C in Fig. 1) are iso- states have finite, nontrivial stabilizer; regions C [17], F [20]), lated, despite the fact that their stabilizer is nontrivial red (generic states have trivial stabilizer; D (see Th. 1), G [17]. The MES is of full-measure. Moreover, SEP 6= [18]), grey (generic states have finite stabilizer [19, 23]; un- LOCC for deterministic transformations of 3-qutrit pure known if it is trivial; region E). The implications of these and states [17]. other results in entanglement theory are summarized in the Regarding three-partite states we show that, already main text. for 4−, 5−, or 6-level systems (in region D in Fig. 1), almost all pure states have trivial stabilizer and are therefore isolated (see Theorem 1). We further derive the op- V. CONCLUSION timal probabilistic protocol for transformations between generic states and find that SEP = LOCC for these con- In this work, we used methods from geometric invari- versions. An open question is whether these results ex- ant theory and the theory of Lie groups to prove that tend to tripartite systems of any local dimension d > 3 almost all pure (n > 3)-qudit states and almost all three (region E in Fig. 1) or not. d-level states, for d = 4, 5, 6, have trivial stabilizer. Com- Four-qubit pure states (region F in Fig. 1) generically bined with the characterization of local transformations have finite, non-trivial stabilizer and their MES is of full of states with trivial stabilizer provided in [18], this has measure [20, 21]. Furthermore, SEP = LOCC for trans- profound implications in entanglement theory. It allows formations among generic pure states, which were char- us to characterize all transformations via LOCC and via acterized in [22]. However, almost all states are isolated SEP among almost all (n > 3)-qudit pure states. We [20]. find that these transformations are extremely restricted. Finally, our work shows that almost all qudit states In fact, almost all (n > 3)-qudit pure states are isolated. of n > 3 qudits (region D in Fig. 1) have trivial stabi- Due to the results presented here, the simple expression lizer (see Theorem 1) and are therefore isolated. That for the optimal success probability for probabilistic lo- is, almost all qudit states are in the MES. We further cal transformations presented in [18] is shown to hold determine the optimal protocol for probabilistic transfor- among generic states. The optimal SEP protocol is a so mations among these states and find that SEP = LOCC called one-successful-branch protocol (OSBP), i.e. a sim- holds in these cases. This shows in particular that the ple protocol for which only one branch leads to the final results of [18], which is devoted to (n > 4)-qubit systems state, which can also be implemented via LOCC. Fur- (region G in Fig. 1), can be generalized to arbitrary local thermore, we discussed implications of our result for the dimension. construction of entanglement measures for pure multi- qudit states. All of these results also hold for three d-level systems, where d = 4, 5, 6. This work shows that, in the context of local state transformations, only a zero-measure subset of the exponentially large space of (n > 3)-qudit states is 12 physically significant. That is, the most powerful states Applying this protocol to k copies of |ψi leads to a are very rare. This is consistent with investigations in state |Ψ(x)i ≡ |φ(x1)i ⊗ ... ⊗ |φ(xk)i with probabil- other fields of physics, e.g. condensed matter physics, ity p(x) = p(x1) . . . p(xk), where we used the notation where it has been shown that under certain conditions x = (x1, . . . , xk). For any > 0 and k ∈ N we define the only a zero-measure subset of all quantum states is following set of states, physically relevant [7] . These results therefore suggest n o that the physically relevant zero-measure subset of T (, k) ≡ |Ψ(x)i | x ∈ {0, 1}k, 2−k(H+) ≤ p(x) ≤ 2−k(H−) , states, such as matrix-product states [7], projected- (A2) entangled pair states [42] (with low bond dimension), or stabilizer states [4], should be investigated more deeply. where H = −p(0) log(p(0)) − p(1) log(p(1)) and the loga- As transformations between fully entangled states of rithm is taken to base two. In classical information the- homogeneous systems are almost never possible, it ory the sequences x ∈ {0, 1}k corresponding to states would moreover be interesting to study transformations |Ψ(x)i ∈ T (, k) are referred to as -typical sequences of generic states of heterogeneous systems. The methods (see e.g. [1]). A well-known result is the following (see developed in Sec. III B can be applied for arbitrary e.g. [1], Theorem 12.2, (1)), multipartite systems. However, interestingly, for certain X heterogeneous systems, one can show that generic lim p(x) = 1. (A3) states always have non-trivial local symmetries [43]. k→∞ x:|Ψ(x)i∈T (,k) Our results further suggest that more general local transformations should be considered. This includes the Hence, for any > 0 the probability that a state in T (, k) more-copy case and transformations between states of is created in the LOCC transformation described above different local dimensions or number of subsystems e.g. approaches one for large k. transformations from n-qubit states to (n − k)-qubit Now, let 0 < r < p(0). It is easy to see that there is state, where 1 ≤ k ≤ n − 1. Finally, the fact that an ˜ > 0 (that only depends on r) such that, for suffi- almost all qudit states have trivial stabilizer and the ciently large k, any state in T (, k) contains at least brkc mathematical tools that we developed to prove this copies of |φ(0)i = |φi and can therefore be transformed could also be relevant in other fields of physics, such as to |φi⊗brkc via LOCC. Combining this Eq. (A3) we see condensed matter physics. that, in the limit of k → ∞, the state |ψi⊗k is deterministically transformed to some state in T (˜, k), which can Acknowledgments:— G.G. research is supported by the then be transformed via LOCC to |φi⊗brkc. That is, for Natural Sciences and Engineering Research Council of any 0 < r < p(0) there exists a sequence of LOCC maps, Canada (NSERC). The research of D.S. and B.K. was ∞ {Λr,k}k=1, s.t. funded by the Austrian Science Fund (FWF) through ⊗k ⊗br0kc Grants No. Y535-N16 and No. DK-ALM:W1259-N27. lim kΛr,k(|ψihψ| ) − |φihφ| k = 0. (A4) k→∞ Since this holds for any 0 < r < p(0) = p (|ψi → |φi) Appendix A: Proof of Lemma 7 max it follows from the definition of the asymptotic conversion rate (see Eq. (6)) that R(|ψi → |φi) ≥ pmax(|ψi → In this appendix we proof Lemma 7, which we restate |φi). here to improve readability.

Lemma 7. Let |ψi, |φi ∈ Hn be two multipartite entan- Appendix B: Kempf-Ness theorem and Principal gled states and let pmax(|ψi → |φi) denote the optimal orbit type theorem success probability to transform |ψi into |φi via LOCC. Then the asymptotic LOCC conversion rate from |ψi to In this Appendix we review two theorems that are |φi fulfills, central to our work, the Kempf-Ness theorem and the principal orbit type theorem, and discuss where they R(|ψi → |φi) ≥ p (|ψi → |φi). (A1) max are used in the main text. We also briefly review some implications of the Kempf-Ness theorem in the context of entanglement theory. For further details the interested reader is referred to [18] and [23]. Proof. We show that the single copy LOCC protocol with success probability pmax(|ψi → |φi) applied to many Let us first review the Kempf-Ness theorem. In the copies individually achieves the rate given as lower bound main text we introduced the set of critical states in Eq. in Eq. (A1). This LOCC protocol transforms one copy (16) as of |ψi into the desired final state |φ(0)i ≡ |φi with Crit(H ) ≡ {φ ∈ H | hφ|X|φi = 0, ∀X ∈ Lie(G)}. probability p(0) = pmax(|ψi → |φi) and into a sep- n n arable state |φ(1)i with probability p(1) = 1 − p(0). (B1) 13

Note that a state |ψi ∈ Hn is critical iff all of its lo- We further say that H/Hψ has a lower type than cal density matrices are proportional to the identity [19]. H/Hφ, denoted as That is, a state is critical if every subsystem is maximally entangled with the remaining subsystems. As men- H/Hψ ≺type H/Hφ, (B3) tioned in the main text, many well-known quantum states if Hφ is conjugate in H to a subgroup of Hψ. It is easy are critical, and the union of the G-orbits of all critical to see that ≺type induces a preorder on the set of all states is dense and of full-measure in Hn. Critical states H-stabilizers. have many other interesting properties mentioned below. That this preorder also induces a preorder on set of Some of these can be derived from the Kempf-Ness theo- all H-orbits can be seen as follows. Note that H|ψi is rem, which provides a characterization of critical states. isomorphic to the left coset of Hψ in H for all |ψi, namely Theorem 14. [27] The Kempf-Ness theorem ∼ H|ψi = H/Hψ. (B4) 1. |φi ∈ Crit(H ) iff kg|φik ≥ k|φik for all g ∈ G. n We can therefore say that H|ψi is of lower type than H|φi, denoted as 2. If |φi ∈ Crit(Hn) and g ∈ G then kg|φik ≥ k|φik with equality iff g|φi ∈ K|φi. Moreover, if g is H|ψi ≺ H|φi, (B5) positive definite then the equality condition holds type iff g|φi = |φi. if H/Hψ ≺type H/Hφ holds. The following theorem, called principal orbit type the- 3. If |φi ∈ Hn then G|φi is closed in Hn iff Gφ ∩ orem (POT theorem), shows that under certain very gen- Crit(Hn) 6= ∅. eral conditions this preorder possesses a maximal ele- The second part of the theorem implies that each ment. This key theorem can be found in [31], as a com- SLOCC orbit contains (up to local unitaries) at most bination of Theorem 3.1 and Theorem 3.8. one critical state. Thus, critical states are natural Theorem 15. [31] The principal orbit type theo- representatives of SLOCC orbits. They are the unique rem Let C be a compact Lie group acting differentiably states in their SLOCC orbits for which each qubit is on a connected smooth manifold M (in this paper we maximally entangled to the other qubits [19]. The assume M ⊂ Hn). Then, there exists a principal or- Kempf-Ness theorem was also important in the proof bit type; that is, there exists a state |φi ∈ M such that of [18] to show that g ∈ G˜ iff g† ∈ G˜ for a critical ψ ψ C/Cψ ≺type C/Cφ for all |ψi ∈ M. Furthermore, the ˜ state |ψi. Together with the fact that Gψ is Z-closed set of |ψi ∈ M such that Cψ is conjugate to Cφ is open (which follows from the definition) this shows that and dense in M with complement of lower dimension and n G˜ψ is a symmetric subgroup of GL(d ) (see e.g. hence of measure 0. [23] for the definition of the Zariski topology). This The following example illustrates how powerful the property is central to the proof of Lemma 11 in this work. POT theorem is. Suppose |ψi ∈ Hn is a (not necessarily In order to state the principal orbit type theorem we critical) state with trivial unitary stabilizer, Keψ = {1l}. ˜ first introduce some definitions and notation. We further Then the POT theorem applied to C = K and M = Hn discuss how a subgroup H ⊂ GL(Hn) induces a preorder directly implies that the set of states with trivial unitary on the the set of all H-orbits of states in Hn. The prin- stabilizer is of full measure in Hn. cipal orbit type theorem then provides conditions under However, in this work we show that the stabilizer which this preorder gives rise to a maximal element. in Ge is generically trivial. As Ge is a noncompact Lie Let |ψi, |φi ∈ Hn be two states. Then Hψ and Hφ group, the POT theorem cannot be applied directly. It are said to have the same type if there exists a h ∈ H is nevertheless central to the proof of Theorem 12, where −1 ˜ such that Hφ = hHψh , i.e. if they are conjugate in H. we applied it to the compact Lie group C = K that acts Clearly, the stabilizer of |ψi and h|ψi are conjugate for differentiably on the connected smooth manifold M = C any h ∈ H, namely (see Lemma 8, (iii)).

−1 hHψh = Hhψ. (B2)

Hence, Hψ and Hφ are of the same type iﬀ there exists Appendix C: Criticial states with trivial stabilizer h ∈ H such that Hφ = Hhψ. However, the fact that Hψ and Hφ have the same type does not imply that there In this appendix we provide examples of critical states is a h ∈ H such that |ψi = h|φi, i.e. it does not imply with trivial stabilizer for n = 3 and d = 4, 5, 6, for n = 4 that they are in the same H-orbit. For example, the Ge- and d > 2 and for n ≥ 5 and d ≥ 2. That is, we give stabilizer of generic 4-qubit states has the same type as examples of criticial states with trivial stabilizer for all ⊗4 3 {σi }i=0 [20], despite the fact that two 4-qubit states Hilbert spaces described in Theorem 1, and for (n > 4)- are generically SLOCC inequivalent and thus not in the qubit systems. Combined with Theorem 12 this com- same Ge-orbit [40]. pletes the proof of Theorem 1 that almost all pure states 14 in these Hilbert spaces have trivial stabilizer. For n > 3 a. A critical n-qudit state, n = 5, n > 6, with local we present these states in Sec. C 1. The states with n = 3 dimension d ≥ 2 and trivial stabilizer have to be constructed diﬀerently and are presented in

Sec. C 2. In this section we show that the critical state |Ψn,di introduced Eq. (29) of Sec. III C is well-defined and has trivial stabilizer for n = 5, n > 6 and d ≥ 2. That is, we 1. Critical (n > 3)-qudit states with trivial prove Lemma 13 of the main text. stabilizer Let us first recall the following definitions made in In this section we present critical states with trivial Sec. III C of the main text. For |φi ∈ Hn,d we define stabilizer for n = 4, d > 2 and n > 4, d ≥ 2. As we will the set of all distinct permutations of |φi as π(|φi) = see, it is easy to show that these states are indeed crit- {Pσ|φi | σ ∈ Sn} and the symmetrization of |φi as ical, i.e. that their single-subsystem reduced states are P |π(|φi)i = |χi∈π(|φi) |χi. Using this notation, we de- proportional to the completely mixed state. In contrast fine the (unnormalized) state to that, the proof that their stabilizer is trivial is more involved and the details of the proof depend on n and ⊗k ⊗n−k |Dk,n(j)i = |π(|ji |j − 1i )i, (C2) on d. However, since we consider only permutationally symmetric states the main steps of this proof are the for 0 ≤ k ≤ n and j ∈ {1, . . . , d − 1}. These states fulfill same for all n > 3. For the sake of readability we outline these four main steps before we present the details in 0 n hD (j)|D 0 (j )i = δ 0 δ 0 . (C3) the subsequent subsections. The main ingredients to k,n k ,n k k,k j,j show that a permutationally symmetric, critical state considered here, say |ψn,di ∈ Hn,d, has trivial stabilizer, For l ∈ {1, . . . , n − 1} we can express |Dk,n(j)i in the are the following. bipartite splitting of any l subsystems and the remaining n − l subsystems as

(1) Since |ψn,di is critical it is suﬃcient to show that min{l,k} K = {1l} holds, i.e. that |ψ i has a trivial X eψn,d n,d |Dk,n(j)i = |Dq,l(j)i|Dk−q,n−l(j)i. (C4) unitary stabilizer, as Lemma 11 states that then q=0

also Geψn,d = {1l} holds. In Sec. III C we then defined for n = 5, n > 6 and (2) We show that a unitary B fulfills B ⊗ B−1 ⊗ ⊗n−2 d ≥ 2 the state 1l |ψn,di = |ψn,di iff B = c1l for some phase d−1 d−1 c. It was shown in [34] that then any v ∈ Kψ e n,d X ⊗n X can be express as v = u⊗n for some u ∈ U(d) (see |Ψn,di = cj|ji + |Dk,n(j)i, (C5) also Lemma 17 below for details). j=0 j=1 (3) It remains to show that for any unitary u ∈ U(d) q where c = n−1 + 1, c = 1 for 0 < i < d − 1 and the equation 0 k−1 i qn−1 ⊗n cd−1 = + 1, with k the smallest natural number u |ψn,di = |ψn,di (C1) k such that 3 ≤ k ≤ n − 2, n 6= 2k and gcd(n, k) = 1. implies that u⊗n = 1l. The corresponding equation for the reduced state of the first two subsystems, Let us first show that k as described above always ex- (1,2) ρn,d = Tr3,...,n(|Ψn,dihψn,d|) reads, ists for n = 5, n > 6 and that |Ψn,di is therefore well- defined. For n ∈ N the Euler totient function φ(n) is (1,2) † † (1,2) (u ⊗ u)ρn,d (u ⊗ u ) = ρn,d . defined as the number of all natural numbers j that are smaller than n and fulfill gcd(n, j) = 1, i.e. This equation can be used to show that u has to be diagonal. However, the details of this proof depend φ(n) = |{j ∈ N| j < n, gcd(n, j) = 1}|. (C6) on n and d. (4) In the last step we show that the only diagonal It is straightforward to see that k as defined below Eq. unitary u that fulfills Eq. (C1) also fulfills u⊗n = 1l. (C5) always exists if φ(n) ≥ 5. We now prove that k exists if n = 4 and n > 6. If n is not divisible by 3 This shows that K = {1l} and completes the eψn,d then k = 3. If n is divisible by 3 and 5 then Euler’s proof. formula for φ(n) (cf. [35]) implies that φ(n) ≥ 5. Fi- The remainder of this section is devoted to the details nally, if n is divisible by 3 but not 5 and n ≥ 9 then k = 5. of this proof. In Sec. C 1 a we consider the case n = 5 and n > 6 and d ≥ 2. In Sec. C 1 b we consider the case Let us now show some properties of |Ψn,di that will n = 4, d > 2 and in Sec. C 1 c the case n = 6 and d ≥ 2. be useful in the proof that it has trivial stabilizer. Note 15

ﬁrst that, due to Eq. (C4), we can express |Ψn,di in the In the case of k = n − 2 we get bipartition of any l subsystems with the rest as (B ⊗ B−1)|0i⊗2 = |0i⊗2, d−1 −1 ⊗2 ⊗2 X ⊗l ⊗n−l (B ⊗ B )(ci|ii + |i − 1i ) |Ψn,di = cj|ji |ji ⊗2 ⊗2 j=0 = ci|ii + |i − 1i for i ∈ {1, . . . , d − 1}. d−1 min{l,k} X X + |Dq,l(j)i|Dk−q,n−l(j)i. (C7) It is then straightforward to see that these equations can Pd−1 j=1 q=0 only be fulﬁlled if B is diagonal, i.e. B = i=0 bi|iihi|.

⊗2 Note further that the following useful lemma on sym- Analogously we can apply 1l ⊗ hDk−1,n−2(i)| to both metric states has been shown in [34]. sides of Eq. (C10) for i ∈ {1, . . . , d − 1} and see that

Lemma 16. [34] Let |ψi be symmetric. Suppose |ψi has −1 the property that (B ⊗ B )|D1,2(i)i = |D1,2(i)i for i ∈ {1, . . . , d − 1}.

B ⊗ B−1 ⊗ 1l⊗n−2|ψi = |ψi iff B = b1l, (C8) Using that |D1,2(i)i = |ii|i − 1i + |i − 1i|ii it is easy to see that these equations imply bi = bj = b for all ˜ ⊗n i, j ∈ {0, . . . , d − 1} and hence B = b1l. for some b ∈ C\{0}. If g ∈ Gψ then g = h for some h ∈ GL(d, C). With these results we are in the position to proof This result can be easily understood as follows. Let Lemma 13 in the main text, namely that |Ψn,di has |ψi be symmetric and let P(1,2) denote the operator that trivial stabilizer for n > 4, n 6= 6, d > 2. We first prove permutes subsystems 1 and 2. Note that if g = g1 ⊗...⊗ this result for n > 6 (see Lemma 18). After that, we −1 gn ∈ Geψ then g |ψi = |ψi and P(1,2)gP(1,2)(P(1,2)|ψi) = provide the proof for n = 5 (see Lemma 19)) which is P(1,2)|ψi hold. Using that P(1,2)|ψi = |ψi this implies slightly different. −1 that g P(1,2)gP(1,2)|ψi = |ψi, i.e. We first consider the case of n > 6 qubits and prove −1 −1 ⊗n−2 the following lemma. g1 g2 ⊗ g2 g1 ⊗ 1l |ψi = |ψi. (C9)

Now, if Eq. (C8) holds, this implies that there is a c1,2 ∈ Lemma 18. For n > 6 and d ≥ 2 the stabilizer of |Ψn,di \{0} such that g1 = c1,2g2. As |ψi is symmetric the C is trivial, i.e. GeΨ = {1l}. same argument can also be used to show that there is a n,d ci,j ∈ C\{0} such that gi = ci,jgj for all i 6= j and thus ⊗n Proof. Due to Lemma 11 and Lemma 17 it remains to g = h for some h ∈ GL(d, C). Note that if g ∈ Ke −1 show that for any unitary u ∈ U(d) the equation then g1 g2 in Eq. (C9) is unitary. Hence, in order to show that any unitary symmetry v ∈ Keψ is of the form ⊗n v = u⊗n for some u ∈ U(d) it is thus sufficient to show u |Ψn,di = |Ψn,di (C11) that Eq. (C8) holds for any unitary B. We use this to ⊗n prove the following lemma. implies that u = 1l. We first show that Eq. (C11) implies that u is diagonal and then that u⊗n = 1l. Lemma 17. If v ∈ Ke fulfills v|Ψn,di = |Ψn,di then v = Considering the reduced state of the first two subsys- u⊗n for some u ∈ U(d). (1,2) tems, ρn,d = Tr3,...,n(|Ψn,dihΨn,d|), Eq. (C19) implies that Proof. |Ψn,di is permutationally symmetric. Due to Lemma 16 it is thus sufficient to show that the only so- (u ⊗ u)ρ(1,2)(u† ⊗ u†) = ρ(1,2). (C12) lution to n,d n,d Note that for n > 6 we have that k < n − 2, where B ⊗ B−1 ⊗ 1l⊗n−2|Ψ i = |Ψ i, (C10) n,d n,d k is defined below Eq. (C5) [49]. This fact simplifies (1,2) where B ∈ U(d), is B = b1l for some complex number the computation of ρn,d (in contrast to the case n = 5, b 6= 0. where k = 3; see Lemma 19). It is then easy to see that

d−2 In order to show this, we first apply for i ∈ {0, . . . , d − (1,2) ⊗2 ⊗n−2 ⊗2 ⊗2 X ⊗2 1} the operator 1l ⊗hi| to both sides of Eq. (C10). ρn,d = α(|0ih0| + |d − 1ihd − 1| ) + β |jihj| Decomposition (C7) for l = 2 is very useful in calculating j=1 the resulting equation. For k < n − 2 we obtain d−1 X + γ |D1,2(j)ihD1,2(j)| (C13) −1 ⊗2 ⊗2 (B ⊗ B )|ii = |ii for i ∈ {0, . . . , d − 1}. j=1 16 with implies that, there exist a, b ∈ Z such that an + bk = 1. Hence, we have n − 1 n − 2 α = + + 1, an+bk k − 1 k ui = ui = 1. (C18) n − 2 n − 2 β = + + 1, We hence proved that ui = 1 for all i, which implies k k − 2 u⊗n = 1l. This proves the assertion. n − 2 γ = . k − 1 In the proof for n > 6 (where k < n − 2) we used Eq. (C13). Let us now consider the case n = 5 (where Using Eq. (C12) we now prove that u is diagonal. In k = 3 = n − 2) for which we need to prove the statement order to do so, we first show that u|0i = u0|0i for some differently. phase u . As Eq. (C13) is the spectral decomposition of 0 Lemma 19. The stabilizer of |Ψ i is trivial for d ≥ 2, ρ(1,2) and since α > β > γ (in fact, α = β + γ, as the 5,d n,d i.e. G = {1l}. state is critical), the following equation must hold, eΨn,d Proof. Due to Lemma 11 and Lemma 20 it is again suf- u ⊗ u|00i = φ0|00i + φd−1|d − 1d − 1i, (C14) ficient to show that any unitary u that fulfills

⊗5 for some coefficients φ0, φd−1. Note that the local oper- u |Ψ5,di = |Ψ5,di (C19) ator u ⊗ u cannot change the Schmidt rank of a state. Hence, the state on the right-hand side of Eq. (C14) fulfills u⊗5 = 1l. must have Schmidt rank 1. That is, either u|0i = u0|0i or u|0i = u0|d − 1i holds for some phase u0. It is easy We first consider the necessary condition to see that only the former can fulfill Eq. (C11). Hence, (1,2) † † (1,2) u|0i = u0|0i holds. (u ⊗ u)ρ5,d (u ⊗ u ) = ρ5,d , (C20)

(1,2) Next, we show that if u|ki = uk|ki for k < i then also where ρ5,d = Tr3,4,5(|Ψ5,dihΨ5,d|). Let Kd denote the u|ii = u |ii holds, for i ∈ {1, . . . , d − 1}. We consider i kernel of ρ(1,2) and let K⊥ denote the orthogonal com- the eigenspace to eigenvalue γ, which is spanned by the 5,d d ⊥ d−1 plement of Kd. Clearly, Kd and K have to be invariant states {|D1,2(i)i}i=1 . Using that u|i − 1i = ui−1|i − 1i d we obtain, under u ⊗ u. In fact, |ψi ∈ Kd iﬀ (u ⊗ u)|ψi ∈ Kd (and ⊥ similarly for Kd ). In the following we use this to prove u ⊗ u|D1,2(i)i = ui−1(u|ii|i − 1i + |i − 1iu|ii), that u is diagonal. Before that we have to characterize ⊥ Kd and Kd . It is straightforward to see that the follow- for i ∈ {1, . . . , d − 1}. This state has to be an element ing holds. of the eigenspace to eigenvalue γ. It is easy to see that this is only possible if u|ii = ui|ii. Combined with Kd = Q ⊕ S−, (C21) u|0i = u0|0i we have that u is diagonal. ⊥ Kd = P ⊕ S+, (C22) We show next that for any diagonal u fulﬁlling Eq. where (C11) it must hold that u⊗n = 1l. Using the notation Pd−1 Q = span{|ii|ji|0 ≤ i, j ≤ d − 1, |i − j| > 1}, (C23) u = i=0 ui|iihi|, it is straightforward to show that Eq. (C11) is equivalent to d−1 S− = span{|ii|i − 1i − |i − 1i|ii}i=1 . (C24) n P = span{|ii|ii}d−1, (C25) ui = 1 for i ∈ {0, . . . , d − 1}, (C15) i=0 d−1 k n−k S+ = span{|D1,2(i)i} . (C26) ui ui−1 = 1 for i ∈ {1, . . . , d − 1}. (C16) i=1

Let us ﬁrst prove that u|0i = u0|ui for some phase u0. Note that u is only uniquely determined up to a global ⊥ ⊥ m Let π denote the projector onto Kd , i.e. phase ωn , where ωn = exp(2πi/n) and m ∈ N. As n u0 = 1 we can choose u0 = 1 without loss of general- d−1 d−1 X 1 X ity. Equations (C15-C16) then reduce to π⊥ = |iihi|⊗2 + |D (i)ihD (i)|. (C27) 1,2 2 1,2 1,2 n k n−k i=0 i=1 ui = 1 and ui ui−1 = 1 for i ∈ {1, . . . , d − 1}. (C17) As K⊥ is invariant under u ⊗ u it holds that We now prove inductively that these equations together d with u = 1 imply that u = 1 for all i. That is, we ⊥ † † ⊥ 0 i (u ⊗ u)π1,2(u ⊗ u ) = π1,2. (C28) show that ui−1 = 1 implies that ui = 1. Due to Eq. k (C17) we have that ui−1 = 1 implies that ui = 1. As For d = 2 it is easy to see that this equation and Eq. gcd(n, k) = 1, a well known result form number theory (C19) can only be fulﬁlled if u|0i = u0|ui for some phase 17

⊥ ⊥ ˜ u0. For d > 2 let π1 = Tr2(π1,2) denote the reduced Lemma 20. If v ∈ K fulfills v|Φ4,di = |Φ4,di for d > 2 ⊥ then v = u⊗4 for some u ∈ U(d). state of π1,2, i.e. The proof is similar to the proof of Lemma 17 and will d−2 3 X be omitted. Using this lemma, we prove the following π⊥ = (|0ih0| + |d − 1ihd − 1|) + 2 |iihi|. (C29) 1 2 lemma. i=1 Lemma 21. For d > 2 the stabilizer of |Φ4,di is trivial, Then Eq. (C28) implies that i.e. GeΦ4,d = {1l}. ⊥ † ⊥ uπ1 u = π1 . (C30) Proof. Due to Lemma 11 and Lemma 17 it remains to show that for any unitary u ∈ U(d) the equation It is easy to see that Eq. (C30) can only be satisfied ⊗4 if u|0i ∈ span{|0i, |d − 1i}. Combining this with the u |Φ4,di = |Φ4,di (C33) ⊥ fact that (u ⊗ u)|00i ∈ Kd , we see that (u ⊗ u)|00i ∈ span{|00i, |d−1d−1i}. However, as u⊗u cannot change implies that u⊗4 = 1l. We first show that Eq. (C19) ⊗4 the Schmidt rank of a state either u|0i = u0|0i or u|0i = implies that u is diagonal and then that u = 1l. u0|d − 1i for some phase u0. It is easy to see that the Considering the reduced state of the first two subsys- (1,2) former cannot fulfill Eq. (C19) and thus u|0i = u0|0i tems, ρ4,d = Tr3,4(|Φ4,dihΦ4,d|), Eq. (C33) implies that holds. Next, we show that if u|ki = uk|ki holds for k < i, (1,2) † † (1,2) (u ⊗ u)ρ4,d (u ⊗ u ) = ρ4,d . (C34) then also u|ii = ui|ii holds, where i ∈ {1, . . . , d − 1}. Using that u|i − 1i = ui−1|i − 1i we get that (1,2) ⊥ Let us denote by Kd the kernel of ρ4,d and by Kd (u ⊗ u)|D (i)i = u (u|ii|i − 1i + |i − 1iu|ii) ∈ K⊥. the orthogonal complement of Kd. In Observation 22 1,2 i−1 d ⊥ we show that Kd = Q ⊕ S− and Kd = P ⊕ S+, where The state on the right-hand side can only be an element Q, S−,P,S+ are given in Eqs. (C23-C26). The proof ⊥ that u is diagonal can then be completed as in the proof of Kd if u|ii = ui|ii. Combined with the fact that u|0i = u0|0i we see that u is diagonal. of Lemma 19 and will therefore be omitted.

d−1 ⊗5 P That u = 1l holds for diagonal u can then be proven Using that u = i=0 ui|iihi|, Eq. (C33) is equivalent in the same way as it was done in the proof of Lemma to 18 for n > 6. 4 ui = 1 for i ∈ {0, . . . , d − 1}, (C35) 3 1 Note that this proof method could also be used to show ui ui−1 = 1 for i ∈ {1, . . . , d − 1}. (C36) that |Ψn,di has trivial stabilizer for n > 6. However, we u2u2 = 1 for i ∈ {1, . . . , d − 1}. (C37) think that the proof for n > 6 presented after Lemma 18 i i−1 is more concise. Similar to the proof of Lemma 13 we set, without loss of generality, u0 = 1. It is then straightforward to see that Eqs. (C35-C37) only have the solution ui = 1 for b. A critical (n = 4)-qudit state with local dimension d > 2 i ∈ {0, . . . , d − 1}. Hence, we proved that u⊗4 = 1l and trivial stabilizer holds.

In this section we consider 4-qudit systems with local Here, we prove the following observation used in the dimension d > 2. We define the (unnormalized) state proof of Lemma 21. √ |Φ i = 15c |0i⊗4 Observation 22. Let Q, S−,P and S+ be as defined in 4,d 0 (1,2) d−1 Eqs. (C23 - C26). Then the kernel of ρ4,d is Kd = X ⊗4 Q ⊕ S− and the orthogonal complement of the Kernel is + ci |D3,4(i)i + |D2,4(i)i − 3|ii , (C31) ⊥ K = P ⊕ S+. i=1 d where Proof. Note that |ψi ∈ Kd iff

s 1,2hψ|Φ4,di1,2,3,4 = 0, (C38) 1 4 d−i c = 1 − − , for i ∈ {0, . . . , d − 1}. (C32) i 3 15 where we explicitly labeled on which subsystems these states are deﬁne. Using this, it is easy to see that d d It is easy to show that this state is critical. In what Q⊕S− ⊂ Kd. As C ⊗C = Q⊕S−⊕P ⊕S+ it is therefore follows we prove that |Φ4,di has only trivial symmetries. suﬃcient to show that P ⊕ S+ does not contain any non- trivial element of Kd, i.e. (P ⊕ S+) ∩ Kd = {0}, in order Note that the following lemma holds. to prove the observation. That is, we have to show that 18

Pd−1 ∗ Pd−1 ∗ an element |ψi = k=0 αk|ki|ki + i=1 βk|D1,2(k)i ∈ linear equations P ⊕ S+ fulﬁlls Eq. (C38) iﬀ αk, βk = 0 for all k. First, it is easy to see that Eq. (C38) implies that M ~α = 0, (C39) βk = −αk/2 for k ∈ {1, . . . , d − 1}. Using this and T the notation ~α = (α0, . . . , αd−1) , it is straighforward to show that Eq. (C38) is equivalent to the system of where

√  15c0 c1 0 0 ... 0 0 0 0  c1 −4c1 c2 0 0 ... 0 0 0     0 c2 −4c2 c3 0 ... 0 0 0    M =  . . . . .  .  ......     0 0 0 0 0 . . . cd−2 −4cd−2 cd−1  0 0 0 0 0 ... 0 cd−1 −4cd−1

P As M is a tridiagonal matrix its determinant det(M) can where |π(|ψi)i = |φi∈π(|ψi) |φi and π(|ψi) = be computed via the following reccurence relation (see {Pσ|ψi | σ ∈ Sn} as in the main text. For d > 3 we e.g. [44]). then introduce the (unnormalized) critical state

2 f(k) = −4ck−1f(k − 1) − ck−1f(k − 2), (C40) r r d−3 for k ∈ {2, . . . , d}, 194 11 X |Φ i = |0i⊗6 + |D (1)i + |ji⊗6 √ 6,d 5 5 5,6 f(1) = 15c0, f(0) = 1, (C41) j=2 √ √ d−1 d−1 ⊗6 ⊗6 X where det(M) = f(d). Using the definition of {ck}k=0 + 21|d − 2i + 51|d − 1i + |φ6(j)i. d one can show that {|f(k)|}k=1 is monotonically in- j=2 creasing and hence det(M) = f(d) 6= 0 holds. That (C43) is, M is invertible, (P ⊕ S+) ∩ Kd = {0} and therefore Kd = Q ⊕ S−, which proves the assertion. Note that this state is not defined for d = 2 and not critical for d = 3. However, for these case we can define the critical states, c. A critical (n = 6)-qudit state with trivial stabilizer for d ≥ 2 ⊗6 |Φ6,2i = 2|0i + |D1,6(1)i + |D3,6(1)i, √ d ⊗6 ⊗6 1 ⊗6 In this section we present a state in (C ) , d ≥ 2, |Φ6,3i = 3|0i + |D5,6(1)i + √ |φ6(2)i + 15|2i . with trivial stabilizer. Before that, we introduce for 2 ≤ 2 j ≤ d − 1 the (unnormalized) state

⊗3 ⊗2 Note that, for d > 3, the state |Φ6,di can also be ex- |φ6(j)i = |π(|ji|j − 1i |j − 2i )i, (C42) pressed as

d−3 d−1 ⊗2 ⊗4 ⊗2 ⊗4 X ⊗2 ⊗4 X ⊗2 |Φ6,di = c0|0i |0i + c1|1i |D3,4(1)i + c1|D1,2(1)i|1i + |ji |ji + {|j − 2i |D1,4(j)i j=2 j=2 ⊗2 + |j − 1i |αji + |D1,2(j)i|D2,4(j − 1)i + (|ji|j − 2i + |j − 2i|ji)|D3,4(j − 1)i + |D1,2(j − 1)i|βji} ⊗2 ⊗4 ⊗2 ⊗4 + cd−2|d − 2i |d − 2i + cd−1|d − 1i |d − 1i . (C44)

⊗2 q q Here, we defined |αji = |π(|ji|j − 1i|j − 2i )i, |βji = ⊗2 194 11 |π(|ji|j −1i |j −2i)i and c0 = 5 , c1 = 5 , cd−2 = 19 √ √ √ 2 21 and cd−1 = 51. This decomposition will be conve- 5 (41 ± 881). As a consequence, we can directly get a nient in later considerations. basis of√ all eigenspaces with eigenvalues different from 2 Let us now show that |Φ6,di has trivial stabilizer for 5 (41 ± 881), 0 from Eq. (C47) . For example, the d > 2. For that, the following lemma is useful. states |0i⊗2, |d − 2i⊗2, |d − 1i⊗2 span the 1-dimensional eigenspaces of ρ(1,2) with eigenvalues 214 , 33, 51, respec- Lemma 23. If v ∈ K˜ fulfills v|Φ6,di = |Φ6,di then v = 6,d 5 u⊗6 for some u ∈ U(d). tively. Note further that u ⊗ u maps an eigenspace of (1,2) ρ6,d to a given eigenvalue to itself. Combining these The proof is similar to the proof of Lemma 17 and will observations one can easily see that be omitted. We are now in the position to prove that |Φ6,di has trivial stabilizer for d > 2. u|0i = u0|0i, (C48)

Lemma 24. For d ≥ 2 the stabilizer of |Φ6,di is trivial, u|d − 2i = ud−2|d − 2i, (C49) i.e. GeΦ6,d = {1l}. u|d − 1i = ud−1|d − 1i, (C50) Proof. Due to Lemma 11 and Lemma 23 it is again suf- for some phases u , u , u . For d = 4 this already ficient to show that any unitary u that fulfills 0 d−1 d−2 implies that u is diagonal. For d > 4 we can consider the u⊗6|Φ i = |Φ i (C45) eigenspaces to nonzero eigenvalues that have dimensions 6,d 6,d larger than 1 to show that u is diagonal. fulfills u⊗6 = 1l. We divide the proof into three parts. In parts (a) and (b) we consider the cases with d > 3. In Using that u|d − 2i = ud−2|d − 2i (see Eq.(C49)) it part (c) we provide a proof of the theorem for d = 3. remains to show that if u|ki = uk|ki holds for k > i, then also u|ii = ui|ii holds, where i ∈ {2, . . . , d − 3}. (a) u is necessarily diagonal for d > 3 Using that u|i + 1i = ui+1|i + 1i we get the following, We consider the case d > 3. From Eq. (C45) we get the following necessary condition, (u ⊗ u)|D1,2(i + 1)i = ui+1(|i + 1iu|ii + u|ii|i + 1i). (C51) (1,2) † † (1,2) (u ⊗ u)ρ6,d (u ⊗ u ) = ρ6,d , (C46) (1,2) Note that u ⊗ u maps the eigenspace of ρ6,d to eigen- (1,2) where ρ = Tr3,...,6(|Φ6,dihΦ6,d|). Let us first deter- d−2 6,d value 18, spanned by {|D1,2(j)i}j=2 , to itself. Hence, (1,2) mine ρ6,d . Eq. (C51) can only be fulfilled if u|ii = ui|ii for some With the help of decomposition (C44) one can easily phase ui. Combined with Eqs. (C48-C50) we have that show that the 2-subsystem reduced state of |Φ6,di for u|ii = ui|ii for i ∈ {0} ∪ {2, . . . , d − 1}. As u is unitary, d > 3 reads this also implies that u|1i = u1|1i for some phase u1 and therefore u is diagonal. d−3 (1,2) 214 ⊗2 X ⊗2 ρ = |0ih0| + 17 |jihj| ⊗6 6,d 5 (b) u = 1l is the only solution for d > 3 j=2 P Using that u = i=0 ui|iihi| for d > 3 it is easy to see + 33|d − 2ihd − 2|⊗2 + 51|d − 1ihd − 1|⊗2 that Eq. (C45) is equivalent to d−2 71 X 6 + |D (1)ihD (1)| + 18 |D (j)ihD (j)| ui = 1 for i ∈ {0, . . . , d − 1} (C52) 5 1,2 1,2 1,2 1,2 j=2 5 u0u1 = 1 (C53) d−1 3 2 X uiui−1ui−2 = 1 for i ∈ {2, . . . , d − 1}. (C54) + 6|D1,2(d − 1)ihD1,2(d − 1)| + 4 (|ji|j − 2i j=3 As in the proof of Lemma (13) we can set, without loss + |j − 2i|ji)(hj|hj − 2| + hj − 2|hj|) of generality, u0 = 1. According to Eqs. (C52-C54) this 5 3 + R, (C47) implies u1 = 1 and u2u1 = 1. Taking the 6-th power 6 18 3 of the second equation we obtain u2u1 = u1 = 1. The 3 5 where only solution to u1 = u1 = 1 is u1 = 1. Using that u = u = 1 in Eq. (C54) we finally obtain u = 1 for 2 0 1 i R = (4c1 + 16)|11ih11| i ∈ {0, . . . , d − 1}. Hence, u⊗6 = 1l for d > 3. + 4c1 [|11i(h20| + h02|) + c.c.] + 4(|20i + |02i)(h20| + h02|), (c) u⊗6 = 1l is the only solution for d = 2 (1,2) Using that u ⊗ u leaves eigenspaces of ρ6,2 invariant it where c.c. denotes the complex conjugate. Note that is straigthforward to see that u is diagonal. Reinserting (1,2) ⊗6 the only part of ρ6,d that is not yet diagonalized is this into Eq. (C45) shows that indeed u = 1l. the operator R. The only nonzero eigenvalues of R are 20

(d) u⊗6 = 1l is the only solution for d = 3 Using that (A ⊗ 1l)|φ+i = (B ⊗ 1l)|φ+i iﬀ A = B, we can In this case it is easy to see that rewrite (C58) as

ρ(1,2) = 11|0ih0|⊗2 + 10|1ih1|⊗2 + 15|2ih2|⊗2 X (1) (2)† (3) ∗ 6,3 Vij Uj = V Ui(V ) ∀i. (C59) j + 7|D1,2(1)ihD1,2(1)| + 3|D1,2(2)ihD1,2(2)| + 2(|02i + |20i)(h02| + h20|) √ These equations are equivalent to Eq. (C57). Let us use (2)† (3) ∗ + 2 2[(|02i + |20i)h11| + c.c.] (C55) the notation Wi ≡ V Ui(V ) . Then a direct consequence of Eq. (C59) is the following necessary condition, (1,2) Using that u ⊗ u maps eigenspaces of ρ6,d with a given † X (1) (1)∗ † eigenvalue to themselves it is easy to see that u|0i = u0|0i WiWi = 1l + Vij Vik UjUk = 1l ∀i. (C60) and u|2i = u2|2i for some phases u0, u2. As u is unitary j6=k P2 this implies that u = i=0 ui|iihi|. This form can then be reinserted into Eq. (C45) to verify that u⊗6 = 1l is in That is, fact the only solution of this equation. X (1) (1)∗ † Vij Vik UjUk = 0 ∀i (C61) j6=k 2. A way to construct critical tripartite states with trivial stabilizer for d > 3 has to hold.

In this section we present a method to look for critical ˜ tripartite states with trivial stabilizer in G. It is partic- b. Special critical states with trivial stabilizer ularily useful if one wants to find nonsymmetric states with these properties. We then employ this approach From now on, we consider critical states for which Eq. to explicitly construct such a state for local dimension (C59) admits a particularily simple form. More precisely, d = 4, 5, 6. we consider states for which U0 = 1l and for which the † unitaries {UiUj }i6=j are linearly independent. The sec- a. On the symmetries of certain tripartite states ond condition implies that Eq. (C61) can only be fulfilled (1) (1)∗ if Vij Vik = 0 for all i and for all j 6= k, i.e. only if (1) Let |ψi ∈ (Cd)⊗3 be a critical state. Due to Lemma 11 V has exactly one nonzero entry in each row. That (1) we know that |ψi does not have any nontrivial symme- iφi is, only if Vij = e δi,σ(i), where σ ∈ Sd is a permuta- tries iff it does not have any nontrivial unitary symme- tion. In the particular case we are considering, Eq. (C59) tries. In what follows we derive necessary and sufficient therefore simplifies to conditions for some |ψi that allow to determine when this † ∗ is the case. We consider the following critical states in iφi (2) (3) e Uσ(i) = V UiV ∀i. (C62) (Cd)⊗3, † Due to Eq. (C62) it holds that V (2) = 1 d−1 X + iφ −1 (3)T † |ψi = √ |ji ⊗ (Uj ⊗ 1l)|φ i, (C56) e σ (0) V U −1 . Reinserting this into Eq. (C62) d σ (0) j=0 ˜ ˜ and using the notation U ≡ Uσ−1(0) and φi ≡ φσ−1(0) −φi d−1 we obtain + √1 P d−1 where |φ i = i=0 |iii, and the operators {Ui}i=0 d ˜ T ∗ † iφi (3) † (3) U = e V U˜ UiV ∀i. (C63) are unitaries that fulfill Tr(Ui Uj) = dδij, i.e. that are σ(i) orthogonal. Suppose V1 ⊗ V2 ⊗ V3 is a unitary symmetry, i.e. In the following section we use the insights gained in this section to explain in detail how a tripartite state V (1) ⊗ V (2) ⊗ V (3)|ψi = |ψi. (C57) with local dimension d = 4 and trivial stabilizer can be explicitly constructed. After that we show how the same (1) P (1) method can be applied to d = 5 and d = 6. We use the notation V = ij Vij |iihj|. With the help of decomposition (C56) and using that 1l ⊗ A|φ+i = T + A ⊗ 1l|φ i for all matrices A it is then easy to see that c. A critical tripartite state with local dimension d = 4 and Equation (C57) is equivalent to trivial stabilizer

X (1) (2) (3)T + + Vij (V UjV ⊗ 1l)|φ i = (Ui ⊗ 1l)|φ i ∀i. Let us construct a critical tripartite state with local j dimension d = 4 and trivial stabilizer. Our goal is to (C58) find unitaries {U0,U1,U2,U3} such that Eq. (C62) is 21 only fulfilled for σ = id and V (3) = eiα3 1l for some α ∈ R. straightforward to see that Eq. (C63) can only be ful- (1) (2) (3) † These conditions then imply that V ⊗ V ⊗ V = 1l filled if for any i ∈ {0, 1, 2, 3} the spectrum of U˜ Ui is as we show now. proportional to the spectrum of Uσ(i). It is however easy (1) iφi Recall that Vij = e δi,σ(i). As σ = id the unitary to see that this necessary condition cannot be fulfilled for (1) (3) iα ˜ ˜ ˜ † V is a phase gate. Using that V = e 3 1l in Eq. U 6= 1l. For example, for U = U1 the spectrum of U U3 is (C62) it is moreover easy to see that V (2) = e−i(φi+α3)1l. not proportional to the spectrum of any of the unitaries iφ ˜ Hence the phases e i cannot depend on i and therefore in {Ui}. Consequently, U = 1l is the only way to satisfy iφi iα1 (1) iα1 fulfill e = e for some α1 ∈ R. Thus, V = e 1l. Eq. (C63), which then simplifies to Hence, we have that V (1) ⊗ V (2) ⊗ V (3) = 1l. T ∗ iφ˜i (3) (3) Uσ(i) = e V UiV ∀i. (C68) In the following we show that the conditions σ = id (3) iα and V = e 3 1l are fulfilled if we choose As the spectra of the {Ui} are not proportional to each other, the only way to fulfill Eq. (C68) is if σ = id. It is ˜ ˜ U0 = 1l, (C64) moreover easy to see that eiφ0 = 1 and eiφj ∈ {−1, +1} 1 † for j ∈ {1, 2, 3}. U1 = T1diag(6 + 8i, −6 − 8i, −6 + 8i, 6 − 8i)T1 , ˜ 10 As eiφ1 ∈ {−1, +1} holds, we square Eq. (C68) for (C65) i = 1 and obtain 1 † T ∗ U2 = 2 T2diag(96 + 28i, 96 − 28i, −96 + 28i, −96 − 28i)T2 , 2 (3) 2 (3) 10 U1 = V U1 V . (C69) (C66) 2 1 Note that the spectrum of U1 is degenerate as we have U3 = 3 × 10 1 diag(936 + 352i, −936 − 352i, −936 + 352i, 936 − 352i), U 2 = − T diag(7 − 24i, 7 − 24i, 7 + 24i, 7 + 24i)T †. 1 25 1 1 (C67) (3)T where Hence, Eq. (C69) can only be fulfilled if V is of the form  1 1 1 1 T 1 1 −i −1 i V (3) = T BT †, (C70) T =   , 1 1 1 2 −1 1 −1 1 −1 −i 1 i where the unitary matrix B is a block diagonal matrix,  i i 1 −1  i.e. B = diag(B1,B2), where B1,B2 are unitary 2 × 2 matrices. Using the form of V T given in Eq. (C70) it 1+√ i − 1√−i − 1+√ i 1√−i 3 1  2 2 2 2  iφ˜ T2 =   . is easy to show that Eq. (C68) with e j ∈ {−1, +1} 2 −i i 1 1   iφ˜j 1+√ i 1√−i 1+√ i 1√−i for all j can only be fulfilled if e = 1 for all j and if 2 2 2 2 T V (3) = c1l for some c 6= 0. Note that T1 and T2 transform the computational basis into the eigenbasis of the generalized Pauli operators In summary, we showed that σ = id and V (3) = c1l for 2 3 X4Z4 and X4 Z4, respectively, where some c 6= 0. As outlined at the beginning of this section these conditions can then be used to show that indeed 3 (1) (2) (3) X V ⊗ V ⊗ V = 1l is the only solution. Hence, the X = |k + 1 mod 4ihk|, 4 state corresponding to the unitaries {U0,U1,U2,U3} in k=0 Eqs. (C64 - C67), i.e. the state Z4 = diag(1, i, −1, −i). 4 1 X + Hence, the unitaries in {Ui} have the same eigenbases |ψi = |ji ⊗ (Uj ⊗ 1l)|φ i, (C71) 2 3 2 as the matrices in {1l,X4Z4 ,X4 Z4,Z4}. However, their j=0 2 3 spectra are different. The choice {1l,X4Z4 ,X4 Z4,Z4} would give rise to nontrivial solutions of Eq. (C62) and has trivial stabilizer in G˜. therefore to nontrivial symmetries. One can show that the same happens if one choses any other subset of gen- 3 eralized Pauli operators to define the unitaries {Ui}i=0. d. Tripartite states with d = 5, 6 and trivial stabilizer It is straightforward to show that the unitaries {U0,U1,U2,U3} fulfill the requirements necessary for Eq. In this section we use the techniques presented in (C63) to be valid. That is, they are all mutually orthog- the previous section to prove that there exist critical † onal and the unitaries {UiUj }i6=j are linearly indepen- tripartite states with local dimension d = 5, 6 and trivial dent. In what follows we show that the only choice of stabilizer. ˜ ˜ U = Uσ−1(0) that can fulfill Eq. (C63) is U = 1l. It is 22

d−1 As in the case of d = 4 we explicitly construct unitaries The unitaries {Ui}i=0 for which we show that the state d−1 {Ui}i=0 for d = 5, 6 that are diagonal in the eigenbasis in Eq. (C72) is critical and has trivial stabilizer in Ge are of certain generalized Pauli operators. We then prove then the following. that the {Ui} are orthogonal and show that the unitaries † {UiUj }i6=j are linearly independent. This shows that Eq. (C62) can be used to prove that the corresponding critical d = 5 : quantum state has trivial stabilizer. Analogously to the case of d = 4, we show that σ = id and V (3) = c1l for U0 = 1l, some c 6= 0 is the only solution of Eq. (C62) for these iβ1 −iβ1 iα1 −iα1 † U1 = U5,1diag(e , e , e , e , −1)U5,1, choices of unitaries {U }. Since the matrices {U U †} i i j i6=j U = diag(−1, eiβ2 , e−iβ2 , eiα2 , e−iα2 ), are linearly independent this then shows that the state 2 U3 = S5,(1,3), d−1 1 X + U4 = S5,(3,1), |ψi = √ |ji ⊗ (Uj ⊗ 1l)|φ i, (C72) d j=0 is critical and has trivial stabilizer in G˜. with α1 = π/3, α2 = π/6 and βi = arccos(1/2 − cos(αi)) for i = 1, 2. d−1 Let us now present these unitaries {Ui}i=0 for d = 5, 6. Recall that for every d ≥ 2 and for any k = (k1, k2) ∈ 2 {0, . . . , d − 1} a generalized Pauli operator is deﬁned as d = 6 :

k1 k2 U0 = 1l, Sd,k = Xd Zd , (C73) 1 U = U diag(6 + 8i, 6 − 8i, −6 + 8i, −6 − 8i, i, −i)U † , where 1 10 6,0 6,0 1 d−1 U = diag(96 + 28i, −96 − 28i, i, −i, 96 − 28i, −96 + 28i), X 2 100 Xd = |k + 1 mod dihk|, k=0 U3 = S6,(1,1),

d−1 U3 = S6,(2,3), X k Zd = ωd |kihk|, U3 = S6,(4,2). k=0 and ωd = exp(2πi/d). As shown in [41] the matrix Ud,t transforms the computational basis into the eigenbasis of Sd,(1,t) for t ∈ {0, . . . , d − 1}, where It is easy to verify that for these choices of {Ui} the † unitaries {UiUj }i6=j are linearly independent. Following d−1 1 X j Pd−1 the same argument as in the previous section one can d−i −t l=i l Ud,t = √ (ωd) (ωd ) |iihj|. (C74) then show that |ψi (given in Eq. (C72)) is critical with d i,j=0 trivial stabilizer.

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