Construction of Quantum Subspace Codes of the Simplex Type for - PowerPoint PPT Presentation

Construction of Quantum Subspace Codes of the Simplex Type for Quantum Network Coding Gabriella Akemi Miyamoto Joint work with Wanessa Gazzoni and Reginaldo Palazzo Jr. July 27, 2018

Contents I 1 Mathematical Concepts 2 Construction of Simplex Subspace Codes 3 Labeling the Subspaces of SSC by Classical Codes Leading to qSSC Codes

Mathematical Concepts Consider | ψ � as an arbitrary pure state with n -qubits given by | ψ � = α 0 | 00 · · · 0 � + α 1 | 00 · · · 1 � + · · · + α 2 n − 2 | 11 · · · 0 � + α 2 n − 1 | 11 · · · 1 � , s =0 | α s | 2 = 1 . As shown in [5], the where α 0 , ..., α 2 n − 1 ∈ C and � 2 n − 1 content of each ket of | ψ � consists of a binary sequence of length n . The space containing all the binary sequences of length n (binary n -tuples) is known as the Hamming space , denoted by H n 2 and the Hamming distance is denoted by d H . Let A ψ be the set consisting of M binary n -tuples associated with the content of the kets of | ψ � .

The Meyer-Wallach entanglement measure, [4], is given by 1 n Q ( | ψ � ) = 4 � D ( ι j (0) | ψ � , ι j (1) | ψ � ) , n j =1 where D ( ι j (0) | ψ � , ι j (1) | ψ � ) = � ψ | ι j (0) , ι j (0) | ψ �� ψ | ι j (1) , ι j (1) | ψ � − |� ψ | ι j (0) , ι j (1) | ψ �| 2 , for every j ∈ { 1 , · · · , n } . Q ( | ψ � ) = 0 if and only if | ψ � is a separable state, and Q ( | ψ � ) = 1 if and only if | ψ � is a pure state with maximum global entanglement , [4]. 1 Consider ι j ( b ) : ( C 2 ) ⊗ n − → ( C 2 ) ⊗ n − 1 as a linear map defined as the action on its product basis: ι j ( b ) ( | x 1 � ⊗ · · · ⊗ | x n � ) = δ bx j | x 1 � ⊗ · · · ⊗ | x j − 1 � ⊗ | x j +1 �⊗ ⊗ · · · | x n � where x i ∈ { 0 , 1 } and b ∈ { 0 , 1 } .

Example 1 | ψ GHZ � = √ ( | 000 � + | 111 � ) : 2 1 √ ι 1 (0) | ψ GHZ � = ι 2 (0) | ψ GHZ � = ι 3 (0) | ψ GHZ � = ( | 00 � ) , 2 1 √ ι 1 (1) | ψ GHZ � = ι 2 (1) | ψ GHZ � = ι 3 (1) | ψ GHZ � = ( | 11 � ) . 2

D = � ψ GHZ | ι j (0) , ι j (0) | ψ GHZ �� ψ GHZ | ι j (1) , ι j (1) | ψ GHZ � |� ψ GHZ | ι j (0) , ι j (1) | ψ GHZ �| 2 − √ √ √ � 2 � 2 � 2 � � � |� 00 | 11 �| 2 = 1 / 2 � 00 | 00 � 1 / 2 � 11 | 11 � − 1 / 2 √ √ � 4 � 2 � � = 1 / 2 ( � 0 | 0 �� 0 | 0 � ) ( � 1 | 1 �� 1 | 1 � ) − 1 / 2 ( � 0 | 1 �|� 0 | 1 � ) √ � 4 � = 1 / 2 = 1 / 4 , j = 1 , 2 , 3. 3 3 Q ( | ψ GHZ � ) = 4 D ( ι j (0) | ψ GHZ � , ι j (1) | ψ GHZ � ) = 4 1 � � 4 = 1 . 3 3 j =1 j =1

After some algebraic manipulations it is possible to rewrite Q ( | ψ � ), obtaining n Q ′ ( | ψ � ) = 4 1 � ( z j · ( M − z j )) , (1) n M 2 j =1 where z j denotes the number of n -tuples in A ψ with “0” in the j -th position and M denotes the number of kets in the state | ψ � . Note that equation (1) is a new description of the Meyer-Wallach entanglement measure for arbitrary pure states with equal amplitudes.

Based on the equivalence between Q and Q ′ , the next step is to establish the conditions that A ψ has to satisfy such that a pure state (with equal amplitudes) achieves Q ′ = 1. In this direction, we have, from (1), that n n ( z j · ( M − z j )) = nM 2 Q ′ ( | ψ � ) = 4 1 � � ( z j · ( M − z j )) = 1 n M 2 4 j =1 j =1 n � 2 � z j − M � → = 0 . 2 j =1

Theorem An arbitrary pure state | ψ � with n-qubits is a state with maximum global entanglement, if and only if, the associated set A ψ , with d H > 1 , satisfies z j = M / 2 , for every j ∈ { 1 , · · · , n } .

All classical binary linear codes satisfy the condition z j = M / 2 for every j ∈ { 1 , · · · , n } [6]. Therefore, every classical binary linear code may be associated with a pure state which achieves maximum global entanglement . On the other hand, the set of codewords of a nonlinear code is not closed under the mod 2 sum operation. However, it is known that some classes of binary nonlinear codes satisfy the condition z j = M / 2, for every j ∈ { 0 , · · · , n } .

Theorem All binary error-correcting codes (linear and nonlinear codes) satisfying the condition z j = M / 2 , for every j ∈ { 1 , · · · , n } may be associated with pure states achieving maximum global entanglement.

Examples Simplex 2 , Reed-Muller, Nordstrom-Robinson and Preparata codes. 2 The binary simplex code is the dual of a Hamming code, that is, the generator matrix G of the binary simplex code is the same as the parity-check matrix of the Hamming code.

Example The code defined by { 000 , 111 } has M = 2, z 1 = z 2 = 1 and the condition z j = M / 2 is satisfied, for j = 1 , 2. This code can be 1 associated to | ψ � = 2 ( | 000 � + | 111 � ) = | ψ GHZ � , a maximum √ entanglement state: Q ( | ψ GHZ � ) = 1 .

Definition A signal set S is geometrically uniform if, given any two signals s and s ′ in S , there exists an isometry u s , s ′ taking s to s ′ while leaving S invariant, that is, u s , s ′ ( s ) = s ′ u s , s ′ ( S ) = S . Intuitively, a signal set is geometrically uniform if all of its signals are spread evenly on the surface of an n -dimensional sphere.

The set of all vector subspaces of F n q is denoted by P q ( n ). The set of all k -dimensional subspaces of F n q is called a Grassmannian and it is denoted by G q ( n , k ), where 0 ≤ k ≤ n . Definition A subspace code C is a non-empty set of P q ( n ), C is called an ( n , d ) q code, where n is the dimension of the vector space F n q . In the case that the subspace code is contained within a single Grassmannian, G q ( n , k ), i.e., all the subspaces have the same dimension, this subspace code is called a constant dimension code and its parameters are given by ( n , k , d ).

The set of all vector subspaces of F n q is denoted by P q ( n ). The set of all k -dimensional subspaces of F n q is called a Grassmannian and it is denoted by G q ( n , k ), where 0 ≤ k ≤ n . Definition A subspace code C is a non-empty set of P q ( n ), C is called an ( n , d ) q code, where n is the dimension of the vector space F n q . In the case that the subspace code is contained within a single Grassmannian, G q ( n , k ), i.e., all the subspaces have the same dimension, this subspace code is called a constant dimension code and its parameters are given by ( n , k , d ).

Definition Let U and V be subspaces of F n q . The subspace distance, d S , between U and V is given by: d S ( U , V ) = dim ( U ) + dim ( V ) − 2 dim ( U ∩ V ) .

Construction of Simplex Subspace Codes (SSC) Step 1 Determining the first codeword (a vector subspace): choose this codeword in such way that the distances between all the codewords will be constant. For small parameters n , k and d S 3 the determination of such codeword is very easy. However, for parameters sufficiently large, it can be very hard. So we developed an algorithm in order to help to provide the first codeword. Step 2 Determining the remaining codewords: each vector in each codeword is a cyclic shift of vectors in the first one. 3 n is the dimension of the space F n 2 , k is the Grassmannian dimension and d S is the code distance.

Example Consider the vector space F 15 2 . The subspace code C with parameters (15 , 7 , 8), that is, C 1 ⊂ G 2 (15 , 7) and the code distance is d S = 8, has the following codewords 4 T 0 = � 0 � T 8 = � e 8 , e 9 , e 10 , e 12 , e 13 , e 1 , e 3 � T 1 = � e 1 , e 2 , e 3 , e 5 , e 6 , e 9 , e 11 � T 9 = � e 9 , e 10 , e 11 , e 13 , e 14 , e 2 , e 4 � T 2 = � e 2 , e 3 , e 4 , e 6 , e 7 , e 10 , e 12 � T 10 = � e 10 , e 11 , e 12 , e 14 , e 15 , e 3 , e 5 � T 3 = � e 3 , e 4 , e 5 , e 7 , e 8 , e 11 , e 13 � T 11 = � e 11 , e 12 , e 13 , e 15 , e 1 , e 4 , e 6 � T 4 = � e 4 , e 5 , e 6 , e 8 , e 9 , e 12 , e 14 � T 12 = � e 12 , e 13 , e 14 , e 1 , e 2 , e 5 , e 7 � T 5 = � e 5 , e 6 , e 7 , e 9 , e 10 , e 13 , e 15 � T 13 = � e 13 , e 14 , e 15 , e 2 , e 3 , e 6 , e 8 � T 6 = � e 6 , e 7 , e 8 , e 10 , e 11 , e 14 , e 1 � T 14 = � e 14 , e 15 , e 1 , e 3 , e 4 , e 7 , e 9 � T 7 = � e 7 , e 8 , e 9 , e 11 , e 12 , e 15 , e 2 � T 15 = � e 15 , e 1 , e 2 , e 4 , e 5 , e 8 , e 10 � 4 Let e i = { 00 ... 010 ... 0 } be the vector having “1” in the i -th coordinate, and “0” in the remaining ones and let � V � represents the linear subspace generated by a set V .

Proposition Let C = { T 1 , T 2 , ..., T n } be a simplex subspace code as described in the construction above. The map f i , j : C → C of the metric space ( P n q , d S ) defined by  f i , j ( T i ) = T j  f i , j ( T j ) = T i f i , j ( T k ) = T k , k � = i , j  is an isometry for 1 ≤ i , j ≤ n. Proposition C is a geometrically uniform code with the isometry defined above. Proposition For k = 1 and any given value of n, there exists a SSC with subspace distance d S = 2 .

Proposition Let C = { T 1 , T 2 , ..., T n } be a simplex subspace code as described in the construction above. The map f i , j : C → C of the metric space ( P n q , d S ) defined by  f i , j ( T i ) = T j  f i , j ( T j ) = T i f i , j ( T k ) = T k , k � = i , j  is an isometry for 1 ≤ i , j ≤ n. Proposition C is a geometrically uniform code with the isometry defined above. Proposition For k = 1 and any given value of n, there exists a SSC with subspace distance d S = 2 .