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 = α0|00 · · · 0+α1|00 · · · 1+· · ·+α2n−2|11 · · · 0+α2n−1|11 · · · 1, where α0, ..., α2n−1 ∈ C and 2n−1

s=0 |αs|2 = 1. As shown in [5], the

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 Hn

2 and

the Hamming distance is denoted by dH. Let Aψ 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 by1 Q(|ψ) = 4 n

n

• j=1

D(ιj(0)|ψ, ιj(1)|ψ), where D(ιj(0)|ψ, ιj(1)|ψ) = ψ|ιj(0), ιj(0)|ψψ|ιj(1), ιj(1)|ψ − |ψ|ιj(0), ιj(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].

1Consider ιj(b) : (C2)⊗n −

→ (C2)⊗n−1 as a linear map defined as the action

• n its product basis: ιj(b) (|x1 ⊗ · · · ⊗ |xn) = δbxj |x1 ⊗ · · · ⊗ |xj−1 ⊗ |xj+1⊗

⊗ · · · |xn where xi ∈ {0, 1} and b ∈ {0, 1}.

Example

|ψGHZ = 1 √ 2 (|000 + |111) : ι1(0)|ψGHZ = ι2(0)|ψGHZ = ι3(0)|ψGHZ = 1 √ 2 (|00), ι1(1)|ψGHZ = ι2(1)|ψGHZ = ι3(1)|ψGHZ = 1 √ 2 (|11).

D = ψGHZ|ιj(0), ιj(0)|ψGHZψGHZ|ιj(1), ιj(1)|ψGHZ − |ψGHZ|ιj(0), ιj(1)|ψGHZ|2 =

• 1/

√ 2 2 00|00

• 1/

√ 2 2 11|11 −

• 1/

√ 2 2 |00|11|2 =

• 1/

√ 2 4 (0|00|0) (1|11|1) −

• 1/

√ 2 2 (0|1|0|1) =

• 1/

√ 2 4 = 1/4, j = 1, 2, 3. Q(|ψGHZ) = 4 3

3

• j=1

D (ιj(0)|ψGHZ, ιj(1)|ψGHZ) = 4 3

3

• j=1

1 4 = 1.

After some algebraic manipulations it is possible to rewrite Q(|ψ),

• btaining

Q′(|ψ) = 4 n 1 M2

n

• j=1

(zj · (M − zj)) , (1) where zj denotes the number of n-tuples in Aψ 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 Aψ has to satisfy such that a pure state (with equal amplitudes) achieves Q′ = 1. In this direction, we have, from (1), that Q′(|ψ) = 4 n 1 M2

n

• j=1

(zj · (M − zj)) = 1

n

• j=1

(zj · (M − zj)) = nM2 4 →

n

• j=1
• zj − M

2 2 = 0.

Theorem

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

All classical binary linear codes satisfy the condition zj = 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 zj = M/2, for every j ∈ {0, · · · , n}.

Theorem

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

Examples

Simplex2 , Reed-Muller, Nordstrom-Robinson and Preparata codes.

2The 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, z1 = z2 = 1 and the condition zj = M/2 is satisfied, for j = 1, 2. This code can be associated to |ψ =

1 √ 2 (|000 + |111) = |ψGHZ, a maximum

entanglement state: Q(|ψGHZ) = 1.

Definition

A signal set S is geometrically uniform if, given any two signals s and s′ in S, there exists an isometry us,s′ taking s to s′ while leaving S invariant, that is, us,s′(s) = s′ us,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 Fn

q is denoted by Pq(n). The set

• f all k-dimensional subspaces of Fn

q is called a Grassmannian and

it is denoted by Gq(n, k), where 0 ≤ k ≤ n.

Definition

A subspace code C is a non-empty set of Pq(n), C is called an (n, d)q code, where n is the dimension of the vector space Fn

• q. In

the case that the subspace code is contained within a single Grassmannian, Gq(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 Fn

q is denoted by Pq(n). The set

• f all k-dimensional subspaces of Fn

q is called a Grassmannian and

it is denoted by Gq(n, k), where 0 ≤ k ≤ n.

Definition

A subspace code C is a non-empty set of Pq(n), C is called an (n, d)q code, where n is the dimension of the vector space Fn

• q. In

the case that the subspace code is contained within a single Grassmannian, Gq(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 Fn

• q. The subspace distance, dS,

between U and V is given by: dS(U, V ) = dim(U) + dim(V ) − 2dim(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 dS 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.

3n is the dimension of the space Fn 2, k is the Grassmannian dimension and

dS is the code distance.

Example

Consider the vector space F15

2 . The subspace code C with

parameters (15, 7, 8), that is, C1 ⊂ G2(15, 7) and the code distance is dS = 8, has the following codewords4 T0 = 0 T8 = e8, e9, e10, e12, e13, e1, e3 T1 = e1, e2, e3, e5, e6, e9, e11 T9 = e9, e10, e11, e13, e14, e2, e4 T2 = e2, e3, e4, e6, e7, e10, e12 T10 = e10, e11, e12, e14, e15, e3, e5 T3 = e3, e4, e5, e7, e8, e11, e13 T11 = e11, e12, e13, e15, e1, e4, e6 T4 = e4, e5, e6, e8, e9, e12, e14 T12 = e12, e13, e14, e1, e2, e5, e7 T5 = e5, e6, e7, e9, e10, e13, e15 T13 = e13, e14, e15, e2, e3, e6, e8 T6 = e6, e7, e8, e10, e11, e14, e1 T14 = e14, e15, e1, e3, e4, e7, e9 T7 = e7, e8, e9, e11, e12, e15, e2 T15 = e15, e1, e2, e4, e5, e8, e10

4Let ei = {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 = {T1, T2, ..., Tn} be a simplex subspace code as described in the construction above. The map fi,j : C → C of the metric space (Pn

q, dS) defined by

   fi,j(Ti) = Tj fi,j(Tj) = Ti fi,j(Tk) = Tk, 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 dS = 2.

Proposition

Let C = {T1, T2, ..., Tn} be a simplex subspace code as described in the construction above. The map fi,j : C → C of the metric space (Pn

q, dS) defined by

   fi,j(Ti) = Tj fi,j(Tj) = Ti fi,j(Tk) = Tk, 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 dS = 2.

Proposition

Let C = {T1, T2, ..., Tn} be a simplex subspace code as described in the construction above. The map fi,j : C → C of the metric space (Pn

q, dS) defined by

   fi,j(Ti) = Tj fi,j(Tj) = Ti fi,j(Tk) = Tk, 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 dS = 2.

Labeling the Subspaces of SSC by Classical Codes Leading to qSSC Codes

Let H be the simplex code with parameters (15, 4, 8). Its generator matrix is given by GH =     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1     ← →     1 2 4 8     where the equivalence denotes the radix 2 integer representation of the subscript of the corresponding subspace.

The SSC code C, may be identified with and labeled by the simplex code H, since both codes have n = 15, d = 8 and 16 codewords. The corresponding qSSC consists of the states labeled by the codewords of a Hadamard code A16 whose generator matrix is GH. T0 ≈ 000000000000000 T8 ≈ 101010101010101 T1 ≈ 000000011111111 T9 ≈ 101010110101010 T2 ≈ 000111100001111 T10 ≈ 101101001011010 T3 ≈ 000111111110000 T11 ≈ 101101010100101 T4 ≈ 011001100110011 T12 ≈ 110011001100110 T5 ≈ 011001111001100 T13 ≈ 110011010011001 T6 ≈ 011110000111100 T14 ≈ 110100101101001 T7 ≈ 011110011000011 T15 ≈ 110100110010110

Since the SSC can be labeled by a binary error-correcting code and by Theorem 3, it is possible to associate the SSC with pure states achieving maximum global entanglement, generating a quantum simplex subspace code, given by |ψ = 1 4{|T0 + |T1 + |T2 + |T3 + |T4 + |T5 + |T6 + |T7 + |T8 + |T9 + |T10 + |T11 + |T12 + |T13 + |T14 + |T15}

Ahlswede, R., Cai, N., Li, S.R., Yeung, R.W.: Network information flow. IEEE Trans. Inf. Theory, vol. 46, pp. 1204-1216, (2000). MacWilliams, F.J., Sloane, N.J.: The Theory of Error-Correcting Codes. Amsterdam, Netherlands: North-Holland Publishing Company, (1977). Forney Jr, G.: Geometrically uniform codes. IEEE Trans.

• Inform. Theory, vol. 37, pp. 1241-1260, (1991).

Koetter, R., Kschischang, F.: Coding for errors and erasures in random network coding. IEEE Trans. Inform. Theory, vol. 54,

• no. 8, pp. 3579–3591, (2008).

Khaleghi, A., Silva, D., Kschischang, F.R.: Subspace codes. Lecture Notes in Computer Science, 5921, pp. 1-21, (2009).

Nilsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge, MA: Cambridge University Press, (2000). Bruβ, D.: Characterizing entanglement. In: J. Math. Phys. 43,

• pp. 4237-4251.

Coffman, V., Kundu, J., Wooters, W.K.: Distributed entanglement, Phys. Rev. A 61, pp. 052306-052311. Meyer, D.A., Wallach, N.R.: Global entaglement in multiparticle systems. Journal of Mathematicasl Physics, 43,

• no. 9, pp. 4273-4278, (2002).

Yu, C.S., Song, H.S.: Global entanglement of multipartite quantum states. Physical Review A 73, pp. 022325-1-5. Lin, S., Costello Jr, D.: Error Control Coding: Fundamentals and Applications. New Jersey, Prentice Hall, (2004).

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(Doumo!)