Designing locally maximally entangled quantum states with arbitrary local symmetries

One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. We show how to design critical states with arbitrarily large local unitary symmetry. We explain that such states can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. Then, local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Therefore, such a group of symmetries is naturally protected against errors that occur in a physical realisation of mode operators. We also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries. Our main technical result states that the $N$th tensor power of any irreducible representation of $\mathrm{SU}(N)$ contains a copy of the trivial representation. This is established via a direct combinatorial analysis of Littlewood-Richardson rules utilising certain combinatorial objects which we call telescopes.


Introduction
Multipartite entangled states [1,2] play important roles in different areas of physics including quantum computation (e.g. measurement based [3]), quantum communication [4,5] and quantum metrology [6] as well as condensed matter physics [7]. One can take on an operational point of view on multipartite entanglement and look at the problem of convertibility of quantum states under Local Operations assisted by Classical Communication (LOCC). One of the key points of such a formulation is that LOCC operations cannot increase entanglement [2]. Hence, entanglement plays the role of a resource in quantum information processing [8].
Another important motivation for our work comes form the fact that any operationally useful quantum state must have local symmetries [9]. Let us next briefly review the main reasons why it is the case. To this end, we will be considering not only LOCC operations, but also certain larger and more tractable classes of operations called SLOCC (Stochastic LOCC) and local separable transformations (SEP), see Fig. 1. We focus on systems of N qudits described by Hilbert spaces of the form H := C d 1 ⊗ . . . ⊗ C d N and restrict our attention only to pure states. SLOCC operations are mathematically described by group G := SL(d 1 , C) × · · · × SL(d N , C) and the action is given by the tensor product (g 1 , . . . , g N ). |Ψ : Orbits of this action correspond to sets of SLOCC-equivalent states. Similarly, orbits of the natural action of the local unitary group (LU) K := SU(d 1 , C) × · · · × SU(d N , C) ⊂ G define classes of LU-equivalent states. Furthermore, the set of SEP operations consists of all operations given by Kraus operators in the product form. In other words, a state with density matrix ρ 1 is equivalent to a state with density matrix ρ 2 under SEP if and the product form means that, for every m, we can write K m = ⊗ N n=1 K m,n for some K n,m . Two states, |Ψ 1 and |Ψ 2 , that are connected via SEP are necessarily connected via SLOCC. This means we can write |Ψ 1 = g 1 |Ψ /||g 1 |Ψ || and |Ψ 2 = g 2 |Ψ /||g 2 |Ψ ||, where g i ∈ G. A sufficient condition for SEP conversion from |Ψ 1 to |Ψ 2 was given in [9] and it boils down to satisfying the following equality where L ∈ N, for some probabilities p k ≥ 0, L k=1 p k = 1 and, what is important for us, some choice of operators S k ∈ G |Ψ , where G |Ψ is the stabiliser of |Ψ defined by G |Ψ := {g ∈ G| g. |Ψ ∝ |Ψ } < G.
Thus, we have established two desirable properties of quantum states that make them useful for general quantum information protocols.
2. State |Ψ should be highly symmetric. The larger the stabiliser G |Ψ , the greater the freedom of conversion |Ψ 1 → |Ψ 2 via SEP.
This motivates us to look for highly entangled states with large stabilisers. A canonical example of such a family of states are the celebrated stabiliser states (aka graph states) [3,10,11] that found use in proposals for quantum computers that are naturally robust against decoherence [12]. Recently, symmetries of stabiliser states have been completely classified [13,14]. Importantly, some new symmetries of stabiliser states have been found showing that there exist new potential applications of such states in quantum information protocols [13]. In this work, we take the above approach to the extreme and take up the task of constructing new families of maximally entangled quantum states that have very large local symmetries. In contrast with stabiliser states, our proposed states always have continuous (infinite) symmetries. More specifically, we restrict our attention to so-called Locally Maximally Entangled states (LME), also called critical states. These are states for which all reduced one-qudit reduced density matrices are maximally mixed (proportional to the identity matrix). It is well-known that LME states of two qudits have large symmetry groups (see e.g. [9]). Namely, for |Ψ LME = d i=1 |ii , stabiliser G |Ψ consists of matrices of the form A −1 ⊗ A T for any A ∈ SL(d, C). Furthermore, the GHZ states, W states and cluster states [3] have stabilisers of dimension greater than one. However, this is not a generic property of quantum states, as a generic quantum state (so also a generic LME state) of N > 4 qubits and N > 3 qudits has a trivial stabiliser [15]. Therefore, it is a non-trivial and fundamentally important task to identify quantum states that have nontrivial and possibly large stabilisers.

A brief exposition of the main result
Another reason for considering LME states is that SLOCC entanglement classes containing LME states form an open and dense subset of the space of normalised pure quantum states (which is geometrically the projective space denoted by P(H)). This open and dense subset is called the set of semistable states and denoted by P(H) ss . The remaining states, i.e. those that belong to the null cone N := P(H) \ P(H) ss can of course have large symmetry groups but we will not discuss them in this paper (see [16,17,18,19,20,21,22] for a detailed geometric and information-theoretic analysis of the null-cone). Interestingly, not all quantum systems of many qudits have LME states. There are some exceptional combinations of local qudit dimensions for which LME states do not exist (for details, see [23]). However, in our paper we focus on the case when d 1 = . . . = d N := d, i.e. H = C d ⊗N where LME states always exist. Following [23], we will say that a state |Ψ ∈ C d 1 ⊗ C d 2 ⊗ . . . ⊗ C d N has a diagonal H-symmetry if for some nontrivial representations π 1 , π 2 , . . . , π N of group H π k : H → SU(d k ), 1 ≤ k ≤ N, we have π 1 (h) ⊗ π 2 (h) ⊗ . . . ⊗ π N (h) |Ψ = |Ψ for every h ∈ H.
Moreover, because π k for every k is an irreducible representation of H, by the virtue of Schur's lemma, ρ k (|Ψ ) must be proportional to identity. Thus |Ψ is LME.
To recapitulate, we have the following result.
Corollary 1. If the tensor product of irreducible representations π k : H → SU(d k ) of H, π 1 ⊗ . . . ⊗ π N , contains a copy of the trivial representation of H, then the corresponding representation space, contains LME states with diagonal H-symmetry.
In our paper we mainly focus on the specific problem concerning N qudits, Our main result is Theorem 4 which asserts that if an irreducible representation E λ of SU(m) (associated to Young diagram λ) corresponds to the Hilbert space of a qudit (i.e. E λ C d ), then the system with N = m qudits, H = (E λ ) ⊗m , contains a copy of the trivial representation of SU(m). Consequently, (E λ ) ⊗m will contain LME states with diagonal SU(m)symmetry and so-called strictly semistable states that can be asymptotically transformed to them (see Lemma 2). It is crucial to note that for a fixed local qudit dimension d group SU(m) (2 ≤ m ≤ d) has an irreducible representation (irrep) of dimension d only for some particular values of m. The admissible m can be found for instance using the Weyl dimension formula or Gelfand-Tsetlin patterns (see e.g. [24]). Example 1. Consider a system of N qudits with single-particle space of dimension six, H = (C 6 ) ⊗N . What are the values of N for which H contains states with diagonal SU(N )-symmetries, 2 ≤ N ≤ 6? Using the above corollary, it suffices to show that SU(N ) has an irreducible 6dimensional representation. Clearly, it is true for N = 2 (by treating one qudit as a particle with spin 5/2) and for N = 6 (the natural representation of SU (6)). On top of that, there is an irreducible representation of SU(3) of dimension six (see Subsection 2.2). There are no other possible diagonal SU(N )-symmetries as SU(4) and SU (5) have no nontrivial irreps of dimension 6.
The problem we solve is related to recent developments concerning the so-called saturation problem in decomposition of tensor products of representations of SU(m) due to Knutson and Tao as well as Klyachko and Totaro. The exposition to these results (and more) can be found in the work of Fulton [26]. In particular, Proposition 14 in [26] points out another criterion (which we do not invoke here in full detail) for the existence of a trivial component in a tensor product of representations corresponding to (possibly different) Young diagrams λ(1), . . . , λ(N ). The criterion requires checking a set of inductively constructed inequalities involving integer partitions related to Young diagrams λ(1), . . . , λ(N ). Satisfying the criterion is equivalent to the existence of the trivial component in the product E λ(1) ⊗ . . . ⊗ E λ(N ) . Our work circumvents the necessity of checking this set of inequalities in the special case when λ(1) = λ(2) = . . . = λ(N ) := λ by performing a straightforward multiplication of Young diagrams for any λ. To our best knowledge, besides our Theorem 4, there is no comparably simple and general criterion allowing one to determine the existence of LME states with diagonal SU(m)-symmetries.
The paper is organised as follows. In Section 2 we present potential physical setups where one could construct our proposed LME states with arbitrary diagonal LU-symmetry. We discuss systems of distinguishable particles as well as systems of distinguishable traps containing bosons or fermions. We also show how one can use our main result, Theorem 4, to tell whether a given system contains so-called strictly semistable states. In Section 3 we recall some basic facts about our main technical tool which are Young diagrams. We briefly review the relationship between Young diagrams and irreducible representations of SU(m) and explain Littlewood-Richardson rules that concern the problem of decomposing a product of representations into irreducible components. Finally, in Section 3 we formulate our main result, i.e. Theorem 4, and give a sketch of its proof. For the sake of clarity of our presentation, the main technical weight of the proof is contained in the Appendix. Section 4, discusses alternative approaches and considers the problem of finding stated with diagonal H-symmetry with H being a general compact semisimple Lie group. The methods we work out are subsequently applied to find LME states with diagonal SO(d) symmetry. Section 5 contains a summary and a discussion of possible ways to extend our work.

Physical settings and other applications
In this section we study in detail possible physical settings for which our construction gives quantum states that possess large symmetries. We do this in several steps, starting with the most straightforward construction of multi qudit states with complete diagonal LU-symmetry (Subsection 2.1). We then move to a slightly more complicated setting that involves distinguishable traps with bosons (Subsection 2.2) and end up with the most general setting involving traps with fermions possessing internal degrees of freedom (Subsection 2.3). In Subsection 2.4 we show that our results have deep implications for the geometric structure of entanglement classes of multi qudit states in terms of geometric invariant theory.

Qudit states with complete diagonal LU-symmetry
Here we analyse the most straightforward case of N qudits, H = C d ⊗N with diagonal SU(d)symmetry which we call the complete diagonal LU-symmetry. Here, representation π := π 1 = . . . = π N from (6) is just the natural representation of SU(d) on C d given by the Young diagram consisting of a single box, i.e. λ = (1). State |Ψ is symmetric with respect to this action, i.e. has complete diagonal LU-symmetry (6), if and only if |Ψ is annihilated by all generators of SU(d). Recall that these generators are defined via projectors More specifically, the Lie algebra su(d) is generated by for all i, j such that 1 ≤ i < j ≤ d. The above generators are diagonally represented on C d ⊗N as follows for any A ∈ su(d). Hence, |Ψ has complete diagonal LU-symmetry if and only if The above set of equations can be simplified by using the following commutation relations: Therefore, it is enough to solve The above simplification is a manifestation of a more general fact which will be described in Section 4. The dimension of the solution space of (15) is the multiplicity of the trivial representation of SU(d) in the product π ⊗N . The resulting dimensions for small d and N are shown in Table 1.  There are two key features of states with complete diagonal LU-symmetry that can be readily seen from Table 1. • States with complete diagonal LU-symmetry exist if and only if N = kd for some integer k. This can be seen as a consequence of Littlewood-Richardson rules introduced in Section 3.
• The dimension of the space of states with complete diagonal LU-symmetry increases with N . In fact, it is given by multidimensional Catalan numbers, as explained in Subsection 4.1.
If N = kd, it is straightforward to write down an example of the corresponding LME state with complete diagonal LU-symmetry. It has the form of the tensor product where state |Ψ i is the completely antisymmetric state with respect to permutations of qudits di + 1 through d(i + 1). Using the antisymmetrisation operator where S d is the permutation group on d elements, we can write down the state |Ψ as

A system of traps with bosons
Consider a quantum system consisting of three distinguishable traps enumerated by i = 1, 2, 3, each containing two bosons that can occupy three modes denoted by creation operators (Fig.2). The Hilbert space of a single trap is which will be treated as one qudit with six degrees of freedom. The total Hilbert space consists of three such qudits The set of all LOCC operations is represented as triples of 6 × 6 matrices A 1 , A 2 , A 3 that act on |Ψ ∈ H as |Ψ → A 1 ⊗A 2 ⊗A 3 |Ψ . Physically, realising such operations requires introducing some interactions between bosons within a single trap to obtain two-body operators of the form We are interested in finding an LME state with the following symmetries. Symmetries are the local diagonal mode special unitary operations, i.e.
which is effectively a 6 × 6 unitary matrix when written in the basis {|n a , n b , n c }. This corresponds to the situation where π from equation (6) is the representation of SU(3) corresponding to the Young diagram that consists of one row with two boxes, λ = (2).
A state |Ψ ∈ H is symmetric with respect to local diagonal mode operators if for all operators U we have This in turn happens when |Ψ is annihilated by all generators of such operations. This boils down to the following set of six linear equations.
The solution is a state which is symmetric with respect to permutations of traps. Using the symmetrisation operator S the solution is written as The symmetrisation operator is defined as the following sum over all three-element permutations For so defined state, the norm reads Ψ|Ψ = 3 √ 2. State |Ψ is a state which is locally maximally entangled with respect to the three traps, i.e. ρ i = diag( 1 6 , . . . , 1 6 ) for i = 1, 2, 3. It can be checked directly or simply deduced from the fact that |Ψ has a diagonal SU(3)symmetry. Let us emphasise once again that ρ i is the single-qudit reduced density matrix, coming from the reduction of ρ = |Ψ Ψ| with respect to the remaining two qudits (traps). This is not to be confused with the reduced density matrix of a single boson.
Notice that, since each trap is effectively a qudit with d = 6, and SU(3) has an irrep in C 6 , from Corollary 1, H had to contain some LME state with diagonal SU(3)-symmetry.
One may check the multiplicity of a trivial representation in H as a carrier space of local diagonal mode special unitary operations. A straightforward calculation involving Young diagrams shows that the multiplicity is 1.
The above physical setting can be generalised to a system of m distinguishable traps with each trap containing n bosons occupying m modes. In that case, a single trap realises a qudit with d = m+n−1 n . This scenario realises representation E λ of SU(m) with a single-row diagram λ = (n) as a diagonal LU-symmetry of the corresponding state. The explicit form of the state can be found by solving where a (k) i is the annihilation operator of kth mode in ith trap.

Spinful fermions and beyond
In this subsection, we introduce the most general construction capturing LME states with a diagonal LU-symmetry given by an arbitrary representation of SU(m). We start with the singleparticle Hilbert space which describes a particle with s internal degrees of freedom (s-charge) that can occupy m modes in a single trap In particular, if s = 2, then we think of a single electron (spin-1/2 fermion) occupying m orbitals. Next, we confine N such fermions in one trap. The total Hilbert space for the system of N fermions in one trap is given by the antisymmetric product Hilbert space H N is equipped with the natural action of SU(m) × SU(s) that changes the basis of orbitals and internal degrees of freedom of each particle simultaneously. Let us next consider an abstract (possible interacting) hamiltonian of the above system of trapped N fermions and assume that it commutes with the total s-charge operator. In the representation-theoretic language, we require the hamiltonian to commute with the Casimir operator of SU(s). For instance, if the considered fermions were ordinary electrons (s = 2), the above symmetry would be the well-known total spin conservation symmetry. If this is the case, Hilbert space H N decomposes into sectors of fixed total s-charge each of which is isomorphic to the tensor product of an irrep of SU(m) with an irrep of SU(s) [25] H where H λ m is the irreducible representation of SU(m) corresponding to Young diagram λ and H λ T s is the irreducible representation of SU(s) corresponding to the transposed Young diagram λ T . The sum is over all Young diagrams with N boxes and at most m − 1 rows and at most s columns. Again, for spinful electrons we require λ to be a two-column diagram and H λ T s is the representation of SU(2) with total spin equal to half of the difference of columns' lengths. Finally, we would like to superselect the particular value of the s-charge. This can be done in general by requiring the hamiltonian to commute with appropriate additional generators of SU(s). In the spin case, we desire the total S z -conservation. In this way, we are able to superselect the sector of H N that consists of states of the form |Ψ = |Φ ⊗ |i , |Φ ∈ H λ m , for fixed i ∈ {1, . . . , s}. This space is isomorphic to H λ m and carries the natural action stemming from the basis change of modes within a single trap. By considering a system of m distinguishable traps, where the Hilbert space for every single trap is the above superselected H λ m , we obtain an abstract physical setting that contains LME states with diagonal SU(m)-symmetry given by the m-fold product of representation π from (31). Finally, let us remark that, similarly to the case of traps with bosons described in Subsection 2.2, every trap is treated as a qudit with local dimension d = dim H λ m . Hence, performing SLOCC operations on a full qudit requires interactions between fermions within one trap.
Example 2 (The doublet space of 3 electrons with m = 3). As an example, consider a system of m = 3 traps, where the Hilbert space of a single trap is given by the doublet space of 3 electrons occupying 3 modes with total S z = +1/2. Such a space comes from the following decomposition where H (2,1) 3 ⊗C 2 is the doublet component corresponding to the total spin of 1/2 and H is the quadruplet component corresponding to the total spin of 3/2. Furthermore, by superselecting the total S z = +1/2 within the doublet space, we obtain the Hilbert space of a single trap which is isomorphic to the irrep of SU(3) given by diagram λ = (2, 1). Space H (2,1) 3 is of dimension 8 and for completeness, we write down its (non-orthonormalised) basis as a subspace of Finally, equations for the desired LME state distributed across three traps |Ψ ∈ H where a (k) i is the annihilation operator of kth mode in ith trap.

Existence of strictly semistable states
According to the Kempf-Ness theorem from geometric invariant theory, a SLOCC class contains an LME state iff it is closed (in the standard complex topology) [27]. We call states within a SLOCC class containing an LME state polystable states. Strictly semistable are states whose SLOCC classes contain LME states only in the closure. It turns out that strictly semistable states exist iff there exists an LME state with more symmetries (in the sense of stabiliser dimension) than generic (i.e. there are at least two LME states with different stabiliser dimensions; see Theorem 1 and Observation 3 from [28] and Fig. 2 therein which explains the relationship between the symmetries of LME states and the structure of SLOCC classes). It is well-known that Hilbert spaces of systems of two qudits and three qubits contain open and dense SLOCC orbits going through LME states, so such systems do not have strictly semistable states. In the remaining cases the generic stabiliser is trivial or discrete [15], so zero-dimensional. On the other hand such systems contain GHZ states that have continuous symmetries (and are LME). Thus, applying the criterion from [28], they contain strictly semistable states. Such strictly semistable states can be asymptotically transformed to the GHZ state. We show that in some cases one may deduce the existence of the special classes of strictly semistable states, namely the ones that can be asymptotically transformed to an LME state with a SU(N )-symmetry (N being number of qudits). Specifically, combining Theorem 4 with Corollary 1 and results from [28], we obtain the following lemma.
. Suppose that there exists a non-trivial irreducible continuous representation π : SU(N ) → SU(d). Then H N contains an LME state |Φ with diagonal SU(N )-symmetry with respect to π ⊗N , as defined in (6). Moreover H N contains strictly semistable states that can be asymptotically transformed to |Φ .
Proof. Recall that G |Ψ denotes the SLOCC stabiliser of |Ψ while K |Ψ denotes the LU stabiliser of |Ψ . Because LME states in H N exist, there exist also polystable states [28]. A generic polystable state in H N , say |Ψ , is known to have no continuous symmetries with respect to the action of G [15] . Let |Ψ c by an LME state in the SLOCC class of |Ψ . We have dimG |Ψc = 0. From Theorem 4, H N contains a copy of the trivial representation of SU(N ). Hence, from Corollary 1, H N contains an LME state |Φ with diagonal SU(N )-symmetry with respect to π ⊗N . It remains to check that dim K |Φ ≥ 1. Indeed, in this case since K < G also dim G |Φ ≥ 1 so |Φ and |Ψ c are two LME states with different G-stabiliser dimensions. Thus, |Φ is a state to which some strictly semistable states can be asymptotically transformed [28]. Let us denote H = SU(N ). Note that H |Φ = H so π(H) < K |Φ . Since K |Φ is closed, it contains the closure π(H) which is an embedded connected Lie subgroup of SU(d). Moreover, dim π(H) ≥ 1 since otherwise π(H) would be discrete and connected, so a singleton. Thus, π would be trivial.
Our work also links to another facet of geometric invariant theory which is the problem of constructing SLOCC-invariant polynomials. Namely, homogeneous SLOCC-invariant polynomials of degree d with respect to an irreducible representation E λ of SU (m) can be viewed as trivial components in the decomposition of S d E λ into irreps. In the spirit of our Theorem 4, one can ask about the smallest d for which S d E λ contains the trivial irrep of SU (m) with non-zero multiplicity. For a way to derive lower bounds for such d, see [43].

Multiplying representations of SU(m)
All irreducible representations of SU(m) are in a one-to-one correspondence with Young diagrams that consist of at most m − 1 rows. The correspondence is established by constructing the so-called Schur module [29], E λ , which gives the canonical form of such a representation. Denote by E = Span{|0 , . . . , |m − 1 } the natural representation of SU(m). Representation E λ of SU(m) described by Young diagram λ = (λ 1 , . . . , λ m−1 ) is a linear subspace of where µ k is the length of kth column of diagram λ. The highest weight vector of E λ is of the form Two straightforward extreme cases are when λ is a single column, i.e.
and when λ is a single row, i.e.
The corresponding representations are The coefficients c λ can be calculated using the Littlewood-Richardson rule (see e.g. [30,31]). When specified for SU(m), we have c λ = 0 iff λ has more than m rows.
The above Facts 1 and 2 allow us to formulate a necessary condition for the N th tensor power of a given representation E λ to contain a trivial representation of SU(m). Namely, representation E λ appears as an irreducible component of E ⊗|λ| . Hence, if the trivial representation appears as an irreducible component of E λ ⊗N , then it appears as an irreducible component of E ⊗|λ|N . Furthermore, product E ⊗|λ|N contains trivial representation E τ with diagram τ = (τ 1 ) m iff |λ|N = mτ 1 . Hence, we have proved the following lemma.
Lemma 3. If the N th tensor power of E λ , an irreducible representation of SU(m), contains a copy of the trivial representation, then |λ|N is an integer multiple of m.
The simplest way to satisfy the above necessary condition is to put N = m. In the remaining part of this section we show that this is, in fact, a sufficient condition (see Theorem 4).
Let us take a closer look at the multiplication rules for Young diagrams. In general, when multiplying two irreducible representations we get where c λη ν is the Littlewood-Richardson (also known as Clebsch-Gordan) coefficient. Representations E ν that appear in the above product with non-zero coefficients can be determined by the following set of rules.

Fact 3. [Littlewood-Richardson rule]
Here we describe how to find the Littlewood-Richardson coefficients by diagram expansions, according to the Littlewood-Richardson rule. We start with two irreducible representations of SU(m) whose corresponding diagrams are λ and η. In order to find the irreducible representations (and their multiplicities) that appear in the product E λ ⊗ E η , we draw the two diagrams next to each other and fill the second diagram, η, with numbers so that boxes of the kth row contain only integer k. To obtain diagram ν that corresponds to an irreducible component of E λ ⊗ E η , we expand the diagram of λ by appending all boxes of diagram η to diagram λ according to the following rules. The boxes can be appended only to the right or to the bottom of λ. Moreover, the following conditions need to be satisfied.
1. The consecutive rows of the resulting diagram have non-increasing lengths, i.e. ν i ≥ ν i+1 . This means they are Young diagrams.
2. In any column of ν, there are no two boxes with the same label.
3. Let # r (m) denote the number of boxes of ν with label m in rows 1 up to r (numerated from top to bottom). Then, for every row r if n < m then # r (m) ≤ # r (n). This is row counting condition.
4. Let # c (m) denote the number of boxes of ν with label m in columns 1 up to c (numerated from right to left). Then, for every column c if n < m then # c (m) ≤ # c (n). This is column counting condition.

5.
Diagram ν has at most m rows.
The Littlewood-Richardson coefficient of a given diagram will be the number of its occurrences among all possible diagrams obtained via above procedure.
It is worth noting that there is another insightful yet equivalent way one can deal with the irreps of SU(m), namely via Gelfand-Tsetlin pattern calculus (see the original paper [32] and the reprinted version in [33]). Using this approach, a numerical algorithm for the explicit calculations of SU(m) Clebsch-Gordan coefficients has been given in [24] (see also [34]).
Example 3. The square of any irreducible representation of SU(2) contains a copy of the trivial representation. The proof is shown on the picture below, where we show that E λ ⊗ E λ ⊃ E (|λ|,|λ|) for any single-row diagram λ.
Example 5. Let E λ be an irreducible representation of SU(4). Then, E λ ⊗4 contains a copy of the trivial representation. The Young diagram of an irreducible representation of SU(4) has at most 3 rows, i.e. λ = (λ 1 , λ 2 , λ 3 ). We proceed similarly as in Examples 3 and 4.
In general, it is hard to prove a theorem which is valid for a Young diagram of any shape. However, we will show that in fact Examples 3, 4 and 5 can be generalised to any dimension N , and Young diagram λ, i.e. the following theorem holds. Proof. Similarly as in Examples 3, 4 and 5 we will give an explicit construction of the sequence of diagram expansions leading to a rectangular diagram with N rows. We will proceed as follows. In every step we first add boxes with label 1, then with label 2 etc. up to label N − 1. Moreover, for every label we add the boxes to the consecutive rows i.e. we start with some top row i and append a certain number of boxes to q ≥ 1 rows from ith to i + q − 1th row. In every step we need to specify the position, label and number of appended boxes. The idea is to append boxes in such a way that the exact number of boxes with a given label p added to a given row is uniquely determined by p and the total number of rows q to which we append boxes with a given label.
In order to keep track of the number and type of boxes appended in every step, we define symbols which can be understood as entries of columns of numbers we call telescopes. The name telescope is motivated by the fact that partial sums of entries of a telescope is a telescopic sum (see Section C.1). A symbol T r p,q denotes the rth entry (counted from bottom) of a telescope T p,q . We call p the label and q the length of a telescope (which is the number of its entries). For example T 2,3 is a telescope with label 2 and length 3. We will represent a telescope T p,q graphically as a column of q boxes with label p, one box for each entry of a telescope. We call such boxes virtual. For example, can be understood as three rows of boxes with labels 2. The rth virtual box (counted from bottom) of a column corresponds to a row of T r p,q boxes with label p. We will encode the way we expand diagrams using virtual boxes. Each column of virtual boxes corresponds to some telescope T p,q . For each step k, we construct a pattern which corresponds to a sequence of telescopes, one for each label m. We append the boxes to the diagram label by label with m increasing. Each virtual box in the column (i.e. entry of a telescope) corresponds to a particular number of boxes appended to some (yet not specified) row l of an expanding diagram. Consecutive virtual boxes correspond to consecutive columns of the diagram. An exact number of boxes m appended to the row l is determined by a number of virtual boxes in a column, the label m of the virtual boxes and the position of a virtual box in the column -according to (43).
For example, the patterns encoding expansions form Examples 4 and 5 are What remains to uniquely define the sequence of diagram expansions for any N is to specify, for every step k and every label m, a row of a diagram with the top virtual of a column, which we denote by beg(m, k), and number of virtual boxes in a column, i.e. the length of a telescope, denoted by len(m, k) (or equivalently -a row with the bottom virtual box end(m, k)). Trying to generalise the patterns for N = 3 and N = 4, we propose the following so that, since end(m, k) = beg(m, k) + len(m, k) − 1, we get For example, in case N = 6 we obtain the patterns presented below. It remains to check that proposed recipe (i.e. the sequence of diagram expansions) is valid and gives a rectangular diagram with N rows and shape (|λ|, |λ|, . . . , |λ|). We need to check the following conditions for every step k ∈ {1, 2, . . . , N }.
1. The result of appending boxes in step k gives a Young diagram.
2. Among boxes appended in step k, there are no two boxes with the same number in any column.
3. Boxes appended in step k satisfy row counting condition.
4. Boxes appended in step k satisfy column counting condition.
Moreover there is additional Condition 5 -the final diagram need to have a rectangular shape.
Since the final diagram consists of N |λ| boxes, Condition 5 follows immediately from Condition 1. The proof of the correctness of proposed recipe is given in the Appendix.

Other approaches and final remarks
Remarkably, our problem of multiplying many Young diagrams may be reformulated so that it becomes equivalent to a problem of multiplication of just two respectively larger diagrams [26]. Let us next revisit main points of this construction whose proof can be found in [26]. Our original problem asks when representation E β of SU(d) appears in the N -fold tensor product E λ ⊗ E λ ⊗ . . . ⊗ E λ . For now, we allow N ≥ d and, as usual, we want β to give the trivial representation, i.e. we require β to be the square diagram with |λ| columns and N rows.
Lemma 5. The multiplicity of E β in E λ ⊗N is equal to the Littlewood-Richardson coefficient c γ αβ where Young diagrams α and γ are constructed from λ as follows. The skew diagram γ/α (the complement of α as a subdiagram of γ) is constructed by arranging n copies of λ so that the top right corner of the ith copy touches the bottom left corner of the (i + 1)th copy.
At the first sight, it may appear that using the above lemma would simplify our proof of Theorem 4. However, the technical difficulty of checking the Littlewood-Richardson rules in the multiplication of such large diagrams remains the same. Nevertheless, Lemma 5 will be very useful in our considerations concerning the complete LU symmetry described in Subsection 4.1.
For general (semi)simple compact Lie groups some techniques analogous to Young diagrams have been developed [39,40,41]. However, their application is not as straightforward as it was in the case of Young diagrams. In this section, we describe the most straightforward way of finding states with diagonal G-symmetry that relies on solving a large set of linear equations derived from generators of G. We test this algorithm for G = SO(d) in subsection 4.2. The algorithm is as follows. The complexification of the Lie algebra of G, g C := g ⊕ ig, has the Cartan decomposition [31] g where h is the maximal commutative subalgebra, ∆ + is the set of positive roots of g C and g C α = Ce α and g C −α = Ce −α with e α and e −α called the positive and negative root operators respectively. For instance, when G = SU(d), the positive root operators are |i j| with i < j while the negative root operators are |i j| with i > j. For G = SO(d), we specify them in Subsection 4.2. The key fact we are using is that given a unitary representationπ : G → U (D), the highest weights in the decomposition of π into irreps are annihilated by all positive root operators [31]. However, we are interested in finding the trivial representation whose highest weight is by definition also the lowest weight, i.e. it is annihilated by all negative root operators. This gives us a set of 2|∆ + | equations. In fact, commutation relations allow us to consider only the simple roots Π ⊂ ∆ + which form a basis of ∆ + . The number of simple roots defines the rank of group G, r G := |Π|. For G = SU(d), the root operators associated with simple roots are |i i + 1| for i = 1, . . . , d − 1. This means in particular that in equation (23)  More formally, we are looking at representationπ : G → U (d N ) defined asπ = π ⊗. . .⊗π with π : G → U (d) being an irrep of G. Algebra g is represented via the derived representation where ∂π : g → u(d) is the derived representation of π. State |Ψ ∈ C d ⊗N spans the trivial representation of G of and only if ∂π(e α ) |Ψ = 0 and ∂π(e −α ) |Ψ = 0 for all α ∈ Π. (50) The above set of equations (50) gives us 2r G linear equations for |Ψ being an element of d Ndimensional Hilbert space. Hence, the complexity of the problem scales polynomially with the rank of G and exponentially with the number of particles.

Complete diagonal LU-symmetry and diagonal SU(2)-symmetry
As promised in Subsection 2.1, here we derive multiplicities of the trivial representation of SU(d) that appear in Table 1 using Lemma 5. Figure 6: Diagrams β and γ/α from Lemma 5 for λ = (1) and N = kd; E λ being the natural representation of SU(d). Figure 6 shows that the multiplicity of the trivial representation E β of SU(d) in the product E λ ⊗kd , λ = (1), is the same as the number of ways one can fill diagram γ/α with labels from tableau β while satisfying Littlewood-Richardson rules. Diagram γ/α has a particularly simple form for which the row counting condition and column counting condition are equivalent to the following condition. Arrange labels of the skew tableau γ/α in a sequence (a 1 , a 2 , . . . , a kd ) where a i is the label of the box in the ith row with row 1 being the top row. Then, for any j ∈ {1, . . . , kd} we require that #{i ≤ j| a i = n} ≤ #{i ≤ j| a i = n − 1} for all n ∈ {1, . . . , d}.
Such sequences are in a bijection with certain combinatorial objects called generalised Dyck paths whose number is given by d-dimensional Catalan numbers [37,36] c (d) As a final part of this subsection, we briefly remark that the problem of finding states of diagonal SU(2)-symmetry is the classical problem of spin composition. More specifically, one views a qudit, C d , as the configuration space of a particle with spin J where d = 2J + 1, i.e.
. In order to find states of N such qudits with diagonal SU(2)-symmetry, we decompose the N -fold product H 1 ⊗ . . . ⊗ H 1 into irreducible SU(2)-components and ask about the multiplicity of the trivial representation which is the singlet (spin 0) space. The multiplicities have been calculated in [38] and they can be expressed by certain combinations of hypergeometric functions. In particular, for d-even the singlet space appears with nonzero multiplicity only when N is even.

States with diagonal SO(d)-symmetry
Here, we specify the general algorithm (50) to G = SO(d). We consider N -fold tensor products of the natural unitary representation of SO(d) on C d . Generators are real d×d real antisymmetric matrices, i.e. A T = −A. In contrast to su(d) C , simple root operators (and their negatives) of so(d) C have slightly more complicated forms whose derivation can be found in [35]. Simple root operators are different for even and odd d. There is a set of d 2 operators that is common for d both even and odd. It consists of block-diagonal operators containing only one nonzero block of size 4 × 4. In the basis |s , s ∈ {0, . . . , d − 1}, their precise forms read where j ∈ {0, . . . , d 2 − 2}. For d even, we add to the above set one more simple root operator (and its negative) which reads In the case when d is odd, we add the following operators According to the general algorithm (50), in order to find a state |Ψ ∈ C d N with the above diagonal SO(d)-symmetry, we need to solve the following set of equations for j ∈ {0, . . . , d 2 − 2}. On top of that, we require for d even and d odd respectively. The dimension of the solution space (the multiplicity of the trivial representation of SO(d)) for small N and d has been calculated in Table 2. For calculations we used a multicore server with 120 GB of RAM. As the size of equations grows exponentially with N , the main limiting resource is the size of RAM. As one can see from Table 2, the method of directly solving equations (60) and (61) is inefficient and allows one to find LME states only for small N and d. However, if one is interested in finding only the dimension of the LME space, one can use more efficient methods for computing multiplicities of irreps. Such methods have been implemented in the Lie-ART 2.0 package [42] that we used here.   (60) and (61). Zeros in italics come from an application of Lemma 6. Numbers in bold were computed using the Lie-ART 2.0 package [42].
Lemma 6 explains the existence of zeros in Table 2 above for even d.  (2) algebra. For a given compact semisimple Lie group G and its irrep π G,d : G → SU(d) assume that Im ∂π d ⊂ Im ∂π G,d . If in H = C d ⊗N there are no LME states with diagonal SU(2)-symmetry given by π d , then there are no LME states with diagonal G-symmetry. In particular, there are no LME states with diagonal SU(d)or SO(d)-symmetry given by natural representations of these groups.
Proof. Because G is semisimple, there is a copy of the trivial representation in the product π G,d ⊗ . . . ⊗ π G,d iff there exists a vector in H that is annihilated by all operators from Im ∂π G,k . Because Im ∂π d ⊂ Im ∂π G,d , any vector annihilated by all operators from Im ∂π G,d is also annihilated by all operators from Im ∂π k , i.e. has a diagonal SU(2)-symmetry. The claim for SU(d) and SO(d) can be verified by a straightforward calculation in standard bases of su(d) and so(d).
Remark 1. One can easily find systems containing LME states with some diagonal G-symmetry but having no LME states with SU(2) symmetry. For instance, consider H = C 8 ⊗ C 8 ⊗ C 8 and take the 8-dimensional representation of SU(3) given by diagram λ = (2, 1). By Theorem 4, E λ ⊗3 contains a copy of the trivial representation of SU(3). However, the third tensor power of the 8-dimensional representation of SU(2) described by diagram µ = (3), E µ = S 7 (C 2 ), does not satisfy the necessary condition given in Lemma 3. Hence, (E µ ) ⊗3 does not contain any trivial SU(2) components.
To end this subsection, we write down exemplary linearly independent solutions of (60) and (61) for N = 4 and any d. We conjecture that this is in fact the full solution space for any d = 4.
Furthermore, if d = N , we have one state which has in fact the complete diagonal LUsymmetry and is of the form (18).

Summary
In this paper we proved a theorem in representation theory and showed how it can be used to design critical states with particular large local symmetries, called diagonal SU(m)-symmetries. Our method may be a source of operationally useful quantum states that can be realised in various systems of distinguishable qudits and traps with bosons or fermions occupying a finite number of modes. Moreover, the criterion given in the paper can be used to identify interesting classes of strictly-semistable states -namely the ones which asymptotically have large local symmetries. One should keep in mind that in the systems with LME states, the dimension of a set of LME states (which contains all states with diagonal H-symmetries) grows exponentially with N , just as the dimension of the Hilbert space of a system. The proof of the main theorem is lengthy and very technical since it involves quite complex combinatorial objects connected with the Littlewood-Richardson rule. However, up to our best knowledge, there is no simple and general criterion allowing one to determine the existence of LME states with diagonal SU(m)-symmetries. In the future, it would be good to consider systems with nonhomogenous local dimensions. We also conjecture that if the manifold of LME states in a multiqudit system H := C d 1 ⊗. . .⊗C d N , up to local unitary equivalence, has dimension at least one, dim(LME/K) ≥ 1, then H contains strictly semistable states. One can also think about better ways of finding explicit forms of LME states with diagonal H-symmetry. For example, in case of compact H it is clear that the set of all such states is the image ImP ⊂ H of the projector where µ is the normalised Haar measure on H. Finally, it would be good to think about concrete forms of Kraus operators realising SEP operations for states with diagonal G-symmetry.

Appendix
In this appendix we prove the correctness of the sequence of diagram expansions proposed in the proof of Theorem 4.

A Auxillary objects for Conditions 1 and 2
A.1 Introduction of ∆ k i and δ k i (m) It is easy to see that the number of boxes with label n appended at step k to row i is given by Numbers ∆ k i for a fixed k describe the shape of the tableau obtained after all steps up to k − 1 -these are the lengths of steps of black "stairs" in Fig. A.1. Precisely, ∆ k i is a difference between the total number of boxes (length) of ith and (i + 1)th row after k − 1 steps. One can think of a matrix ∆ with entries ∆ k i . In order to find the interpretation of δ k i (m), one can leftalign all the boxes appended in step k. Then, δ k i (m) is the distance (number of boxes) between the rightmost box with label m − 1 in ith row and the rightmost box with label m in i + 1th row. We introduce the symbol: For example, in case N = 6 we obtain (see also Fig. A.2):

A.2 Handling # function
The possible values of n, k and i in (A.1) form a 3-dimensional lattice we denote by Λ = {1, 2, . . . , N } ×3 . By an x-layer we mean a subset of Λ with fixed x, where x is n, k or i. By # we denote the function given by #(n, k, i) = (#n) k i . In order to derive explicit formula for # we combine (43) and (A.1). We get four possible nontrivial formulas, denoted A1, A2, B1, B2, each one valid on a different domain of Λ, which define # piecewise. The domains are given by and define # : Λ → N by From now on we will stick to decomposition of S into n-layers. To make calculations more traceable, we introduce the moving distinguished point (slider) denoted by O. Its ik-coordinates (for fixed n-layer) are O = (n, N − n + 1). Fix a column of interest j. We introduce the parameter = j − n, i.e. the distance from O. We have 5 cases.
Let us define the symbol for the sum of elements from jth column which will contribute to ∆ s j (this corresponds to the non-faded points in jth column on Fig. A.5 (b)). The value of n should be clear from the context. Notice that We also introduce the symbol X n i := X(n, k, i), where X ∈ {A1, A2, B1, B2} is one of (A.9) and k is the running parameter (the symbols will be used under summation over k).
In order to avoid splitting equations into cases, with every condition C on n, k and i we associate the subset S(C) ⊂ Λ of all triples (n, k, i) that satisfy C. We define the characteristic function χ(C) : Λ → {0, 1} as the characteristic function of a set S(C), i.e. By a n i , where a ∈ {1, 2, 3, 4}, we denote the formula for (s, j) in case a given by (A.15) up to (A.19). The value of s should be clear from the context.
We say that the n-layer is non-degenerate if it realises all 5 cases (A.10) in a non-degenerate way. Otherwise the layer is called degenerate. Non-degeneracy is equivalent with n < N − 2 and N ≥ 4. From now on we assume that N ≥ 5 to avoid some degenerate cases; For the proof in cases N = 2, 3, 4 we refer to direct constructions given in Examples 3, 4 and 5, which can be easily checked to be valid. We freely use the following properties of (A.9) A1(n, k, i) = A1(n, k ± 1, i ± 1) = A1(n ± 1, k ∓ 2, i) = A1(n ± 1, k, i ± 2), (B.1) A2(n, k, i) = A2(n ± 1, k ∓ 1, i) = A2(n, k, i ± 1), (B.2) B1(n, k, i) = B1(n ± 1, k, i ± 1) = B1(n, k ± 1, i), whenever right-hand side is defined. We also evaluate some telescopic sums which will appear frequently We (B.14) We did not include the arguments N, j and s in the definitions B.7 up to B.14 since they will always be constant. Using already the fact that j < N we can simplify χ j<N = 1.

C Conditions 3 and 4 C.1 Condition 3 (row counting)
Clearly, the partial sum of entries of a telescope T p,q of the form (43) from bottom to top. Note that the entries of (C.1) are determined, up to position q, only by the top entry λ p+q−1 , which in turn depends only on p + q. Let σ n,k be the difference between the rows in which top virtual boxes of columns, for two consecutive labels n and n + 1, appear at a given step k. Note that from (45), for fixed k and m > n, we have beg(m, k) > beg(n, k). Thus, σ n,k is either 1 or 2 and σ n,k = 2 is obtained exactly in step k = N − n. Moreover, from (46), for fixed k and m > n, we have len(m, k) ≥ len(n, k). Hence, the difference τ n,k between lengths of telescopes for two consecutive labels n and n + 1, at given step k, is either 0 or 1. Thus, it suffices to consider three cases, depicted in Since (C.1) is non-increasing from top to bottom, it is clear that Condition 3 is satisfied locally (i.e. for any two telescopes with consecutive labels, in any fixed step) in every case, so Condition 3 is always satisfied.

C.2 Condition 4 (column counting)
We show that Condition 4 in fact follows from what we already have proved. Let#(m) k l denote the number of boxes with label m appended in step k up to the lth column (counted from right to left). Notice that the inequalities proved within Condition 2, ∆ k i ≥ δ k i (m), imply that, for a fixed step k, the blocks with the same label are distributed stair-like (block in row i ends before or just as the block in row i + 1 begins), as shown in Fig. C.2. Moreover, in every row the labels of blocks are in non-decreasing order (from left to right). In order to prove that Condition 4 is satisfied, it suffices to show that whenever it makes sense. We distinguish two cases -either the column l 0 crosses some mblock, say at row i, (Case I) or it does not (Case II). We start with Case I.