Quantum error correction codes and absolutely maximally entangled states
Paweł Mazurek, Máté Farkas, Andrzej Grudka, Michał Horodecki, Michał Studziński
QQuantum error correction codes and absolutely maximally entangled states
Pawe(cid:32)l Mazurek,
1, 2
M´at´e Farkas, Andrzej Grudka, Micha(cid:32)l Horodecki,
1, 2 and Micha(cid:32)l Studzi´nski Institute of Theoretical Physics and Astrophysics,National Quantum Information Centre, Faculty of Mathematics,Physics and Informatics, University of Gda´nsk, 80-308 Gda´nsk, Poland International Centre for Theory of Quantum Information, University of Gda´nsk, 80-308 Gda´nsk, Poland Faculty of Physics, Adam Mickiewicz University, 61-614 Pozna´n, Poland
For every stabiliser N -qudit absolutely maximally entangled state, we present a method for deter-mining the stabiliser generators and logical operators of a corresponding quantum error correctioncode. These codes encode k qudits into N − k qudits, with k ≤ (cid:98) N/ (cid:99) , where the local dimension d is prime. We use these methods to analyse the concatenation of such quantum codes and link thisprocedure to entanglement swapping. Using our techniques, we investigate the spread of quantuminformation on a tensor network code formerly used as a toy model for the AdS/CFT correspon-dence. In this network, we show how corrections arise to the Ryu-Takayanagi formula in the caseof entangled input state, and that the bound on the entanglement entropy of the boundary state issaturated for absolutely maximally entangled input states. I. INTRODUCTION
A powerful connection between theories of gravity andquantum field theory is the so-called AdS/CFT corre-spondence [1]. This relates gravitational theories in theinterior (bulk) of anti-de Sitter (AdS) spaces to confor-mal field theories (CFT) defined on the boundary of thespace. This duality allows for translating calculationsfrom weakly interacting, and thus more tractable, grav-itational models to strongly interacting quantum fieldtheoretical models. A particularly interesting feature ofthis correspondence is the so-called Ryu-Takayanagi for-mula [2], which states that the entropy of a subregion ofthe boundary is proportional to the length of the geodesicconnecting the endpoints of this subregion.Recently the quantum information community is alsofocusing attention on the AdS/CFT correspondence.In [3], the authors present a toy model of the correspon-dence, using a discrete tiling of the AdS space. Theydefine a tensor on every tile, and every neighbouring ten-sor pair is connected by an edge, forming a network oftensors. On every edge, they define a quantum systemof local dimension d . Interestingly, the Ryu-Takayanagiformula holds for every concise region of the boundary,as long as the above tensors are perfect tensors . Thismeans that if we treat the tensors as mappings betweentheir edges, they define an isometry from any of their n edges to any of their remaining m edges, as long as n ≤ m .Perfect tensors are important in quantum informa-tion theory, because they correspond to absolutely max-imally entangled (AME) states, that is, states that aremaximally entangled on every bipartition. AME statesare in close connection with quantum error correctioncodes [4], and they are a necessary resource in multipar-tite quantum communication and quantum secret sharingschemes [5]. From the mathematical perspective, manyAME state constructions can be obtained from combina-torial designs [6, 7].In this work, we exploit the connection between AME states and quantum error correcting codes in order togain understanding on the toy model of the AdS/CFTcorrespondence. We treat the perfect tensors of the net-work as AME states and therefore quantum codes, whichallows us to describe the spread of the encoded quantuminformation in the network using code concatenation andthe stabiliser formalism of quantum codes.We derive the form of logical operators and stabilisergenerators for every code emerging from an n + m quditstabiliser AME state, that encodes n qudits into m quditswith n ≤ m . The proof is valid for local dimensions be-ing prime, and is based on properties of graph states, intowhich every stabiliser AME state can be mapped underlocal unitaries. Apart from this, for a specific perfect ten-sor network, we analyse the spread of entanglement alongthe boundary of the code, with respect to the nature ofinput entanglement. We find that the upper bound of theentanglement entropy of the boundary state is saturatedfor AME input states for every connected bipartition ofthe boundary. We postulate that this optimality is an-other characteristic of AME states.This paper is organised as follows: In Section II wepresent the stabiliser formalism and describe the con-struction and applications of AME states, followed bythe relation between stabiliser AME states and stabiliserquantum error correction codes. In Section III, we de-scribe how to construct tensor networks using quantumcode concatenation and entanglement swapping. Then inSection IV we apply this procedure to analyse the spreadof quantum information on a network used in the toymodels of the AdS/CFT correspondence. Finally, in Sec-tion V we apply our formalism to study the entanglemententropy of the boundary state of the perfect tensor net-work with respect to the initial input state. a r X i v : . [ qu a n t - ph ] O c t II. STABLISER CODES AND AME STABILISERSTATES
We start by showing the connection between statesthat are maximally entangled on a bipartition, and quan-tum error correction codes. We assume that both thestate and the code can be described within the stabiliserformalism, which we outline below.
A. Stabiliser formalism
Let us denote a Hilbert space of N qudits by H = ⊗ Ni =1 H i , where H i ∼ = C d , and the space of linear opera-tors acting on this Hilbert space by B ( H ). On this space,we call a state | Φ (cid:105) a stabiliser state if it is uniquely (up toa global phase factor) characterised by an Abelian sub-group S of the Pauli group P N ⊂ B ( H ) via the equations s | Φ (cid:105) = | Φ (cid:105) ∀ s ∈ S , (1)that is, the group S stabilises | Φ (cid:105) . The Pauli group P N isgenerated by { ω c X a j j Z b j j } a j ,b j ,c =0 ,...,d − j =1 ,...,N , where ω = e π i d is the d -th root of unity, and X j and Z j are the gener-alised Pauli operators acting on the j -th qudit, definedas X = d − (cid:88) k =0 | k + 1 (cid:105)(cid:104) k | Z = d − (cid:88) k =0 ω k | k (cid:105)(cid:104) k | , (2)where, | (cid:105) , . . . , | d − (cid:105) is the computational basis. Notethat for any nontrivial stabiliser state we have that ω c I / ∈S if c (cid:54) = 0, where I is the identity operator on H .It is often convenient to describe the stabiliser group S by its generating set, G ( S ), which is a set of mutuallycommuting operators from P N . If a group S ⊆ P N doesnot define a unique state, but a higher dimensional sub-space, we will call this subspace H log = {| Φ (cid:105) : ∀ s ∈ S , s | Φ (cid:105) = | Φ (cid:105)} a logical subspace , and it defines a quantumcode. In this case, there exist linearly independent oper-ators from P N commuting with S , but linearly indepen-dent from S . We will call such operators logical operators of the code, and, due to the fact that they commute with S , we define them modulo G ( S ). Let D = N − |G ( S ) | ,where |·| denotes the cardinality of the set. Then we havethat dim H log = d D , and H log ∼ = H L, ⊗· · ·⊗H L,D , where H L,j ∼ = C d is the j -th logical subsystem of the code. Theset of operators commuting with S can then be writtenas { X L, , Z L, , . . . , X L,D , Z
L,D } ∪ S , where ( X L,j , Z
L,j )form pairs of logical operators acting non-trivially onlyon H L,j , such that Z L,j X L,j = ωX L,j Z L,j . Therefore,logical operators labeled with different indices commutewith each other.
B. AME states
Consider a state | Φ (cid:105) on a Hilbert space of N qu-dits: H = ⊗ Ni =1 H i , H i ∼ = C d . For the set of indices I = { , . . . , N } numbering the qudits, define a biparti-tion I = A ∪ B into two, non-empty sets of indices. Wewill call a state | Φ (cid:105) maximally entangled (ME) with re-spect to the bipartition H = H A ⊗ H B , if it satisfiesthe following equivalent criteria (where, without loss ofgenerality, we assume that m := | B | ≤ (cid:98) N (cid:99) ):(i) Tr A (cid:0) | Φ (cid:105)(cid:104) Φ | (cid:1) = I B (ii) S (cid:0) Tr A | Φ (cid:105)(cid:104) Φ | (cid:1) = S ( B ) = | B | log d (iii) | Φ (cid:105) = √ d m (cid:80) k ∈ Z md | k (cid:105) B . . . | k m (cid:105) B m | Ψ( k ) (cid:105) A , with (cid:104) Ψ( k ) | Ψ( k (cid:48) ) (cid:105) = δ k,k (cid:48) .Above, Tr A is the partial trace over H A , S ( ρ ) = − Tr ρ log ρ is the von Neumann entropy, and δ m,n is theKronecker delta. If a state | Φ (cid:105) satisfies the above criteriafor every bipartition, it is called absolutely maximallyentangled (AME). From condition (i) we see that if astate is ME with respect to every bipartition such that m = (cid:98) N (cid:99) , then it is also AME.For qubit systems ( d = 2), AME states exist for N = 2parties ( | Ψ (cid:105) ∝ | (cid:105) + | (cid:105) ) and N = 3 parties ( | Ψ (cid:105) ∝| (cid:105) + | (cid:105) ), but for N = 4 qubits AME states donot exist [8]. By numerically minimizing the function (cid:80) B : | B | = (cid:98) N (cid:99) Tr ρ B , where ρ B = Tr A ρ , AME states for N = 5 , N = 7,complementing the earlier result on their absence for N ≥ N , there always existslarge enough d such that AME states exist [5].An interesting application of AME states comes fromtheir connection to isometries. From the defining equa-tion (i) it can be easily seen that any decomposition of anAME state into two subsystems | Φ (cid:105) = (cid:80) a,b T a,b | a (cid:105)| b (cid:105) isuniquely associated with a map T : | b (cid:105) → (cid:80) a T a,b | a (cid:105) thatsatisfies (cid:80) a T † b (cid:48) ,a T a,b = δ b (cid:48) ,b , i.e. that preserves all the in-ner products. That is, transformations associated withAME states are isometries with respect to every divisioninto two subsystems, and are therefore perfect tensors.This property is what allows AME-based tensor net-works to be used as toy models for the AdS/CFT corre-spondence, because it ensures that an equivalent of theRyu-Takayanagi formula holds in the network. However,we will show in Section V that this formula gains somecorrections if the input state of the tensor network is en-tangled.From now on, we will assume that the AME statescorresponding to the tensors of the network are stabiliserstates. To facilitate the description of networks in thestabiliser language, below we describe the stabilisers andlogical operators of quantum codes corresponding to sta-biliser AME states. C. Construction of stabiliser codes from AMEstates
Every N -qudit stabiliser state can be mapped to a so-called graph state by Clifford operations acting locally onthe qudits, whenever d is prime [11]. Since local Cliffordoperations do not affect entanglement between parties,every stabiliser AME state can be mapped to a graphstate using local Cliffords. The generators of the sta-biliser group of an N -qudit graph state are N operatorsof the form g i = X i (cid:81) Nj =1 Z A i,j j . Here, X i and Z j aregeneralised Pauli matrices acting on the i -th ( j -th) qu-dit, and A i,j is the adjacency matrix of a weighted graph,that is, a symmetric matrix with elements from the set { , . . . , d − } , such that all its diagonal elements are 0.Let K = { k , . . . , k m } be any subset of the indices { , . . . , N } , indexing the generators (or the subsystems),such that m ≤ (cid:4) N (cid:5) . We denote by A i /K the i -th row of the matrix A with entries removed from the columnsindexed by K . Theorem 7 of [12] shows that, whenever A is the adjacency matrix of an AME state, the set ofvectors { A i /K } i ∈ K is linearly independent in Z N − md . Letus choose K to encompass the first m rows/columns of A , that is, k i = i .Therefore, the first m rows of A , truncated to theirlast N − m entries, are linearly independent. Thus, if weconstruct a matrix of these truncated rows, we can per-form Gauss-Jordan elimination to bring it to a reducedrow echelon form. Then, by swapping the columns of A , we can move the m columns containing just one ‘1’to the last m positions. Since multiplication of the rowsof the adjacency matrix A corresponds to multiplicationof the generators of the corresponding AME state, theabove operations on A can be directly translated intooperations on the list of stabiliser generators: (1) Gauss-Jordan elimination, (2) reordering or columns: N N (cid:122) (cid:125)(cid:124) (cid:123) X . . . . . . . . . . . . . . . . . . . . . . . .. . . X . . . . . . . . . . . . . . . . . . . . .. . . . . . X . . . . . . . . . . . . . . . . . .. . . . . . . . . X . . . . . . . . . . . . . . .. . . . . . . . . . . . X . . . . . . . . . . . .. . . . . . . . . . . . . . . X . . . . . . . . .. . . . . . . . . . . . . . . . . . X . . . . . .. . . . . . . . . . . . . . . . . . . . . X . . .. . . . . . . . . . . . . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) m. . . . . . X m m (1) −−→ m (cid:122) (cid:125)(cid:124) (cid:123) X . . . X Z Z m (cid:122) (cid:125)(cid:124) (cid:123) . . .X X . . . Z . . .. . . X X Z Z. . . . . . . . . X . . . . . . . . . . . . . . .. . . . . . . . . . . . X . . . . . . . . . . . .. . . . . . . . . . . . . . . X . . . . . . . . .. . . . . . . . . . . . . . . . . . X . . . . . .. . . . . . . . . . . . . . . . . . . . . X . . .. . . . . . . . . . . . . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) m. . . . . . X m m (2) −−→ (2) −−→ m (cid:122) (cid:125)(cid:124) (cid:123) X . . . X Z . . . m (cid:122) (cid:125)(cid:124) (cid:123) Z X X . . . . . . Z . . . X X Z Z. . . . . . . . . X . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . X . . . . . .. . . . . . . . . . . . X . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . X . . .. . . . . . . . . . . . . . . . . . . . . . . . X. . . . . . . . . . . . . . . X (cid:124) (cid:123)(cid:122) (cid:125) m. . . . . . . . . m m (3) −−→ m (cid:122) (cid:125)(cid:124) (cid:123) X . . . X . . . . . . . . . m (cid:122) (cid:125)(cid:124) (cid:123) Z X X . . . . . . . . . . . . Z . . . X X . . . . . . . . . Z. . . . . . . . . X . . . . . . . . . . . . . . .. . . . . . . . . . . . X . . . . . . . . . . . .. . . . . . . . . . . . . . . X . . . . . . . . .. . . . . . . . . . . . . . . . . . X . . . . . .. . . . . . . . . . . . . . . . . . . . . X . . .. . . . . . . . . . . . . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) m. . . . . . X m m (4) −−→ Above, every row corresponds to a different generator,every column to a qudit, and an entry with possible 1 or Z a assignment is marked by “ . . . ”. Note that swappingqudits in (2) may destroy the diagonal structure of the X operators. However, by proper swapping of rows from m + 1 to N , we can recover this structure in step (3).Finally, by properly multiplying the last N − m gen-erators by the first m generators (4), we can completelyeradicate the Z operators from the last m positions ofthe last N − m generators: (4) −−→ m (cid:122) (cid:125)(cid:124) (cid:123) X . . . X . . . . . . . . . m (cid:122) (cid:125)(cid:124) (cid:123) Z X X . . . . . . . . . . . . Z . . . X X . . . . . . . . . Z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X . . . . . . . . . . . . . . . . . . X . . . . . . . . . . . . . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) m X m m Note that here, in some places marked by “ . . . ” inthe first m columns of the last N − m rows, X or XZ operators may also be present. Generally speaking, weobtain a list of N generators with a fixed structure on thelast m qudits (full separation and local diagonalisationwith respect to Z and X operators), and possibly mixedentries on the remaining qudits.We will show that these N generators truncated tothe first N − m qudits provide the generators and logicaloperators of an error correction code encoding m quditsinto N − m qudits. For this, we need to show that all theoperators are linearly independent, and we have N − m commuting operators providing the stabiliser generators.Moreover, the remaining 2 m operators form pairs satis-fying the same commutation relations as X and Z Paulioperators, and they commute with the N − m genera-tors.We prove linear independence of the truncated oper-ators by contradiction: assume that the truncated listis not linearly independent. Then, by multiplying somegenerators with each other, we are able to obtain a trun-cated generator that consists only of identity operators.However, this contradicts with [12, Theorem 7], whichstates that any m generators of an AME state truncatedto length N − m are linearly independent (and this prop-erty is preserved under multiplying the generators with each other).To see commutativity, notice that the truncated gen-erators from rows m + 1 , . . . , N − m − m positions that were truncated. These truncated gen-erators are therefore stabiliser generators of an error cor-rection code. On the other hand, the truncated operatorsfrom the first and last m rows commute with the newlyestablished generators of the code, and for every trun-cated operator from the first m rows, there is exactlyone truncated operator from the last m rows with whichit commutes the same way as Pauli X and Z operators.Also notice that each of these operators commuteswith all other truncated operators apart from their pairs.Therefore, the truncated operators from the first and last m rows form logical operators of the above quantum errorcorrection code. As m ∈ { , . . . , (cid:4) N (cid:5) } , every stabilizerAME state of N qubits generates at least (cid:4) N (cid:5) differentstabiliser codes. Each of these codes encodes m logicalqudits into N − m physical ones.We will illustrate the connection between AME statesand quantum error correction codes by an example of a6-qubit AME state [13], characterised by the followinglist of stabiliser generators: X Z Z X
X Z Z X X X Z Z Z X X Z X X X X X XZ Z Z Z Z Z −→ − Y Z Y Z − Z X Z
XY Y Z Z Z Z X X − Z Y Y Z
X Z Z X · · · · · · · · · ·
41 (3)Here, even without referring to local Clifford transfor-mations, we can obtain a desired, alternative list of sta-biliser generators by multiplying them with each other (the numbers on the right side of each new generator de-note which original generators have to be multiplied toobtain it). From the reasoning presented above for thegeneral case, it follows that the obtained form explicitlyexpresses stabiliser generators and logical operators for1 →
5, 2 →
4, and 3 → → G ( S ) = { Y Y Z Z, ZZX X, − ZY Y Z , XZZX } (4)and Z L, = − Y ZY , X L, = − ZXZ , (5)for the 2 → G ( S ) = {− ZY Y Z, XZZX } (6)and Z L, = − Y ZY , X L, = − ZXZ ,Z L, = Y Y Z , X L, = ZZX , (7)and finaly for the 3 → G ( S ) = ∅ (8)and Z L, = − Y ZY, X L, = − ZXZ,Z L, = Y Y Z, X L, = ZZX,Z L, = − ZY Y, X L, = XZZ. (9)
III. CONCATENATION OF CODES ANDENTANGLEMENT SWAPPING BETWEENSTATES
Having achieved the above standard form of the list ofstabiliser generators, we are ready to describe a partic-ular measurement process performed on AME states. Itwill enable us to understand the construction of perfect tensor networks, and associated error correction codes, asa concatenation of codes corresponding to AME states.First, let us recall the rules of updating the list of gen-erators of a stabiliser state after performing a measure-ment [14, Chapter 10.5]. If the measured observable canbe constructed from the generators, then we do not needto update the list. If the measured observable cannotbe constructed as a product of generators, but commuteswith all of them, then we add it to the list of generatorswith a phase factor depending on the measurement out-come. Finally, if the observable cannot be constructedas a product of generators and does not commute withsome number of generators, then it replaces one generatorwith which it does not commute (possibly multiplied bya phase factor depending on the measurement outcome),while we multiply the other non-commuting generatorsby the generator that we removed from the list – thisprocess assures that all the new generators are linearlyindependent and mutually commuting.Using these rules and our previous machinery, we pro-vide a connection between concatenation of stabilisercodes and entanglement swapping. Let us start with twostabiliser states, which, by the position of their stabilisergenerators on the common stabiliser list (see below), wewill call left and right states, and denote by L and R ,respectively. The left state is defined on N L qudits andis arbitrary, while the right one is defined on N R qudits,and we take it to be an AME state. Each stabiliser of the L state can be represented in the form L i ⊗ σ i , where L i is a tensor product of Pauli operators on qudits labeledfrom l to l N L − , while σ i is a single qudit Pauli oper-ator on qudit l N L , and i = 1 , . . . , N L . Similarly, everystabiliser of the state R is of the form σ j ⊗ R j , with R j acting on qudits r , . . . , r N R , σ j acting on qudit r , and j = 1 , . . . , N R . The list of N L + N R stabilisers of thejoint state, being a product of the left and right states,takes the form presented in the first table below: L X . . . L Z . . . L . . . L X . . . . . .L N L Z . . . . . . Z R . . . X R . . . R . . . R . . . . . . R N R l l . . . l N L r r . . . r N R XX −−→ L X . . . L Z Z R L . . . L X . . . . . .L N L Z Z R . . . X X . . . . . . X R . . . R . . . R . . . . . . R N R l l . . . l N L r r . . . r N R ZZ −−→ L X X R L Z Z R L . . . L X X R . . .L N L Z Z R . . . X X . . . . . . Z Z . . . . . . R . . . R . . . . . . R N R l l . . . l N L r r . . . r N R −→ −→ L R L R L . . . L R . . .L N L R . . . X X . . . . . . Z Z . . . . . . R . . . R . . . . . . R N R l l . . . l N L r r . . . r N R → L R L R L . . . L R . . .L N L R . . . R . . . R . . . . . . R N R l l . . . l N L − r . . . r N R Note that in the first table we already exploited thefact that the R state is AME. Therefore, its stabilisergenerators can be written such that on an arbitrarily se-lected qudit there are only two generators acting non-trivially: one as X , and the other as Z . In the firststep, we measure the observable XX on qudits l N L and r . A non-commuting generator in the seventh line ofthe table is replaced by the measured operator (we as-sume here that the measurement outcome was +1), andthe other non-commuting generators from the second andsixth lines are multiplied by the removed generator fromthe seventh line (all marked with red in the first table).In the next step, we measure the observable ZZ onthe same pair of qudits and apply the same update pro-cedure (the newly modified generators are marked withblue, the generator in the eighth line will be replaced bythe operator being measured, while the first and fourthgenerators will be multiplied by it, all marked with red).At the end, we see that the qudits labeled by l N L and r are maximally entangled with each other, and are corre-lated with no other part of the system. We trace themout by erasing the two corresponding columns and rowsof the table.The resulting table, representing the stabiliser gener-ators of the state formed in the process of entanglementswapping on a pair of qudits shared between the state L and the AME state R , consists of two groups: theoriginal stabilisers of the state R , trivially extended twothe whole system (marked in orange), and the stabilisersof the state L , multiplied by truncated stabilisers of thestate R (marked in purple). Note that this multiplicationis conditioned on the l N L -th component of the stabiliserof the L state: if it is X ( Z ), the stabiliser will be multi-plied by R ( R ). Due to the initial AME structure of thestate R , this construction can be extended to the case ofarbitrary number of ( XX, ZZ ) measurements performedon up to min {(cid:98) N L / (cid:99) , (cid:98) N R / (cid:99)} , arbitrarily selected, dis-joint pairs of qudits.Noting that R and R are the logical X and Z opera-tors of the code described by the AME state R , encoding1 qudit into N R − L is AME as well, then the list of sta-biliser generators of the state resulting from entangle-ment swapping is composed simply from the originalgenerators of states R and L , acting non-trivially onlywithin their original domains, and two generators beingthe product of logical X and Z operators: Z L, ⊗ Z L, and X L, ⊗ X L, (truncated to qubits which do not be-long to the qubit pair on which the measurements wereperformed). Clearly, such a state does not have to beAME. IV. SPREAD OF QUANTUM INFORMATIONIN AME-BASED NETWORKS
Having connected entanglement swapping with theconcatenation procedure, we are now able to exploit thestabiliser formalism to investigate the spread of quantuminformation in perfect tensor networks based on stabiliser
FIG. 1: A multiqubit state (left) vs. an AME state, specif-ically a 3-qubit GHZ state (right). Vectors corresponding toevery uncontracted leg (a qubit) are taken from the corre-sponding column of the matrix formed by the list of genera-tors of the two states. Therefore, different entries of a givenvector describe the action of different stabilisers on the corre-sponding qubit.FIG. 2: After measuring the adjacent qubits in the Bell basis(contracting the leg connecting the polygon with the triangle),the state on the uncontracted legs (qubits) is fully describedby the updated generators. Note that the generators of thepolygon do not change (apart from the one associated withthe contracted leg), while the generators formerly belongingto the AME state get updated according to the vectors of thequbits that are contracted.
AME states. The final code can be formed by a gradualexpansion of the network through entanglement swap-ping, where in each step at least one of the states is AME.Note that we can extend the procedure presented in theprevious section to describe the case when more than twoAME states are the subject of entanglement swapping.Due to the restriction min {(cid:98) N L / (cid:99) , (cid:98) N R / (cid:99)} of the num-ber of qubits of a given AME state that can participatein this gradual concatenation, the networks that accom-modate the spread of quantum information must to havea non-decreasing number of branches, starting from thepoint of the origin. Here, we analyse a network investi-gated in [3], based on a tiling of a two dimensional spacewith negative curvature (see Fig. 4). The tiling consists FIG. 3: Example of the emergence of stabilizers throughconcatenation. Generators are given in transparent framesand logical operators are given in blue frames. From the top:2 codes, based on 3-qubit GHZ states, each encoding one qubitinto two qubits. The stabiliser generator of the resulting codesis ZZ . Then, 2 of the resulting qubits get encoded into a 4qubit code, based on 6-qubit AME state. The generators ofthis code appear. On the other hand, the previously existingstabilisers get concatenated through logical operators of the4-qubit code, and the final list of stabilizers is given in thebottom frame. Thus this total set of stabilisers consists oftwo classes: (i) those in the first two lines, which are ”old”stabilizers, just encoded through the 4 qubit code (ii) those inthe last two lines – the ”new” ones – which are the stabilisersof the 4-qubit code. of pentagons, which are connected by edges, representingqubits. Each of the pentagons is associated with the 6-qubit AME state, described in Eq. (3). The associationis that each of the legs coming from an edge of a pen-tagon corresponds to a qubit state, while the sixth qubitis associated with a leg that emerges from the center ofthe pentagon, and stretches above the plane in which thefigure lies.In most cases, we will treat the interior red dots as log-ical qudits, and the outer white dots as physical qudits,encoding the logical qudits into the code space of theemerging quantum code. The encoding of the operator X from one of the interior points of the networks (logicalqudit) to the boundary (physical qudits) can be under-stood in the following way (see Fig. 5): an X operatorthat acts originally on a logical qudit, gets encoded as aresult of entanglement swapping between qudit pairs of FIG. 4: Tiling of a surface with negative curvature into pen-tagons. Red dots inside every pentagon, as well as the whitedots coming out of the pentagons, represent qubits. Each ofthe pentagons represents the 6-qubit AME state. Contractedlegs represent entanglement swapping. consecutive AME states of the tensor network, and theencoded operators on each level are determined by thelogical operators of the corresponding quantum codes.Whether 1 →
5, 2 → → X operator is given modulo stabilisers, thatare defined by the concatenation as well. Note that thefinal form of the logical operators and generators resultsfrom concatenation, that can be viewed in terms of per-forming commuting measurements (entanglement swap-ping on disjoint qudit pairs). Therefore, the order of theconcatenation, governed by the direction of arrows, canonly influence the final form of logical operators and gen-erators up to multiplication by stabilisers, i.e. it cannothave any measurable consequences. V. ENTANGLEMENT CORRECTIONS TO THERYU-TAKAYANAGI FORMULA
In this section we will apply our techniques to analysecorrections to the Ryu-Takayanagi formula in the pen-tagon code in [3]. Namely, as noted in [3], if the inputstate is product, then the Ryu-Takayanagi formula holds,
FIG. 5: Spread of the operator X applied to the red entry(logical qudit) of the largest pentagon. First, it gets encodedin the form of the logical operator X = − ZXZ → A consists of qubits labeled by 1 , . . . , s B −
1, whilethe region B consists of qubits labeled by 13 , . . . , s B . Here, s B = 13. but when it is entangled it may not hold anymore. Herewe provide a simple bound on the entanglement entropyof a connected region of the boundary, and check whetherit is saturated by various input states for the network inFig. 6, which has 2 layers of concatenation.We find that among the selected several states, onlyAME states saturate the bound (15).Let us recall the Ryu-Takayanagi formula, which statesthat if we divide the boundary of the network into twodisjoint but connected regions A and B , then the entan-glement entropy of the boundary state along this cut is S A ∝ | γ A | , (10)where γ A is a geodesic in the interior of the network,connecting the two points where A and B meet. The“volume” of this geodesic, | γ A | , in our case is simply thenumber of legs the geodesic cuts through.In general, the state on the boundary can be describedas | Ψ (cid:105) = (cid:88) i,a,b,k,k (cid:48) ψ k,k (cid:48) P ( k ) a,i | a (cid:105) A Q ( k (cid:48) ) k,i | b (cid:105) B = (cid:88) k,k (cid:48) ψ k,k (cid:48) (cid:88) i | P ( k ) i (cid:105) A | Q ( k (cid:48) ) i (cid:105) B , (11)where A and B are the above mentioned disjoint con-nected regions with some bases {| a (cid:105)} and {| b (cid:105)} definedon them. Any path γ that connects the two points onthe boundary where A and B meet divides the operatorcorresponding to the full tensor network into two oper-ators, P and Q . The index i runs through the quditsthis path cuts through, whereas k and k (cid:48) run throughthe (red) input qudits of P and Q . Therefore, theseoperators are defined via P : | i (cid:105)| k (cid:105) → (cid:80) a P ( k ) a,i | a (cid:105) and Q : | i (cid:105)| k (cid:48) (cid:105) → (cid:80) a Q ( k (cid:48) ) b,i | b (cid:105) . Note that in Eq. (11), we define | P ( k ) i (cid:105) A = (cid:80) a P ( k ) a,i | a (cid:105) A and | Q ( k (cid:48) ) i (cid:105) B = (cid:80) b Q ( k (cid:48) ) k,i | b (cid:105) B .When the input state (cid:80) k,k (cid:48) ψ k,k (cid:48) | k (cid:105)| k (cid:48) (cid:105) = | k ∗ (cid:105)| k (cid:48) ∗ (cid:105) isproduct with respect to the bipartition into inputs of P and Q , by adjusting the basis we can simply write | Ψ (cid:105) = (cid:88) i | ˜ P i (cid:105) A | ˜ Q i (cid:105) B , (12)where | ˜ P i (cid:105) = | P k ∗ i (cid:105) and | ˜ Q i (cid:105) = | Q k (cid:48)∗ i (cid:105) . Now, if P and Q are isometries, we have (cid:104) P ki | P k (cid:48) i (cid:48) (cid:105) = δ i,i (cid:48) δ k,k (cid:48) = (cid:104) Q ki | Q k (cid:48) i (cid:48) (cid:105) ,and therefore (cid:104) ˜ P i | ˜ P i (cid:48) (cid:105) = δ i,i (cid:48) = (cid:104) ˜ Q i | ˜ Q i (cid:48) (cid:105) . Therefore,Tr B | Ψ (cid:105)(cid:104) Ψ | ∝ I A , and the entanglement entropy is thelogarithm of the rank of the state (which is d | γ | ), i.e. S (Tr B | Ψ (cid:105)(cid:104) Ψ | ) = log ( d ) · | γ | , (13)which is precisely the Ryu-Takayanagi formula if we min-imise over γ . If P and Q are not isometries, the aboveconstitutes an upper bound on the entropy. In the gen-eral case, when the input state might be entangled, wehaveTr B | Ψ (cid:105)(cid:104) Ψ | = (cid:88) k,k (cid:48) ,i, ˜ k, ˜ k (cid:48) , ˜ i ψ k,k (cid:48) ψ ˜ k, ˜ k (cid:48) (cid:104) Q k (cid:48) i | Q ˜ k (cid:48) ˜ i (cid:105)| P ki (cid:105)(cid:104) P ˜ k ˜ i | , (14)and therefore the entropy of the reduced state can surpassthe product state bound, as we get S (Tr B | Ψ (cid:105)(cid:104) Ψ | ) ≤ log ( d ) · ( | γ | + | P | ) , (15)where | P | is the number of input states of P . The abovebound is valid for any choice of the path γ , and thereforewe choose it to minimise | γ | + | P | (or just | γ | for the prod-uct case, in which case γ = γ A is a geodesic connectingthe two points where A and B meet). For the encoding setting presented in Fig. 6, we inves-tigate the entropy of the reduced state of the boundarywith respect to different bipartitions (qubit 20 vs. therest, qubits 19 and 20 vs. the rest, and so on), and differ-ent input states (6-qubit AME state, 6-qubit GHZ state,one singlet pair (otherwise product), and a fully productstate). We use the methods from the previous section togenerate the list of stabilisers of the code on the bound-ary, as well as the encoded version of generators thatstabilise the code state: X Y X X Y X Y Z Y
Y Z Y Y Z Y Y Z Y Z X Z
Z X Z Y Z Y X Y X X Y X Y Z
Z X Z Y Z Y Y Z Y Z X Z
Z Y Y Z
X Z Z X
Z Y Y Z
X Z Z X
Z Y Y Z
X Z Z X
Z Y Y Z
X Z Z X
Z Y Y Z
X Z Z X
Z X
Z X X X X X X Y Y X Y Y X Z Z X X Z X X Z X Z Z X Y Y X X X X
Z X
Z X X Y YY Y X Z Y Z Y Y Y X Z Z Z Z Z Y Z Y X X X X X X X X X X X X X X X X X X X XZ Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z (16)
Above, columns correspond to qubits on the boundary,with the leftmost column corresponding to qubit 1, andthe rightmost column to qubit 20. The first 14 rows arethe stabiliser generators of the boundary state, while thelast 6 rows are the encoded versions of the generators ofthe 6-qubit AME state (LHS of (3)).For different input states, the last 6 rows are replacedwith encoded versions of the respective stabiliser genera-tors: for the 6-qubit GHZ state these will correspond tothe encoded versions of
XXXXXX , ZZ ZZ ZZ
11 , 111 ZZ
1, and 1111 ZZ . For the “singlet pair”input state, we take the encoded versions of 111 XX ZZ X X X
111 and 11111 X (qubits4 L and 5 L are entangled), while for a product state wetake the encoded versions of X X X X
11, 1111 X X . Above, we take the orderof the input qubits to be 1 L , . . . , L from Fig. 6.The entanglement of the reduced state can be easilydeduced from the list of the generators [15], via S (Tr B | Φ (cid:105)(cid:104) Φ | ) = | S AB | , (17)where | S AB | is the minimal number of generators actingnon-trivially on both A and B .This minimisation is over all possible representationsof the generators of the stabiliser group. From [15], wesee that this representation can be found in the followingway: Let us denote by P B the projection mapping g A ⊗ g B to I A ⊗ g B , and the list of generators by { g i } i . If theset { P B ( g i ) } i contains n pairs of operators that do notcommute with each other, but otherwise all operatorscommute, then we have | S AB | = 2 n .The above representation can be achieved by manip-ulating the initial generator list, such as (16): For a se-lected cut A : B , we check the commutation of everyelement with all other elements on the subsystem B . If0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ s B ● AME ■ GHZ ● singlet ■ product FIG. 7: Entanglement entropy S (Tr B | Ψ (cid:105)(cid:104) Ψ | ) of the bound-ary states of the code from Fig. 6 for different input states(AME, GHZ, singlet or product, see the text for details), andfor boundary points of the region B being qubits s B and 20. two elements do not commute, then they will form a non-commuting pair. Then, we update the list by multiplyingall the elements that do not commute with one elementof the pair by the other element of the pair. For example,the list of the generators of the state stabilized by (16),and for region B comprising qubits 18, 19 and 20, can bebrought to the following form on the last three qubits: Y YX Z Z Y Z X XY X ZZ Y Y , (18) from which we infer S (Tr B = { , , } | AM E (cid:105) enc (cid:104)
AM E | enc ) = 3 . (19)We present the value of the entanglement entropy fordifferent cuts and different encoded states in Fig. 7.We see that for an arbitrary cut the AME state andthe product state give limiting values of the entangle-ment entropy. The increased values of entropy (comparedto product state input) for the “singlet state” input arepresent only for cuts that leave the entangled qubits, 4 L and 5 L , on the opposite sides of the shortest path. How-ever, we observe the difference between entanglement en-tropy even in cases when the same amount of entangled qubits is left on the opposite side of the optimal cut,i.e. between encoded GHZ and AME states. For a par-ticular cut, the bound (15) is set by the geometry of theproblem (Fig. 6), and one can check that the boundarystate encoding the AME state saturates it. For example,for an AME input state and the division of the boundarydetermined by s B = 13, the shortest path in the bulkis shown in Fig. 6 by a dash-dotted line. Dash-dottedcircles surround logical inputs that account for the cor-rection to the standard Ryu-Takayanagi formula.Note that the corrections to the Ryu-Takayanagi for-mula correspond to the entanglement entropy of the in-put state with respect to the bipartition defined by thegeodesic. This is maximised by AME states, therefore weconjecture that AME input states saturate the bound. VI. CONCLUSIONS
We have shown that each stabilizer AME state on N qudits leads to a quantum error correction code encod-ing k logical qudits into N − k physical qudits, whenever k = 0 , , . . . , (cid:98) N/ (cid:99) . We provide an algorithm to deter-mine the stabiliser generators and logical operators of theemerging code. We have also shown that entanglementswapping between two states such that one of them isAME enables to easily calculate the spread of quantuminformation in AME-based tensor networks, by concate-nating stabilisers and logical operators.Moreover, we have shown that the Ryu-Takayanagi for-mula acquires corrections in the case of tensor networkcodes with entangled inputs. In the particular case ofthe pentagon code with two layers of concatenation, wereport that the bound on the corrections to the formulais saturated by AME input states for any bipartition ofthe boundary. We conjecture that AME input states sat-urate the entropy bound for boundary states of perfecttensor structures. Acknowledgments.