Constructing -uniform states of non-minimal support Zahra Raissi, - - PowerPoint PPT Presentation

Constructing 𝑙-uniform states of non-minimal support

Zahra Raissi, Adam Teixidó, Christian Gogolin, and Antonio Acín

Quantum Information and String Theory (Japan), June 2019

ICFO - The Institute of Photonic Sciences

Zahra Raissi

There is a fundamental question to ask, which states are useful for quantum information applications?

𝑙-uniform states and Absolutely Maximally Entangled (AME) states

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Zahra Raissi

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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What are AME states?

1 2 3 4

𝜔 =

Zahra Raissi

What are AME states?

1 2 3 4

𝜔 =

Tr*1,2+𝑑 𝜔 𝜔 ∝ 𝟚

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Zahra Raissi

What are AME states?

1 2 3 4

𝜔 =

Tr*1,3+𝑑 𝜔 𝜔 ∝ 𝟚

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Zahra Raissi

What are AME states?

1 2 3 4

𝜔 =

Tr*1,4+𝑑 𝜔 𝜔 ∝ 𝟚

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Zahra Raissi

What are AME states?

A pure state of 𝑜 parties with local dimension 𝑟 is AME if for all 𝑇 ⊂ 1, … , 𝑜

𝑇 ≤ 𝑜 2 ⟹ 𝜍𝑇 = Tr𝑇𝑑 𝜔 𝜔 ∝ 𝟚

1 2 3 4

𝜔 =

Tr*3,4+𝑑 𝜔 𝜔 ∝ 𝟚

𝐵𝑁𝐹(𝑜, 𝑟):

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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[2] A. J. Scott, Phys. Rev. A, 69, 052330 (2004). [3] F. Huber, O. Gühne, and J. Siewert, Phys. Rev. Lett. 118, 200502 (2017). [1] A. Higuchi, A. and Sudbery, Phys. Lett. A 273,213 (2000).

Existence of AME states

Zahra Raissi

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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• Still fundamental questions open.

[2] A. J. Scott, Phys. Rev. A, 69, 052330 (2004). [3] F. Huber, O. Gühne, and J. Siewert, Phys. Rev. Lett. 118, 200502 (2017). [1] A. Higuchi, A. and Sudbery, Phys. Lett. A 273,213 (2000).

Existence of AME states

Zahra Raissi

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝜚+ = 00 + 11

𝐵 𝐶

𝜍𝑗 = 𝟚 ∀𝑗

• Still fundamental questions open.
• For qubits, (𝑟 = 2):

𝐻𝐼𝑎 = 000 + 111

𝐵 𝐶 𝐷

𝜍𝑗 = 𝟚 ∀𝑗

𝑜 = 2, 3

[2] A. J. Scott, Phys. Rev. A, 69, 052330 (2004). [3] F. Huber, O. Gühne, and J. Siewert, Phys. Rev. Lett. 118, 200502 (2017). [1] A. Higuchi, A. and Sudbery, Phys. Lett. A 273,213 (2000).

Existence of AME states

Zahra Raissi

𝑜 = 2, 3, 4, 5, 6, 7, 8, 9, …

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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[1,2,3] 𝜚+ = 00 + 11

𝐵 𝐶

𝐻𝐼𝑎 = 000 + 111

𝐵 𝐶 𝐷

• Still fundamental questions open.
• For qubits, (𝑟 = 2):

𝜍𝑗 = 𝟚 ∀𝑗 𝜍𝑗 = 𝟚 ∀𝑗

Existence of AME states

Zahra Raissi

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝜍𝑗𝑘 = 𝟚 ∀𝑗, 𝑘 𝐵𝑁𝐹(4,3) = 𝑗, 𝑘, 𝑗 + 𝑘, 𝑗 + 2𝑘

2 𝑗,𝑘=0

• By increasing the local dimension 𝑟, we can find AME state
• For qutrits, (𝑟 = 3):

𝐵 𝐶 𝐷 𝐸

𝑜 = 2, 3, 4, 5, 6, 7, 8, 9, …

[1,2,3]

• For qubits, (𝑟 = 2):
• Still fundamental questions open.

modulo(3)

𝐵𝑁𝐹(𝑜, 𝑟) states:

A pure state 𝜔 of 𝑜 parties with local dimension 𝑟 is AME if for all 𝑇 ⊂ *1,2, … , 𝑜+,

𝑇 ≤ 𝑜 2 ⟹ Tr𝑇𝑑 𝜔 𝜔 ∝ 𝟚

• Since AME states may not always exist, one can loosen the criteria for maximal mixedness,

𝑙-UNI(𝑜, 𝑟) states:

A pure state 𝜔 of 𝑜 parties with local dimension 𝑟 is 𝑙-uniform if for all 𝑇 ⊂ *1,2, … , 𝑜+,

𝑇 ≤ 𝑙 ⟹ Tr𝑇𝑑 𝜔 𝜔 ∝ 𝟚

1 2 3 4

𝜔 =

Zahra Raissi

𝑙-uniform states

𝑜

. . .

• Obviously, an AME state is a 𝑙 =

𝑜 2 -uniform state.

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Zahra Raissi

Why are 𝑙-uniform states interesting?

• Resource for multipartite parallel teleportation and quantum secret sharing [1]

[2] A.J. Scott, Phys. Rev. A 69, 052330 (2004). [1] W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.K. Lo, Phys. Rev. A, 86, 052335 (2012).

• Natural generalization of EPR and GHZ states

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝜒 = 𝑏 0 + 𝑐 1

𝜚+ 𝐵𝐶 = 00 + 11

𝐵 𝐶 𝐵 𝐶

• 𝑙-uniform states are a type of quantum error correcting codes having the maximal distance

allowed by the Singleton bound (optimal codes) [2,3]

[3] M. Grassl and M Rötteler, IEEE Int. Symp. Inf. Teory (ISTT), 1108 (2015) .

Zahra Raissi

Why are AME states interesting?

[2] F. Patawski, B. Yoshida, D. Harlow, and J. Preskill, HEP, 06, 149 (2015). [1] Z. R., C. Gogolin, A. Riera, A. Acín, J. Phys. A, 51, 7 (2018)

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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AME states of minimal support Classical error correcting codes Quantum error correcting codes Perfect tensors

[1]

• Holographic models implementing the AdS/CFT correspondence [2]

Zahra Raissi

Content of this talk

Classical error correcting codes 𝑙-uniform states of minimal support Constructing 𝑙-uniform state of non-minimal support:

Classical part quantum part

⋮ ⋮ All terms of a state of min support complete orthonormal basis

1. 2.

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝜔 𝜚 𝜔 𝜚

LU equivalent

Graph states: 3. … …

Zahra Raissi

𝑙-uniform states of minimal support

Classical error correcting codes 𝑙-uniform states of minimal support

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Zahra Raissi

𝑙-uniform states of minimal support

• Classifying the 𝑙-uniform states according to the number of their terms ⟶ they are

expanded in product basis. 𝜔 = 𝑑

𝑘1,…,𝑘𝑜 𝑘1, … , 𝑘𝑜 𝑟−1 𝑘1,…,𝑘𝑜=0

𝑟𝑙 ≤ # terms ≤ 𝑟𝑜

• # terms = 𝑟𝑙 : states with this number of terms or local unitary equivalent to this state

are called 𝑙-uniform of minimal support.

• # terms > 𝑟𝑙 : states with this number of terms are 𝑙-uniform of non-minimal support.

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Alice Bob 1

Word

Channel

Zahra Raissi

Classical error correcting codes

F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1.

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Alice Bob

SEND RECEIVE

𝑞 1 − 𝑞

1 1

Word

Zahra Raissi

Classical error correcting codes

error Channel

F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1.

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Zahra Raissi

Classical error correcting codes

Alice Bob

Word

1

Encoding

000 111

(Codewords) error Channel

F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1.

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Alice Bob

Word

1

Encoding

000 111 010 101

Error

(Codewords) error Channel

Zahra Raissi

Classical error correcting codes

F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1.

Constructing 𝑙-uniform states of non-minimal support - study the graph states

10

Alice Bob

Word

1

Encoding

000 111 010 101

Error Correction

000 111

(Codewords)

1

Decoding

error Channel

Zahra Raissi

Classical error correcting codes

F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1.

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Alice Bob

Word

1 𝑛 = (𝑦1, … , 𝑦𝑙)

Encoding

000 111 010 101

Error Correction

000 111

Decoding

(Codewords)

1

F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1.

error Channel

Zahra Raissi

Classical error correcting codes

𝑙 = 1

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Alice Bob

Word

1 𝑛 = (𝑦1, … , 𝑦𝑙)

Encoding

000 111 010 101

Error Correction

000 111

Decoding

(Codewords)

1

F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1.

error Channel

Zahra Raissi

Classical error correcting codes

𝑑 = (𝑦1, … , 𝑦𝑙 , 𝑦𝑙+1, … , 𝑦𝑜)

Message symbols Check symbols

𝑙 = 1 𝑜 = 3

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Alice Bob

Word

1 𝑛 = (𝑦1, … , 𝑦𝑙)

Encoding

000 111 010 101

Error Correction

000 111

Decoding

(Codewords)

1

F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1.

error Channel

Zahra Raissi

Classical error correcting codes

𝑑 = (𝑦1, … , 𝑦𝑙 , 𝑦𝑙+1, … , 𝑦𝑜)

Message symbols Check symbols

𝑒𝐼

𝑒𝐼 = 2𝑢 + 1

𝑙 = 1 𝑜 = 3

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Alice Bob

Word

1

Encoding

000 111 010 101

Error Correction

000 111

Decoding

(Codewords)

1

F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1.

error Channel

Zahra Raissi

Classical error correcting codes

𝑒𝐼

𝑒𝐼 = 2𝑢 + 1

𝑙 = 1 𝑜 = 3

,𝑜 = 3, 1𝑙 = 1, 𝑒𝐼= 3-𝑟=2

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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[1] F.J. MacWilliams, N.J.A. Sloane, The theory of error-correction codes (1977).

𝑙 𝑜

𝑛 𝐻𝑙×𝑜 = 𝑑

Word Generator Matrix Codeword

[2] R. Singleton, IEEE Trans. Inf. Theor., 10, 116 (2006).

• Constructing linear codes ,𝑜, 𝑙, 𝑒𝐼-𝑟 [1]

MDS codes

Zahra Raissi

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝑒𝐼 = 2𝑢 + 1

[1] F.J. MacWilliams, N.J.A. Sloane, The theory of error-correction codes (1977).

𝑙 𝑜

𝑛 𝐻𝑙×𝑜 = 𝑑

Word Generator Matrix Codeword

[2] R. Singleton, IEEE Trans. Inf. Theor., 10, 116 (2006).

MDS codes

Zahra Raissi

• Constructing linear codes ,𝑜, 𝑙, 𝑒𝐼-𝑟 [1]

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝑒𝐼 = 2𝑢 + 1

• Singleton bound for any linear code: 𝑒𝐼 ≤ 𝑜 − 𝑙 + 1 [2]

[1] F.J. MacWilliams, N.J.A. Sloane, The theory of error-correction codes (1977).

𝑙 𝑜

𝑛 𝐻𝑙×𝑜 = 𝑑

Word Generator Matrix Codeword

[2] R. Singleton, IEEE Trans. Inf. Theor., 10, 116 (2006).

MDS codes

Zahra Raissi

• Singleton bound for any linear code: 𝑒𝐼 ≤ 𝑜 − 𝑙 + 1 [2]

Optimal Code ⟹ Maximum Distance Separable (MDS) code

• Constructing linear codes ,𝑜, 𝑙, 𝑒𝐼-𝑟 [1]

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝑒𝐼 = 2𝑢 + 1

Constructing 𝑙-uniform state from MDS codes

Zahra Raissi

Classical MDS codes 𝑙-uniform states of minimal support

• From an MDS code a k-uniform state can be constructed ⟶ taking the equally weighted

superposition of all the codewords 𝑜 ≤ 𝑟 + 1 𝑜 ≤ 𝑟 + 2 If 𝑟 is even, 3-uniform states 𝑙-uniform states

• The existence of the MDS codes and hence a set of k-uniform states of minimal support

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝑟 is a power of prime

𝜔 = all codewords

An example of 𝑙-uniform state

Zahra Raissi

• Generator matrix of an MDS code ,4,2,3-3

𝐻2×4 = 1 1 1 1 1 2

• AME(4, 3):

𝜔 = 𝑛 𝐻2×4 = 𝑗, 𝑘, 𝑗 + 𝑘, 𝑗 + 2𝑘

2 𝑗,𝑘=0

(All additions and multiplications modulo 𝑟 = 3.) 𝑛 = (𝑗, 𝑘) , = 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 + 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 1 2 2 1

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝐵 𝐶 𝐷 𝐸

𝑑 = (𝑗, 𝑘, 𝑗 + 𝑘, 𝑗 + 2𝑘) ,

Basis

Zahra Raissi

• Given a k-uniform state of minimal support 𝜔 = 𝑛𝐻𝑙×𝑜

form a complete orthonormal basis 𝜔𝑏 ≔ 𝑁 𝑏 𝜔 𝑁 𝑏 ≔ 𝑁 𝑏 𝑎 ⊗ 𝑁 𝑏 𝑌 = 𝑎𝑏1 ⊗ 𝑎𝑏𝑙+1 ⊗ ⋯ ⊗ 𝑎𝑏𝑙

𝑙

⊗ 𝑌𝑏𝑤𝑙+1 ⊗ 𝑌𝑏𝑙+2 ⊗ ⋯ ⊗ 𝑌𝑏𝑜

𝑜−𝑙

,

𝜔 𝑁 𝑏 †𝑁 𝑐 𝜔 = 𝜀𝑏𝑗,𝑐𝑗

𝑗

∀𝑏𝑗 ∈ *0, … , 𝑟 − 1+

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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#states= qn

𝑌 𝑘 = 𝑘 + 1 mod 𝑟 𝑎 𝑘 = 𝜕𝑘 𝑘 𝜕 ≔ e

2𝜌𝑗 𝑟

𝑌𝑟 = 𝑎𝑟 = 𝟚 Generalized Pauli operators

Zahra Raissi

Constructing 𝑙-uniform states of non-minimal support

A systematic method to construct a set of non-minimal support states.

Classical part quantum part

⋮ ⋮ All terms of the ℓ -uniform minimal support complete orthonormal basis

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝑙-uniform states of non-minimal support

Zahra Raissi

⋮ ⋮ All terms in the computational basis Complete orthonormal basis

Classical part quantum part

𝜔 = 𝑤 𝐻𝑙×𝑜

𝑤

𝜔′𝑏 ≔ 𝑁 𝑏 𝜔′

𝑤 1 𝐻𝑙×𝑜 𝑤 2 𝐻𝑙×𝑜 𝑤 3 𝐻𝑙×𝑜

𝜔′𝑏1 𝜔′𝑏2 𝜔′𝑏3

ℓ − UNI (𝑜cl, 𝑟) state ℓ′ − UNI (𝑜q,𝑟) basis

𝑜𝑑𝑚 𝑜𝑟

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝑙-uniform states of non-minimal support

Zahra Raissi

𝑜𝑑𝑚 𝑜𝑟 𝑜 = 𝑜𝑑𝑚 + 𝑜𝑟

⋮ ⋮ All terms in the computational basis Complete orthonormal basis

Classical part quantum part

𝜔 = 𝑤 𝐻𝑙×𝑜

𝑤

𝜔′𝑏 ≔ 𝑁 𝑏 𝜔′

𝑤 1 𝐻𝑙×𝑜 𝑤 2 𝐻𝑙×𝑜 𝑤 3 𝐻𝑙×𝑜

𝜔′𝑏1 𝜔′𝑏2 𝜔′𝑏3

ℓ − UNI (𝑜cl, 𝑟) state ℓ′ − UNI (𝑜q,𝑟) basis

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝜚 ≡

𝑙-uniform states of non-minimal support

Zahra Raissi

𝑜𝑑𝑚 𝑜𝑟 𝑜 = 𝑜𝑑𝑚 + 𝑜𝑟

⋮ ⋮ All terms in the computational basis Complete orthonormal basis

Classical part quantum part

𝜔 = 𝑤 𝐻𝑙×𝑜

𝑤

𝜔′𝑏 ≔ 𝑁 𝑏 𝜔′

𝑤 1 𝐻𝑙×𝑜 𝑤 2 𝐻𝑙×𝑜 𝑤 3 𝐻𝑙×𝑜

𝜔′𝑏1 𝜔′𝑏2 𝜔′𝑏3

# 𝑟ℓ - term # 𝑟𝑜𝑟 - states ℓ = 𝑜𝑟 e.g. ℓ − UNI (𝑜cl, 𝑟) state ℓ′ − UNI (𝑜q,𝑟) basis

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝜚 ≡

𝑙-uniform states of non-minimal support

Zahra Raissi

𝑜𝑑𝑚 𝑜𝑟 𝑜 = 𝑜𝑑𝑚 + 𝑜𝑟

⋮ ⋮ All terms in the computational basis Complete orthonormal basis

Classical part quantum part

𝜔 = 𝑤 𝐻𝑙×𝑜

𝑤

𝜔′𝑏 ≔ 𝑁 𝑏 𝜔′

𝑤 1 𝐻𝑙×𝑜 𝑤 2 𝐻𝑙×𝑜 𝑤 3 𝐻𝑙×𝑜

𝜔′𝑏1 𝜔′𝑏2 𝜔′𝑏3

# 𝑟ℓ - term # 𝑟𝑜𝑟 - states ℓ = 𝑜𝑟 e.g. ℓ − UNI (𝑜cl, 𝑟) state ℓ′ − UNI (𝑜q,𝑟) basis

• 𝜚 which is 𝑙-uniform state in 𝑙 − UNI (𝑜, 𝑟) and 𝑙 = min*ℓ + 1, ℓ′ + 1+

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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𝜚 ≡

𝑙-uniform states of non-minimal support

Zahra Raissi

𝑜𝑑𝑚 𝑜𝑟 𝑜 = 𝑜𝑑𝑚 + 𝑜𝑟

⋮ ⋮ All terms in the computational basis Complete orthonormal basis

Classical part quantum part

𝜔 = 𝑤 𝐻𝑙×𝑜

𝑤

𝜔′𝑏 ≔ 𝑁 𝑏 𝜔′

𝑤 1 𝐻𝑙×𝑜 𝑤 2 𝐻𝑙×𝑜 𝑤 3 𝐻𝑙×𝑜

𝜔′𝑏1 𝜔′𝑏2 𝜔′𝑏3

# 𝑟ℓ - term # 𝑟𝑜𝑟 - states ℓ = 𝑜𝑟 e.g. ℓ − UNI (𝑜cl, 𝑟) state ℓ′ − UNI (𝑜q,𝑟) basis

• 𝜚 which is 𝑙-uniform state in 𝑙 − UNI (𝑜, 𝑟) and 𝑙 = min*ℓ + 1, ℓ′ + 1+

Constructing 𝑙-uniform states of non-minimal support - study the graph states

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Case I. 𝜚 ≡

𝑙-uniform states of non-minimal support

Zahra Raissi

𝑜𝑑𝑚 𝑜𝑟 𝑜 = 𝑜𝑑𝑚 + 𝑜𝑟

⋮ ⋮ All terms in the computational basis Complete orthonormal basis

Classical part quantum part

𝜔 = 𝑤 𝐻𝑙×𝑜

𝑤

𝜔′𝑏 ≔ 𝑁 𝑏 𝜔′

𝑤 1 𝐻𝑙×𝑜 𝑤 2 𝐻𝑙×𝑜 𝑤 3 𝐻𝑙×𝑜

𝜔′𝑏1 𝜔′𝑏2 𝜔′𝑏3

# 𝑟ℓ - term # 𝑟𝑜𝑟 - states ℓ = 𝑜𝑟 e.g. ℓ − UNI (𝑜cl, 𝑟) state ℓ′ − UNI (𝑜q,𝑟) basis

• 𝜚 which is 𝑙-uniform state in 𝑙 − UNI (𝑜, 𝑟) and 𝑙 = min*ℓ + 1, ℓ′ + 1+

Constructing 𝑙-uniform states of non-minimal support - study the graph states

16

Case I. Case II. 𝜚 ≡

𝑙-uniform states of non-minimal support

Zahra Raissi

𝑜𝑑𝑚 𝑜𝑟 𝑜 = 𝑜𝑑𝑚 + 𝑜𝑟

⋮ ⋮ All terms in the computational basis Complete orthonormal basis

Classical part quantum part

𝜔 = 𝑤 𝐻𝑙×𝑜

𝑤

𝜔′𝑏 ≔ 𝑁 𝑏 𝜔′

𝑤 1 𝐻𝑙×𝑜 𝑤 2 𝐻𝑙×𝑜 𝑤 3 𝐻𝑙×𝑜

𝜔′𝑏1 𝜔′𝑏2 𝜔′𝑏3

# 𝑟ℓ - term # 𝑟𝑜𝑟 - states ℓ = 𝑜𝑟 e.g. ℓ − UNI (𝑜cl, 𝑟) state ℓ′ − UNI (𝑜q,𝑟) basis

• 𝜚 which is 𝑙-uniform state in 𝑙 − UNI (𝑜, 𝑟) and 𝑙 = min*ℓ + 1, ℓ′ + 1+

Constructing 𝑙-uniform states of non-minimal support - study the graph states

16

Case I. Case II. Case III. 𝜚 ≡

𝜔 = 𝑗, 𝑘, 𝑗 + 𝑘

𝑗,𝑘

⨂ 𝑎𝑗 ⊗ 𝑌𝑘 𝑛, 𝑛

𝑛

Examples of 𝑙-uniform of non-minimal support

Zahra Raissi

𝑜𝑑𝑚 = 3 𝑜𝑟 = 2

𝜔 =

1 2 3 4&5

• AME 𝑜 = 5, 𝑟 = 2 :

Classical part quantum part

1 𝜚+ 1 𝜔+ 1 1 1 1 𝜔− 𝜚−

Constructing 𝑙-uniform states of non-minimal support - study the graph states

17

+ + +

• D. Goyeneche, Z. R., S. DiMartino, and K. Życzkowski, Phys. Rev. A, 97, 062326 (2018).

𝑟 ≥ 2 𝑟 = 2

New states

Zahra Raissi

𝑜 𝑟

Table of AME states/perfect tensors / multi-unitary matrices, http://www.tp.nt.unisiegen.de/ +fhuber/ame.html

Constructing 𝑙-uniform states of non-minimal support - study the graph states

18

• 𝐵𝑁𝐹(7,4)
• 𝐵𝑁𝐹(11,8)

quantum error correcting codes

𝑙-uniform of non-minimal support vs 𝑙-uniform of minimal support

Zahra Raissi

Constructing 𝑙-uniform states of non-minimal support - study the graph states

19

• We construct states with better parameters compare to the states that are obtained from the

MDS codes ⟶ We found new states

𝑙-uniform of non-minimal support vs 𝑙-uniform of minimal support

Zahra Raissi

• for given 𝑜 and 𝑟:

LU

𝑇 ⊂ *1, … , 𝑜+ ∀𝑇 s. t. S ≤ 𝑙 ⟹ 𝜍𝑇 ∝ 𝟚 ∀𝑇 s. t. S ≤ 𝑙 ⟹ 𝜍𝑇 ∝ 𝟚

Constructing 𝑙-uniform states of non-minimal support - study the graph states

19

• We construct states with better parameters compare to the states that are obtained from the

MDS codes ⟶ We found new states

from an MDS code

𝜔 𝜚

⋮ ⋮

𝜔 = all codewords

𝑙-uniform of non-minimal support vs 𝑙-uniform of minimal support

Zahra Raissi

• for given 𝑜 and 𝑟:

LU

𝑇 ⊂ *1, … , 𝑜+ ∀𝑇 s. t. S ≤ 𝑙 ⟹ 𝜍𝑇 ∝ 𝟚 ∀𝑇 s. t. S ≤ 𝑙 ⟹ 𝜍𝑇 ∝ 𝟚

Constructing 𝑙-uniform states of non-minimal support - study the graph states

19

• Which state is better for teleportation and … ?

∀𝑇 s. t. S = 𝑙 + 1 ⟹ 𝜍𝑇 ∝ 𝟚 ∃𝑇 s. t. S = 𝑙 + 1 ⟹ 𝜍𝑇 ∝ 𝟚

• We construct states with better parameters compare to the states that are obtained from the

MDS codes ⟶ We found new states

𝜔 𝜚

⋮ ⋮

from an MDS code

𝜔 = all codewords

Zahra Raissi

Graph state

Description of the 𝑙-uniform states within the graph state formalism.

Constructing 𝑙-uniform states of non-minimal support - study the graph states

20

Stabilizers formalism within the graph states

Zahra Raissi

• State 𝜔 constructed from an MDS code

Constructing 𝑙-uniform states of non-minimal support - study the graph states

21

1 2 𝑙

…

𝑙 + 1 𝑙 +2 𝑙 +3 𝑜

…

𝑏11 𝑏12 𝑏1𝑜

𝜔 = all codewords

Complete bipartite graph

Stabilizers formalism within the graph states

Zahra Raissi

• State 𝜔 constructed from an MDS code

Constructing 𝑙-uniform states of non-minimal support - study the graph states

21

1 2 𝑙

…

𝑙 + 1 𝑙 +2 𝑙 +3 𝑜

…

𝑏11 𝑏12 𝑏1𝑜

• State 𝜚 constructed from

Classical part quantum part ⋮ ⋮

All terms Complete

• rthonormal basis

⋮ ⋮ ⋮ … …

?

1 2 𝑜

𝜔 = all codewords

Complete bipartite graph

Stabilizers formalism within the graph states

Zahra Raissi

Constructing 𝑙-uniform states of non-minimal support - study the graph states

21 ⋮ ⋮ ⋮ … …

• State 𝜔 constructed from an MDS code

1 2 𝑙

…

𝑙 + 1 𝑙 +2 𝑙 +3 𝑜

…

𝑏11 𝑏12 𝑏1𝑜

• State 𝜚 constructed from

Classical part quantum part ⋮ ⋮

All terms Complete

• rthonormal basis

1 2 𝑜

𝜔 = all codewords

Complete bipartite graph

Zahra Raissi

Summary

Constructing 𝑙-uniform states of non-minimal support - study the graph states

Thank you for your attention!

Classical error correcting codes 𝑙-uniform states of minimal support Constructing 𝑙-uniform state of non-minimal support:

⋮ ⋮ All terms of a state of min support complete orthonormal basis

1. 2. 𝜔 𝜚 𝜔 𝜚

LU equivalent

Graph states: 3. … …

Graph states: Introduction

Zahra Raissi

• 𝑌 and 𝑎 that generalize the Pauli operators to Hilbert spaces of dimension 𝑟 ≥ 2

Constructing 𝑙-uniform states of non-minimal support - study the graph states

1 2 3

• initialize each qudit as the state, + = 0 + ⋯ + 𝑟 − 1 , perform

4

𝐷𝑎𝛽𝛾 = 𝑚 𝑚 𝛽 ⊗ 𝑎𝑚𝛾

𝑟−1 𝑚=0

𝑡1 = 𝑌 𝟚 𝑎 𝑎 𝑡2 = 𝟚 𝑌 𝑎 𝑎2 𝑡3 = 𝑎 𝑎 𝑌 𝟚 𝑡4 = 𝑎 𝑎2 𝟚 𝑌 𝑌 𝑘 = 𝑘 + 1 mod 𝑟 𝑎 𝑘 = 𝜕𝑘 𝑘 𝜕 ≔ e

2𝜌𝑗 𝑟

𝑌𝑟 = 𝑎𝑟 = 𝟚

Graph states: Introduction

Zahra Raissi

• 𝑌 and 𝑎 that generalize the Pauli operators to Hilbert spaces of dimension 𝑟 ≥ 2

Constructing 𝑙-uniform states of non-minimal support - study the graph states

1 2 3

• initialize each qudit as the state, + = 0 + ⋯ + 𝑟 − 1 , perform

4

𝐷𝑎𝛽𝛾 = 𝑚 𝑚 𝛽 ⊗ 𝑎𝑚𝛾

𝑟−1 𝑚=0

𝑡1 = 𝑌 𝟚 𝑎 𝑎 𝑡2 = 𝟚 𝑌 𝑎 𝑎2 𝑡3 = 𝑎 𝑎 𝑌 𝟚 𝑡4 = 𝑎 𝑎2 𝟚 𝑌 𝐵𝑁𝐹(4,3) = 𝑗, 𝑘, 𝑗 + 𝑘, 𝑗 + 2𝑘

2 𝑗,𝑘=0

LU equivalent 𝑌 𝑘 = 𝑘 + 1 mod 𝑟 𝑎 𝑘 = 𝜕𝑘 𝑘 𝜕 ≔ e

2𝜌𝑗 𝑟

𝑌𝑟 = 𝑎𝑟 = 𝟚

⋮ ⋮

Classical part quantum part

𝑤 1 𝐻𝑙×𝑜 𝑤 2 𝐻𝑙×𝑜 𝑤 3 𝐻𝑙×𝑜

𝜔′𝑏1

𝑜𝑑𝑚 𝑜𝑟

𝑤 4 𝐻𝑙×𝑜 𝑤 5 𝐻𝑙×𝑜 𝑤 6 𝐻𝑙×𝑜

𝜔′𝑏2 Zahra Raissi

Constructing 𝑙-uniform states of non-minimal support - study the graph states

• 𝐵𝑁𝐹(7,4)
• 𝐵𝑁𝐹(11,8)

Repetition in the basis

Stabilizers formalism within the graph states

Zahra Raissi

• The minimal support 𝑙-uniform state constructed from the MDS codes
• A complete bipartite graph shows the structure of the graph

𝜔 = 𝑤 𝐻𝑙×𝑜

𝑤

𝐻𝑙×𝑜 = ,𝟚 𝑙 𝐵𝑙×𝑜−𝑙-

Constructing 𝑙-uniform states of non-minimal support - study the graph states

• 𝑙-uniform state, 𝑙 − UNI 𝑜, 𝑟
• Generator matrix of an MDS code 𝑜, 𝑙, 𝑒𝐼 𝑟
• Graph state

1 2 𝑙

…

𝑙 + 1 𝑙 +2 𝑙 +3 𝑜

…

𝑏11 𝑏12 𝑏13 𝑏1𝑜 𝑏𝑗𝑘

≡ CZ𝑏𝑗𝑘

𝑏𝑗𝑘 are matrix elements of 𝐵𝑙×(𝑜−𝑙)

Graph state

Zahra Raissi

⋮ ⋮ All terms in the computational basis Complete orthonormal basis

Classical part quantum part

𝜔 = 𝑤 𝐻𝑙×𝑜

𝑤

𝜔′𝑏 ≔ 𝑁 𝑏 𝜔′

𝑤 1 𝐻𝑙×𝑜 𝑤 2 𝐻𝑙×𝑜 𝑤 3 𝐻𝑙×𝑜

𝜔′𝑏1 𝜔′𝑏2 𝜔′𝑏3

ℓ − UNI (𝑜cl, 𝑟) state ℓ′ − UNI (𝑜q,𝑟) states

𝑜𝑑𝑚 𝑜𝑟 ⋮ ⋮ ⋮ … …

𝜔 𝜔′ 𝑁(𝑏 )

Constructing 𝑙-uniform states of non-minimal support - study the graph states