Invariant tensors are states in the SU(2) tensor product representation that are invariant under SU(2) action. They play an important role in the study of loop quantum gravity. On the other hand, perfect tensors are highly entangled many-body quantum states with local density matrices maximally mixed. Recently, the notion of perfect tensors has attracted a lot of attention in the fields of quantum information theory, condensed matter theory, and quantum gravity. In this work, we introduce the concept of an invariant perfect tensor (IPT), which is an n-valent tensor that is both invariant and perfect. We discuss the existence and construction of IPTs. For bivalent tensors, the IPT is the unique singlet state for each local dimension. The trivalent IPT also exists and is uniquely given by Wigner's j 3 symbol. However, we show that, surprisingly, 4-valent IPTs do not exist for any identical local dimension d. On the contrary, when the dimension is large, almost all invariant tensors are asymptotically perfect, which is a consequence of the phenomenon of the concentration of measure for multipartite quantum states.Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.10 The 3D spatial geometry, quantized by spin-networks, serves as the initial data of 4D gravity. 11 The closure equation generalized to a constant curvature polyhedron has been proposed in [8-10]. © 2017 IOP Publishing Ltd and Deutsche Physikalische Gesellschafti i = + . The spin angular momentum operators have commutation relations given by J J i J , a b abc c = [ˆˆ]ˆ. An n-valent tensor is a vector n y ñ | in n . Let the total spin operator be J J. 1 i n i 1 å = =ˆ( ) 12 In the context of bulk-boundary duality, the Ryu-Takayanagi formula conjectures that the CFT entanglement entropy of a spatial region A is proportional to the minimal area of the bulk codim-2 surface attached to A ¶ [21]. 13 The recent AdS/CFT computation reveals that a black hole should be dual to a quantum system of the fastest scrambling [23], which is consistent with the scrambling feature of perfect tensors.3 4 3 4¢ ¢ is given byThe second equality is true for any element, which means thatJ j M J J d 0 1 2 1 2 ñ {| }implies that the identity operator has the unique decomposition of J M J M , , , d J j M J J 0 2 2 1 + = . For any given intermediate angular momentum J 1 2 , formulating the state as the first scheme: C J M J M , , B 2 J M J J M M J J 0,0 1 2 1 2 1 2 1 2 m m m m J j M J