Application of Complementary Dual AG Codes to Entanglement-Assisted - - PowerPoint PPT Presentation

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Application of Complementary Dual AG Codes to Entanglement-Assisted Quantum Codes

Francisco Revson F. Pereira

joint work with Ruud Pellikaan, Giuliano La Guardia, and Francisco M. de Assis TU/e, the Netherlands

IEEE International Symposium on Information Theory July 12, 2019

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Content

Motivations Algebraic Geometry Codes New QUENTA codes Asymptotically Good QUENTA codes

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CSS Construction Method1

Proposition

Let C1 and C2 denote two classical linear codes with parameters [n, k1, d1]q and [n, k2, d2]q, respectively, such that C ⊥

2 ⊆ C1. Then

there exists a [[n, k1 + k2 − n, d]]q quantum error-correction code with minimum distance d ≥ min{d1, d2}.

1Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and

Quantum Information (2nd ed.). Cambridge: Cambridge University Press

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CSS Construction Method1

Proposition

Let C1 and C2 denote two classical linear codes with parameters [n, k1, d1]q and [n, k2, d2]q, respectively, such that C ⊥

2 ⊆ C1. Then

there exists a [[n, k1 + k2 − n, d]]q quantum error-correction code with minimum distance d ≥ min{d1, d2}. ◮ Constraint: One of the codes needs to be contained

1Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and

Quantum Information (2nd ed.). Cambridge: Cambridge University Press

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CSS Construction Method1

Proposition

Let C1 and C2 denote two classical linear codes with parameters [n, k1, d1]q and [n, k2, d2]q, respectively, such that C ⊥

2 ⊆ C1. Then

there exists a [[n, k1 + k2 − n, d]]q quantum error-correction code with minimum distance d ≥ min{d1, d2}. ◮ Constraint: One of the codes needs to be contained ◮ Possible way out: Entanglement!

1Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and

Quantum Information (2nd ed.). Cambridge: Cambridge University Press

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Entanglement-Assisted Quantum Error Correcting Codes

◮ The first QUENTA code was proposed by Bowen2 ◮ The stabilizer formalism for qubits QUENTA code was done by Brun et al.3 ◮ It was shown that this class of codes can achieve the hashing bound4 and violate the quantum Hamming bound5

2Bowen, G.: Entanglement required in achieving entanglement-assisted channel

• capacities. Physical Review A 66, 052313–1-052313–8 (2006)

3Brun, T., Devetak, I., Hsieh, M.H.: Correcting quantum errors with

• entanglement. Science 314(5798), 436–439 (2006)

4Wilde, M.M., Hsieh, M.H., Babar, Z.: Entanglement-assisted quantum turbo

• codes. IEEE Transactions on Information Theory 60(2), 1203–1222 (2014)

5Li, R., Guo, L., Xu, Z.: Entanglement-assisted quantum codes achieving the

quantum Singleton bound but violating the quantum hamming bound. Quantum Information & Computation 14(13), 1107–1116 (2014)

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Entanglement-Assisted Quantum Error Correcting Codes

◮ Recent research focus on constacyclic and negacyclic codes6 7

8 9

6Fan, J., Chen, H., Xu, J.: Constructions of q-ary entanglement-assisted quantum

MDS codes with minimum distance greater than q + 1. Quantum Information and Computation 16(5& 6), 423–434 (2016)

7Lu, L., Ma, W., Li, R., Ma, Y., Liu, Y., Cao, H.: Entanglement-assisted quantum

mds codes from constacyclic codes with large minimum distance. Finite Fields and Their Applications 53, 309–325 (2018)

8Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS

codes constructed from negacyclic codes. Quantum Information Processing 16(12), 303 (2017)

9Lu, L., Li, R., Guo, L., Ma, Y., Liu, Y.: Entanglement-assisted quantum MDS

codes from negacyclic codes. Quantum Information Processing 17(3), 69 (2018)

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Entanglement-Assisted Quantum Error Correcting Codes

◮ Recent research focus on constacyclic and negacyclic codes6 7

8 9

What about codes with larger length?

6Fan, J., Chen, H., Xu, J.: Constructions of q-ary entanglement-assisted quantum

MDS codes with minimum distance greater than q + 1. Quantum Information and Computation 16(5& 6), 423–434 (2016)

7Lu, L., Ma, W., Li, R., Ma, Y., Liu, Y., Cao, H.: Entanglement-assisted quantum

mds codes from constacyclic codes with large minimum distance. Finite Fields and Their Applications 53, 309–325 (2018)

8Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS

codes constructed from negacyclic codes. Quantum Information Processing 16(12), 303 (2017)

9Lu, L., Li, R., Guo, L., Ma, Y., Liu, Y.: Entanglement-assisted quantum MDS

codes from negacyclic codes. Quantum Information Processing 17(3), 69 (2018)

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Entanglement-Assisted Quantum Error Correcting Codes

◮ Recent research focus on constacyclic and negacyclic codes6 7

8 9

What about codes with larger length? and Is there asymptotically good QUENTA codes?

6Fan, J., Chen, H., Xu, J.: Constructions of q-ary entanglement-assisted quantum

MDS codes with minimum distance greater than q + 1. Quantum Information and Computation 16(5& 6), 423–434 (2016)

7Lu, L., Ma, W., Li, R., Ma, Y., Liu, Y., Cao, H.: Entanglement-assisted quantum

mds codes from constacyclic codes with large minimum distance. Finite Fields and Their Applications 53, 309–325 (2018)

8Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS

codes constructed from negacyclic codes. Quantum Information Processing 16(12), 303 (2017)

9Lu, L., Li, R., Guo, L., Ma, Y., Liu, Y.: Entanglement-assisted quantum MDS

codes from negacyclic codes. Quantum Information Processing 17(3), 69 (2018)

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Content

Motivations Algebraic Geometry Codes New QUENTA codes Asymptotically Good QUENTA codes

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Some Notations

◮ Let F/Fq be an algebraic function field over Fq with genus g

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Some Notations

◮ Let F/Fq be an algebraic function field over Fq with genus g ◮ Let P0, P1, . . . , Pn, P∞ be pairwise distinct rational places of F/Fq and D = P1 + · · · + Pn

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Some Notations

◮ Let F/Fq be an algebraic function field over Fq with genus g ◮ Let P0, P1, . . . , Pn, P∞ be pairwise distinct rational places of F/Fq and D = P1 + · · · + Pn ◮ Let G, G1, G2 be divisors of F/Fq such that

◮ suppG ∩ suppD = ∅ and suppGi ∩ suppD = ∅, for i = 1, 2 ◮ 2g − 2 < deg(G), deg(G1), deg(G2) < n

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Some Notations

◮ Let F/Fq be an algebraic function field over Fq with genus g ◮ Let P0, P1, . . . , Pn, P∞ be pairwise distinct rational places of F/Fq and D = P1 + · · · + Pn ◮ Let G, G1, G2 be divisors of F/Fq such that

◮ suppG ∩ suppD = ∅ and suppGi ∩ suppD = ∅, for i = 1, 2 ◮ 2g − 2 < deg(G), deg(G1), deg(G2) < n

◮ And the Riemann-Roch space associated with G is given by L(G) = {x ∈ F/Fq|(x) ≥ −G} ∪ {0}, where ℓ(G) denotes its dimension

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Algebraic Geometry Codes

Definition of CL(D, G)

The algebraic-geometry (AG) code CL(D, G) associ- ated with the divisors D and G is defined as the im- age

• f

the linear map evD : L(G) → Fn

q

called eval- uation map, where evD(f ) = (f (P1), . . . , f (Pn)); i.e., CL(D, G) = {(f (P1), . . . , f (Pn))|f ∈ L(G)}.

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Algebraic Geometry Codes

Definition of CL(D, G)

The algebraic-geometry (AG) code CL(D, G) associ- ated with the divisors D and G is defined as the im- age

• f

the linear map evD : L(G) → Fn

q

called eval- uation map, where evD(f ) = (f (P1), . . . , f (Pn)); i.e., CL(D, G) = {(f (P1), . . . , f (Pn))|f ∈ L(G)}. ◮ CL(D, G) is a linear [n, k, d]q with parameters k = deg(G) − g + 1 and d ≥ n − deg(G).

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New from Old (Preliminary)

Definition

Let G and H be divisors over F/Fq. If G =

P∈PF νP(G)P and

H =

P∈PF νP(H)P, where P ∈ PF is a place, then the intersection

G ∩ H of G and H over F/Fq is defined as follows G ∩ H =

• P∈PF

min{νP(G), νP(H)}P. In addition, the union is given by G ∪ H =

• P∈PF

max{νP(G), νP(H)}P.

Proposition10

Let G and H be divisors over F/Fq. Then L(G)∩L(H) = L(G ∩H).

10Munuera, C., Pellikaan, R.: Equality of geometric Goppa codes and equivalence

• f divisors. Journal of Pure and Applied Algebra (1993)

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New from Old

Theorem

Let F/Fq be a function field of genus g. If G1 and G2 are two divisors such that deg(G1 ∪G2) < n, then CL(D, G1)∩CL(D, G2) = CL(D, G1 ∩ G2).

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New from Old

Theorem

Let F/Fq be a function field of genus g. If G1 and G2 are two divisors such that deg(G1 ∪G2) < n, then CL(D, G1)∩CL(D, G2) = CL(D, G1 ∩ G2).

Sketch of the Proof

(⇒)Let c ∈ CL(D, G1) ∩ CL(D, G2), then exist g1 ∈ L(G1) and g2 ∈ L(G2) such that c = evD(g1) = evD(g2), which implies in evD(g1 − g2) = 0. Since that g1 − g2 ∈ L(G1 ∪ G2) and deg(G1 ∪ G2) < n, then g1 = g2 and, consequently, c ∈ CL(D, G1 ∩ G2) (⇐)This is a straightforward consequence of Munuera and Pel- likaan’s Proposition

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Algebraic Geometry Codes

Definition of CL(D, G)⊥

The Euclidean dual of the (AG) code CL(D, G) is given by CL(D, G)⊥ = CL(D, G ⊥), where G ⊥ = D − G + (η), and η is a Weil differential such that νPi(η) = −1 and ηPi(1) = 1 for all i = 1, . . . , n.

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Algebraic Geometry Codes

Definition of CL(D, G)⊥

The Euclidean dual of the (AG) code CL(D, G) is given by CL(D, G)⊥ = CL(D, G ⊥), where G ⊥ = D − G + (η), and η is a Weil differential such that νPi(η) = −1 and ηPi(1) = 1 for all i = 1, . . . , n. ◮ CL(D, G)⊥ is a linear [n, k′, d′]q with parameters k′ = n + g − 1 − deg(G) and d′ ≥ deg(G) − (2g − 2).

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Content

Motivations Algebraic Geometry Codes New QUENTA codes Asymptotically Good QUENTA codes

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Euclidean Construction Method

Proposition11

Let C1 and C2 be two linear codes over Fq with parameters [n, k1, d1]q and [n, k2, d2]q and parity check matrices H1 and H2, respectively. Then there is an QUENTA code with parameters [[n, k1 + k2 − n + c, d; c]]q, where d ≥ min{d1, d2}, and c = rank(H1HT

2 ) = dim C ⊥ 1 − dim(C ⊥ 1 ∩ C2)

is the number of required maximally entangled states.

11Galindo, C., Hernando, F., Matsumoto, R., Ruano, D.: Entanglement-assisted

quantum error-correcting codes over arbitrary finite fields. Quantum Information Processing 18(116), 1–18 (2019)

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Euclidean Construction Method

Proposition11

Let C1 and C2 be two linear codes over Fq with parameters [n, k1, d1]q and [n, k2, d2]q and parity check matrices H1 and H2, respectively. Then there is an QUENTA code with parameters [[n, k1 + k2 − n + c, d; c]]q, where d ≥ min{d1, d2}, and c = rank(H1HT

2 ) = dim C ⊥ 1 − dim(C ⊥ 1 ∩ C2)

is the number of required maximally entangled states. An QUENTA code is ◮ MDS if d = (n − k + c)/2 + 1 ◮ Maximal entanglement if c = n − k

11Galindo, C., Hernando, F., Matsumoto, R., Ruano, D.: Entanglement-assisted

quantum error-correcting codes over arbitrary finite fields. Quantum Information Processing 18(116), 1–18 (2019)

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QUENTA codes derived from AG codes

Main Theorem

Let F/Fq be an algebraic function field. If deg(G ⊥

1 ∩ G2) < 0,

then there is an QUENTA code with parameters [[n, deg(G2) − g + 1, d; c]]q, where d ≥ n − max{deg(G1), deg(G2)} and c = n + g − 1 − deg(G1).

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QUENTA codes derived from AG codes

Main Theorem

Let F/Fq be an algebraic function field. If deg(G ⊥

1 ∩ G2) < 0,

then there is an QUENTA code with parameters [[n, deg(G2) − g + 1, d; c]]q, where d ≥ n − max{deg(G1), deg(G2)} and c = n + g − 1 − deg(G1).

Sketch of the Proof

We have the following: ◮ CL(D, Gi) has parameters [n, deg(Gi) − g + 1, di ≥ n − deg(Gi)]q ◮ The dimension of the Euclidean dual of CL(D, G1) is n + g − 1 − deg(G1) ◮ Since deg(G ⊥

1 ∪ G2) < 0, then dim(C ⊥ 1 ∩ C2) = 0 and

c = n − deg(Gi) + g − 1

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QUENTA codes derived from Hermitian AG Codes

Theorem (Explicit Example 1)

Let q be a power of a prime and a1, a2, b1, b2 be positive integers such that b1 ≤ a1, b2 ≤ a2, and q(q −1)−1 ≤ b1 +b2 ≤ a1 +a2 < q3−1. In addition, it is adopted a1 > min{b2, q3+q(q−1)−3−a2}. Then there exists an QUENTA code with parameters [[q3 − 1, b1 + b2−q(q−1)/2+1, q3−1−a1−a2; q3−1−a1−a2+q(q−1)/2−1]]q2.

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QUENTA codes derived from Hermitian AG Codes

Theorem (Explicit Example 1)

Let q be a power of a prime and a1, a2, b1, b2 be positive integers such that b1 ≤ a1, b2 ≤ a2, and q(q −1)−1 ≤ b1 +b2 ≤ a1 +a2 < q3−1. In addition, it is adopted a1 > min{b2, q3+q(q−1)−3−a2}. Then there exists an QUENTA code with parameters [[q3 − 1, b1 + b2−q(q−1)/2+1, q3−1−a1−a2; q3−1−a1−a2+q(q−1)/2−1]]q2. Some examples of parameters: [[7, 3, 3; 3]]4, [[26, 12, 10; 8]]9, and [[63, 40, 18; 13]]16

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QUENTA codes derived from Hermitian AG Codes

Sketch of the Proof

◮ Let F/Fq2 be the Hermitian function field defined by the equation yq + y = xq+1 ◮ It has 1 + q3 rational points and genus q(q − 1)/2 ◮ G1 = a1P0 + a2P∞, and G2 = b1P0 + b2P∞ ◮ As a Weil differential, consider η =

1 xq2−x dx, which has divisor

(η) = −D − P0 + (q3 + 2g − 2)P∞

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QUENTA codes derived from Hermitian AG Codes

Sketch of the Proof

◮ Let F/Fq2 be the Hermitian function field defined by the equation yq + y = xq+1 ◮ It has 1 + q3 rational points and genus q(q − 1)/2 ◮ G1 = a1P0 + a2P∞, and G2 = b1P0 + b2P∞ ◮ As a Weil differential, consider η =

1 xq2−x dx, which has divisor

(η) = −D − P0 + (q3 + 2g − 2)P∞ ◮ Since a1 > min{b2, q3 + q(q − 1) − 3 − a2}, we have that CL(D, G1) and CL(D, G2) are LCD ◮ Thus, c = n − k1 = q3 − 1 − a1 − a2 + q(q − 1)/2 − 1 and the result follows

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QUENTA codes derived from Elliptic AG Codes

Theorem (Explicit Example 2)

Let F/Fq be an Elliptic function field with n + 2 rational points. Let a1, a2, b1, b2 be positive integers such that b1 ≤ a1, b2 ≤ a2, and 1 ≤ b1 + b2 ≤ a1 + a2 < n. In addition, it is adopted a1 > min{b2, n − a1}. Then there exists an QUENTA code with parameters [[n, b1 + b2, n − a1 − a2; n − a1 − a2]]q.

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QUENTA codes derived from Elliptic AG Codes

Theorem (Explicit Example 2)

Let F/Fq be an Elliptic function field with n + 2 rational points. Let a1, a2, b1, b2 be positive integers such that b1 ≤ a1, b2 ≤ a2, and 1 ≤ b1 + b2 ≤ a1 + a2 < n. In addition, it is adopted a1 > min{b2, n − a1}. Then there exists an QUENTA code with parameters [[n, b1 + b2, n − a1 − a2; n − a1 − a2]]q. For a1 = b1 and a2 = b2, the QUENTA code created is almost MDS and maximal entanglement

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QUENTA codes derived from Elliptic AG Codes

Theorem (Explicit Example 2)

Let F/Fq be an Elliptic function field with n + 2 rational points. Let a1, a2, b1, b2 be positive integers such that b1 ≤ a1, b2 ≤ a2, and 1 ≤ b1 + b2 ≤ a1 + a2 < n. In addition, it is adopted a1 > min{b2, n − a1}. Then there exists an QUENTA code with parameters [[n, b1 + b2, n − a1 − a2; n − a1 − a2]]q. For a1 = b1 and a2 = b2, the QUENTA code created is almost MDS and maximal entanglement Some examples of parameters: [[7, 4, 3; 3]]4, [[12, 7, 5; 5]]8, and [[39, 25, 14; 14]]32

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QUENTA codes derived from Elliptic AG Codes

Sketch of the Proof

◮ F/Fq be the elliptic function field with e rational points and genus g = 1 defined by the equation y2 + y = x3 + bx + c, ◮ Assume that G1 = a1P0 + a2P∞ and G2 = b1P0 + b2P∞ ◮ Let η =

dx

• αi ∈S(x+αi), then (η) = (e − 1)P∞ − Pα0 − D

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QUENTA codes derived from Elliptic AG Codes

Sketch of the Proof

◮ F/Fq be the elliptic function field with e rational points and genus g = 1 defined by the equation y2 + y = x3 + bx + c, ◮ Assume that G1 = a1P0 + a2P∞ and G2 = b1P0 + b2P∞ ◮ Let η =

dx

• αi ∈S(x+αi), then (η) = (e − 1)P∞ − Pα0 − D

◮ From a1 > min{b2, n − a1}, we have that CL(D, G1) and CL(D, G2) are LCD ◮ Thus, c = n − k1 = n − a1 − a2 and the result follows

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Content

Motivations Algebraic Geometry Codes New QUENTA codes Asymptotically Good QUENTA codes

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Preliminary

Definition

Let q be a prime power. A family of indexed linear codes Ci, with pa- rameters [ni, ki, di]q, such that ni → ∞ as i → ∞, is called asymp- totically good if δ = limi→∞ di/ni > 0 and R = limi→∞ ki/ni > 0

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Asymptotically Good QUENTA Codes

Theorem

Let q ≥ 3 be a power of a prime and A(q) = lim supg→∞

Nq(g) g

. Then there exists a family of asymptotically good maximal entan- glement QUENTA codes with parameters [[nt, kt, dt; ct]]q, such that lim

t→∞

dt nt ≥ δ, lim

t→∞

kt nt ≥ 1 − δ − 1 A(q), and lim

t→∞

ct nt ∈ [δ, δ + 1/A(q)]. for all δ ∈ [0, 1 − 1/A(q)].

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Asymptotically Good QUENTA Codes

Sketch of the Proof

◮ From Carlet, et al.12, there is a asymptotically good family of LCD codes ◮ Using such family and the result presented, there is a family of asymptotically good QUENTA code ◮ Since the classical codes are LCD, the family of QUENTA codes is also maximal entanglement

12Carlet, C., Mesnager, S., Tang, C., Qi, Y., Pellikaan, R.: Linear codes over Fq are

equivalent to LCD codes for q > 3. IEEE Transactions on Information Theory 64(4), 3010–3017 (2018)

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Comparison between asymptotically QUENTA codes and quantum GV bound13

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Rate (R) Bounds for q = 64

QUENTA codes from Theorem 9 Quantum Gilbert-Varshamov bound 0.0 0.2 0.4 0.6 0.8 1.0

Relative distance (δ)

0.0 0.2 0.4 0.6 0.8 1.0

Entanglement rate (c/n)

Minimum entanglement rate from Theorem 9 Maximum entanglement rate from Theorem 9 Asymptotically Gilbert-Varshamov bound

13Galindo, C., Hernando, F., Matsumoto, R., Ruano, D.: Entanglement-assisted

quantum error-correcting codes over arbitrary finite fields. Quantum Information Processing 18(116), 1–18 (2019)

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Thanks for your attention!

Any questions?

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Communication scheme using an QUENTA code

|0n−k−c |ψ |Φ⊗c E N

sender receiver

id⊗c D |ψ′

n − k − c qudits k qudits c qudits c qudits k qudits n qudits