Modified-Shortening: Modifying method of constructing quantum codes from highly entangled states

There is connection between classical codes, highly entangled pure states (called k-uniform or absolutely maximally entangled states), and quantum error correcting codes (QECCs). This can lead to a systematic method of constructing stabilizer QECCs by starting from a highly entangled state and removing one party. In this method of constructing codes, the description of the stabilizer was on the center of attention so far. But, to build quantum devices we also need a theory instructing us how to decode and encode using a QECC without losing the protection against errors. We show how to find explicit codespace and encoding procedure besides presenting the stabilizer formalism. We then modify the method to produce a new set of stabilizer QECCs with a larger code subspace compared with the existed construction. In this method, we start from a highly entangled state and construct stabilizer QECC without removing any party. More precisely, this construction produces quantum codes with parameters [[n,1,[n/2]]]_q, starting from an absolutely maximally entangled state or alternatively quantum code [[n,0,[n/2]+1]]_q.


I. INTRODUCTION
Quantum error correction is one of the main challenges in the field of quantum computation and one of our attempts to use multipartite entangled states in applications [1], [2]. Investigation of the connection between quantum codes and existing classical error correcting codes led us to understand the structure of quantum codes and their connection to the highly entangled subspaces [3]- [6]. The extra knowledge on code parameters of classical codes provides a great advantage to construct quantum codes [7]. Therefore, one general framework is constructing QECCs from known classical codes and the associated entangled states [8].
It is now well understood that a particular type of highly entangled pure quantum states, called k-uniform states, are a special set of quantum codes [6]. k-uniform (or for simplicity k-UNI) states are genuinely entangled, which refers to the fact that all subsystems of size k are correlated and that the states are not separable concerning any possible splitting of k subsystems. Those that are maximally entangled along with any splitting the parties into two groups are called Absolutely Maximally Entangled (AME) i.e., ⌊n/2⌋ -UNI states.
A given k-UNI state represents stabilizer quantum error correcting code that encodes a logical qudit into a subspace spanned by the entangled state [6]. A large set of quantum codes, called stabilizer QECC, describe in terms of the stabilizer of the codewords. stabilizer is a finite Abelian sub-group of the (generalized) Pauli group that leaves every element from the codespace invariant. One general method of constructing these states is based on the connection between them and a family of classical error correcting codes known as maximum distance separable (MDS) [9], [10].
Besides using classical codes to construct QECCs, another method that simplifies the task of finding quantum codes is using old codes to find new ones. Implementing modification techniques on a given code can produce new code with different parameters [5], [7]. A less trivial manipulation is to remove the last party of a given stabilizer code and convert it into a new code with n − 1 parties. In this method, the derived code from a stabilizer code is a stabilizer code [2].
Combining the two methods of using the highly entangled states associated with the classical error correcting codes and the method of constructing new codes from old ones, leads to constructing a family of QECCs. With this technique, one starts from a k-UNI state and construct a new QECC by taking partial trace over one particle, i.e., code with n − 1 parties. Repeating this technique produces a family of QECCs with a different set of code parameters [11], [12]. This method can be called Shortening which refers to the connection it has with the classical codes and the subspace spanned by the highly entangled states [5], [13]- [15].
The Shortening process is one of the practical methods of constructing stabilizer codes. So far, in the previous literatures, the description of the stabilizer formalism of the codes was in the center of attention [7], [12]. But, we will want to use quantum codes in the operation of communication and quantum computers that behave correctly in the presence of errors, therefore, we will need to describe how to encode and decode. In this paper, we work on constructing a set of QECCs starting from a k-UNI states and using the Shortening process. Unlike previous construction, we present the list of the codewords and the encoding and decoding procedures besides presenting the stabilizer formalism. We discuss the structure of the highly entangled subspace of the quantum stabilizer codes.
Then, this leads us to a new systematic way of constructing quantum codes from old ones. This method that we call modified-Shortening, produce QECCs without removing any party (without taking the partial trace). Therefore, quantum stabilizer codes with larger codespace can be constructed and that improve the achievable rate compared with the existed construction. For this, we start from an AME state and without removing any party we construct a QECC whose codewords are all AME states. More precisely, starting from an AME state or alternatively the quantum code [[n, 0, ⌊n/2⌋+1]] q , we show how to produce a family of quantum error correcting codes [[n, 1, ⌊n/2⌋]]q. We present the codewords as well as presenting the stabilizer formalism.
The structure is as follows: we discuss the connection between classical codes, k-UNI states, and QECCs in Sections II. Then, in Sections III we introduce the Shortening process which provides a family of QECCs starting from a given k-UNI state. This method is different from previous construction as we could list the codewords as well as presenting the stabilizer formalism. In Section IV we show how the modified-Shortening can produce QECCs starting from an AME state without removing any party.

II. CONNECTION BETWEEN CLASSICAL CODES, k-UNIFORM STATES, AND OPTIMAL QUANTUM CODES
We begin with a short review of general definitions of classical error correcting codes, k-UNI states and quantum error correcting codes. Then we describe how these concepts are connected.
Classical error correcting codes. A linear classical code C = [n, k, d H ] q consists of q k codewords with length n over q-level dits. Protection against errors on some of the letters of the codewords is possible only if n > k. The Hamming distance d H between two codewords is defined as the number of positions in which they differ. The large Hamming distance is essential in guaranteeing to recover the original message if noise causes t = ⌊(d H − 1)/2⌋ dits of the code. The Singleton bound provides upper bound on the maximally achievable minimal Hamming distance between any two codewords In general, for the encoding procedure of the linear code C , it is possible to define a generator matrix, in which the codewords are all possible linear combinations of the rows of such matrix. Generator matrix is a k × n matrix over a finite field GF (q), and it can always be written in the standard form [16, Chapter 1] where ½ k is identity matrix with size k ×k, and A ∈ GF (q) k×(n−k) . Codes that satisfy the Singleton bound (1) refer to as maximum-distance separable (MDS) codes. For a given MDS code any subset of up to k columns of G k×n are linearly independent, or equivalently every square submatrix of A is nonsingular [16,Chapter 11] [17]- [19].
k-UNI states. Pure states of n distinguishable qudits, with this property that all k-qudit reductions of the whole system are maximally mixed are called k-UNI states. A k-UNI state |ψ in Hilbert space H(n, q) = C ⊗n q , denote in what follows by k-UNI(n, q), whenever where S c denotes the complementary set of S. The minimal number of terms for which the condition of maximally mixed marginals can be fulfilled is q k , and the corresponding states called k-UNI states of minimal support.
Moreover, the Schmidt decomposition shows that a state can be at most ⌊n/2⌋ -UNI, i.e., k ≤ ⌊n/2⌋. The ⌊n/2⌋ -UNI states are called AME states for short.
A subclass of pure k-UNI states can be constructed by taking equally weighted superposition of all the q k codewords of MDS codes C = [n, k, n − k + 1] q with k ≤ ⌈n/2⌉ in the computational basis i.e., As a side remark, note that it is always possible to find a suitable generator matrix G k×n over finite field GF (q), all one has to do is to take a power of a prime q sufficiently large [16,Chapter 11] [8], [10], [18] (for the existence conditions see [10], [18] A quantum code with maximum possible integer for d, is considered as an optimum code (for further details see [21,Definition 6]). We recall a strict definition of distance d, and the conditions under which a quantum code is a valid code in the next section.
As mentioned, there is a direct correspondence between minimal support k-UNI states and the classical MDS codes. And, an explicit closed form expression for k-UNI state, Eq. (4), can be constructed from a given MDS code [8]- [10]. Also, in particular, a k-UNI state of n particles corresponds to a pure QECC of distance d = k + 1 andk = 0, denotes by C = [[n, 0, k + 1]] q [6]. This connection is coming from the fact that construction of some QECCs can be reduced to that of the classical linear error correcting codes.
The QECCs that associate to k-UNI(n, q) states are pure stabilizer codes [4], [5], [7], [14]. This type of codes are a family of QECC with this property that can be shortened, which means, the existence of a quantum pure stabilizer code (or equivalently a k-UNI state), implies the existence of a stabilizer QECC with larger code dimension whose spanning vectors are k − 1-UNI states, see [ After recalling the theory of the Shortening process, in this section, we show explicitly how to find codewords of the quantum codes. For this, we first discuss the conditions under which a subspace is a QECC. Then, we show explicitly how to construct code space of a quantum code [[n − r, r, d − r]] q , with r > 0, from a given k-UNI(n, q) state, such that the subspace are spanned by k−r-UNI states of n−r parties. This construction requires n−r ≤ q+1 [21], which refers to the existence condition of the classical MDS codes.
To define stabilizer quantum codes we first introduce some notations. Let's start with the definition of Pauli operators which act on a given local element of a product state |j as follows with ω := e i 2π/q the q-th root of unity. X and Z are unitary, traceless operators, and X q = Z q = ½. For We call operators that are tensor products of powers of Pauli operators Pauli strings.
Let {|ψ m } m∈[qk] be a set of orthonormal quantum states of n qudits in which a subspace C is spanned by this set of states. We denote by [qk] a string ofk symbols that range from 0 to q − 1, e.g., for the case thatk = 1 we have, [q] := (0, . . . , q − 1). A stabilizer group S, contains an abelian subgroup of Pauli strings (exclude −½), such that the non-trivial subspace C of H(n, q) is stabilized by S. A stabilized subspace C defines a quantum code space as follows: S is generated by n −k independent stabilizer operators S i , so that the code space C encodesk logical qudits into n physical qudits. The code C with parameters [[n,k, d]] q is a valid QECC if it obeys Knill-Laflamme conditions [14], [20] ∀m, for all E, F with wt(E † F ) < d. wt is the weight of an operator which denotes the number of sites on which it acts non-trivially. And, d is the minimal number of local operations that act on single sites to create a non-zero overlap between any two different states |ψ m and |ψ m ′ .
Every k-UNI state constructed from a classical MDS code is a stabilizer quantum code [[n, 0, k + 1]] q and has the advantage of extra knowledge of the classical codes. Thus, as we discussed, the state |ψ , Eq. (4), is the codeword of the quantum stabilizer code [[n, 0, k + 1]] q . The stabilizer formalism provided in Appendix VI-A. In the following we show how codes with larger code dimensionk > 0, can be obtained by the Shortening process.

A. First step:
As the first step, Shortening procedure can convert the code [[n, 0, k + 1]] q , into a code with parameters [[n − 1, 1, k]] q such that a logical qudit is encoded in a q-dimensional subspace spanned by k − 1-UNI states. In the following we show how to find the code space We start with the generator matrix G k×n = [½ k |A k×(n−k) ], Eq. (2), and remove the last row. Because the generator matrix has the standard form, the k-th column contains only 0s, consider now that we remove this column too.
With these changes the generator matrix now transform to a matrix of size k − 1 × n − 1 where we denote by G i for the result of removing the i-th row and column from the original matrix G. G k contains k − 1 linearly independent columns, therefore, G k is a valid generator matrix to construct MDS code is a k − 1-UNI state. Now, we define the operator M which is a string of the X such that the vector of exponents is the last row of the G matrix that we removed, concretely where we denoted non-zero elements of matrix G k×n by g i,j . In the following lemma, we show how the Pauli string M , defines k − 1-UNI states based QECC.
Lemma 1: Consider k − 1-UNI state |ψ 0 constructed from the generator matrix G k , Eq. (11), and M operator, a Pauli string constructed from elements of the last row of the G k×n matrix, Eq. (12). The subspace ψ m |W |ψ m ′ = 0 , where wt(W ) < d H − 1 = n − k. Note that all set of states |ψ m are k − 1-UNI, as acting with local unitaries does not change the entanglement properties.
Code space C = span{|ψ m } is a QECC if and only if it satisfies two conditions. (i) In the presence of errors one is able to distinguish two different codewords [4], [7]. Considering this, Eq. (15) implies that the minimum number of single-qudit operations that are needed to create a non-zero overlap between any two orthogonal states is n − k.
Therefor, for all errors E and F with the weight of 1 < wt(E † F ) < n − k and all m, The above condition is a direct consequence of the fact that the minimum distance between two different codewords |ψ m and |ψ m ′ is d H − 1 = n − k. (ii) In addition, it is sufficient to distinguish different errors when they act non-trivially on a given codeword |ψ m (see Eq. (28) of [4]). As the states |ψ m for every m ∈ [q], are k − 1-UNI state, then one gets for errors E † F that act non-trivially on any subset of less than k − 1 sites, i.e., wt(E † F ) < k − 1. As we always have n − k ≥ k − 1, then, considering the conditions (i) and (ii) the code distance is d = min(n − k + 1, k) = k. By the definition Eq. (9), one can conclude that the subspace C is a [[n, 1, d = k]] q QECC.
As a side remark, note that if the M operator, (12), contains only k of the X operators with the vector of exponent described before, the distance of the code is still d = k. The stabilizer formalism is presented in Appendix VI-B.

B. Second step:
The second step of the Shortening procedure convert the code [[n − 1, 1, k]] q , into a code with parameters [[n − 2, 2, k − 1]] q with a q 2 -dimensional subspace spanned by k − 2-UNI states. For the code resulting to yield the corresponding codewords, in a similar manner, we remove the k − 1 and the k-th columns and rows of the original generator matrix The structure is the same as the first step, and hence, it is obvious that G k−1, k is generator matrix of an MDS code [n − 2, k − 2, n − k + 1] q . Therefore, k − 2-UNI state |ψ 00 can be constructed via Two Pauli strings M 1 and M 2 that involve X operators can be defined such that the vector of exponents are the k − 1 and the k-th rows of G k×n while both the k − 1 and k-th columns are removed: Finally, the code space C is spanned by k − 2-UNI states By the same argument as before the fact that state |ψ 00 and operators M 1 and M 2 are linear combination of rows of the matrix G k×n (or codewrods of MDS code [n, k, n − k + 1] q ), where two parties are removed, leads us to the first Knill-Laflamme condition (i): with wt(E † F ) < d H − 2 = n − k − 1. The second condition can be satisfied because: (ii) the subspace C is manifestly spanned by orthogonal k − 2-UNI states |ψ m1,m2 so that, for wt(E † F ) < k − 2. This leads that the code distance of the QECC with code space C = span{|ψ m1,m2 }, is d = min(n − k, k − 1) = k − 1. An example of how to find codewrods with the second step of the Shortening process is provided in Appendix VI-C.
In general, the Shortening procedure can be repeated k − 1 times and the codes with less particles and higher code dimensionk can be obtained (see Table I). Note, these codes can be optimal, saturating the quantum Singleton bound, only if k = ⌊n/2⌋. This means that the starting point should be an AME state.

REMOVING PARTIES
Shortening is a method of finding new QECCs from old ones. The structure of the Shortening process is based on removing one particle (taking partial trace) from a given stabilizer QECC at each step. In the previous section, we introduced a different view of constructing new QECCs from a k-UNI state, that allows us to produce codewords and study the structure of the code space. In this section, we discuss a method of constructing new codes from old codes without removing any parties starting from an AME state. We call this method modified-Shortening. To introduce the method, we first review a systematic way of constructing generator matrices to construct classical Starting point: Step 1: Step 2: Step r: Step k − 1: We start by introducing the concept of so-called Singleton arrays, which is a special case of Cauchy matrix and have this property that all its square sub-matrices are non-singular [16,Chapter 11]. For any finite field GF (q), a Singleton array is defined to be where γ is an element of GF (q) called primitive element [ . By taking n = q + 1, one can construct the generator matrix G ⌊n/2⌋×n = [½|A] and hence an MDS code. AME state |φ 0 over GF (q) has closed form expression [8] |φ Then, we consider the ⌊(q + 1)/2⌋ + 1-th row of the Singleton array S q containing ⌈(q + 1)/2⌉ − 1 elements (1, a ⌈q/2⌉ , a ⌈q/2⌉+1 , . . . , a q−2 ). Using this, we define the Pauli string M of length n, such that, the first ⌊n/2⌋ elements are identity matrices (as we have in each row of the generator matrix G ⌊n/2⌋×n ), the vector of exponents of the X operators is the ⌊(q + 1)/2⌋ + 1-th row of S q , and it contains one Z operator as the n-th element, i.e., where n = q + 1.
such that all |φ m are AME states of n parties with n = q + 1.
Proof: Codewords produce by acting the M m operators for m ∈ [q] on the AME state |φ 0 . M contains X operators with the vector of exponents ⌊(q + 1)/2⌋ + 1-th row of S q and one Z operator. More explicitly, M = M X M Z where, For the purpose of the proof we discuss how M X and M Z act on the state |φ 0 separately.
First, we show M m X for m ∈ [q] that acts non-trivially on ⌈n/2⌉ − 1 cites provides states with distance ⌈n/2⌉ − 1. To do this, we take sub-matrix A ′ , the matrix A ′ contains one more row and one less column. Using A ′ , one can construct generator matrix G ′ (⌊n/2⌋+1)×n = [½ ⌊n/2⌋+1 |A ′ ] and MDS code C ′ = [n, ⌊n/2⌋ + 1, ⌈n/2⌉] q . After one step of Shortening we get which is the generator matrix of MDS code [n − 1, ⌊n/2⌋ , ⌈n/2⌉] q , and hence, AME state |ψ ′ 0 of n − 1 parties can be written as is the same generator matrix as G ⌊n/2⌋×n = [½|A] if one deletes the last column, as well as saying, the state |ψ ′ 0 is the same as state |φ 0 , Eq. (27), without considering the last party. As we discussed in the Shortening procedure, the M operator that produces subspace with specific distance d is a string of only X operators with the vector of exponent last row of the generator matrix. In this case, M is the pauli string of X operators with the exponent vector ⌊(q + 1)/2⌋ + 1-th row of S q . Therefore, M contains the first n − 1 Pauli operators in M X , Eq. (30). We can get the set of states The same proof as that of Lemma 1 establishes the distance between every two different states of the above set of states is d H − 1 = ⌈n/2⌉ − 1. This shows that because |ψ m is the same as |φ m if one removes the last party, the distance between every two states |φ m and |φ m ′ is at least d = ⌈n/2⌉ − 1 . Therefore, for two different codewords, we have This is one of the Knill-Laflamme condition Eq. (9), in which, two different codewords should be distinguishable in the presents of errors when act non-trivially on wt (E † F ) < d.

Moreover
, errors E and F should not be able to change an encoded state for the weight wt (E † F ) < d, i.e., for two given codewords it is necessary to have φ m |E † F |φ m = φ m ′ |E † F |φ m ′ . As unitary operator M dose not change the entanglement property, therefore all the states |φ m are AME states, then In Table II we compare QECCs one can construct from AME state |φ 0 Eq.(27), using Shortening and modified-Shortening processes. We can see that the modified-Shortening provides quantum codes with smaller local dimension q given n than previous codes. This method also provides explicit codewords besides stabilizer formalism.

V. CONCLUSIONS
We have studied the remarkable relation between classical optimal codes, maximally multipartite entangled states, and quantum error correcting codes. This study, in general, can lead to the construction of optimal quantum error correcting codes from highly entangled subspaces. We discussed a method that starts from a k-UNI state and by removing one party constructs a set of stabilizer QECCs. Our construction provided the list of codewords besides presenting the stabilizer formalism. Along the way, we have also shown this method can be repeated and how to find the codewords in each step. Then, we extended the connection between classical codes, k-UNI states and quantum codes to provide codes with larger code subspace compared with the existed constructions. We have shown how to modify the method to produce QECCs starting from an AME state without removing any party. Let us first start with more details of the classical codes. For a given linear classical code with parameters [n, k, d H ] q , a generator matrix is a k × n matrix, G k×n , over a finite field GF (q). Given the generator matrix in standard form Eq. (2), is useful to study the condition which is necessary to construct an MDS code. Also it is useful to find a matrix which is called parity check matrix H (n−k)×n = [−A T |½ n−k ]. The two matrices are related with GH T = 0. This shows that the rows of the H matrix specify parity checks that all the codewords must satisfy. As the codewords are all possible linear combinations of the rows of G matrix over GF (q), obviously any linear combination of the rows of H is also a parity check, i.e., in general, there are q n−k parity checks. Let C be an MDS code [n, k, n − k + 1] q , over GF (q), the following statements for the generator matrix G k×n and parity check matrix H (n−k)×n are equivalent (see [16,Chapter 11 The k-UNI state of minimal support constructed from MDS code, Eq. (4), recall is the plus one eigenstate of n stabilizer operators. The generators are divided into two sets, X stabilizers, S X , and Z stabilizers, S Z , where the matrix elements of G k×n are denoted by g i,j and that of the code's parity check matrix H (n−k)×n by h i,j . The first k generators involve the X operators (the X stabilizers). This forms a set of stabilizers, because adding the same codeword to all other codewords is just a relabeling of the terms in the summation. Another set of stabilizers, n − k of them, can be constructed from the Z operators (the Z stabilizers). The action of product of stabilizers S Z leave state |ψ invariant because of the fact that G k×n (H (n−k)×n ) T = 0, (see also [7], [8]).

B. Stabilizers group of the code state space
The stabilizer formalism of the state |ψ 0 , Eq. (11), can be found by taking advantage of the connection to the classical coding theory. Therefore, based on Eq. (38), one can find n − 1 generators of the stabilizers of the state |ψ 0 , where the matrix elements of G k are denoted by g i,j and that of the code's parity check matrix by h i,j .
For the code C := span({|ψ m m∈[q] }) ⊂ C ⊗n−1 q with |ψ m = M m |ψ 0 , Eq. (13) to be a stabilizer code, we need to generate a stabilizer group that stabilizes the given subspace. The set of the stabilizers S C should satisfy the following equality ∀i, m : S C i M m |ψ 0 = M m |ψ 0 .
The above condition implies that every S C i ∈ S C must commute with M (and hence M m ) operator and stabilize the state |ψ 0 . The M operator is a vector of exponents of the X operators. Therefore, the k − 1 generators of the stabilizer group of the state |ψ 0 that involve X operators, S ψ0 X , (first equation of Eq. (43)), commute with M and hence leave the state |ψ m invariant. In order to find the stabilizers S C i that involve the Z operators, we first consider direct computation for any two Pauli strings. For two Pauli strings A and B the commutator follows where A = ( A X , A Z ) and b is defined in the same way and, This implies that the stabilizers of |ψ 0 that involves Z operators,S ψ0 Z (the second equation of Eq. (43)), satisfies the Eq. (40) if for all m it holds that m m X · S ψ0 Z = 0 mod q. This is also equivalent to just having m X · S ψ0 Z = 0 mod q, where the vector m X represent the vector of exponents in the M operator. The vector of exponents S ψ0 Z of the Z stabilizers is constructed from linear combination of the rows of the parity check matrix H (n−k)×(n−1) . Therefore, S ψ0 Z = vH, represent the vector of exponents of the S ψ0 Z , where v ∈ GF (q) n−k . The string of Z operators that leave |ψ m = M m |ψ 0 invariant are those vector of exponents such that m X . vH = 0. In general, the generator S C of the stabilizer groups of C are The number of generators for the stabilizers group that involve X operators is k − 1 and that involve Z operators is n − k − 1, in total, they are n − 2 generators.

C. Example for the construction of Shortening process
As an example of our construction, we start with AME(6, 5) which is constructed from an MDS code [6,3,4] which is the generator matrix of the code [4, 1, 4] 5 . It yields the following closed form expression of the codewords |ψ 0 , which is a 1-UNI state of 4 parties As it is discussed before, in this case there are two M 1 and M 2 operators The set of states |ψ m1,m2 = M m1