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 ICFO - The Institute of Photonic Sciences Quantum Information and String Theory (Japan), June 2019

𝑙 -uniform states and Absolutely Maximally Entangled (AME) states There is a fundamental question to ask, which states are useful for quantum information applications? 1 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

What are AME states? 1 2 3 4 𝜔 = 2 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

What are AME states? 1 2 3 4 𝜔 = Tr *1,2+ 𝑑 𝜔 𝜔 ∝ 𝟚 2 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

What are AME states? 1 2 3 4 𝜔 = Tr *1,3+ 𝑑 𝜔 𝜔 ∝ 𝟚 2 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

What are AME states? 1 2 3 4 𝜔 = Tr *1,4+ 𝑑 𝜔 𝜔 ∝ 𝟚 2 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

What are AME states? 1 2 3 4 𝜔 = Tr *3,4+ 𝑑 𝜔 𝜔 ∝ 𝟚 𝐵𝑁𝐹(𝑜, 𝑟) : A pure state of 𝑜 parties with local dimension 𝑟 is AME if for all 𝑇 ⊂ 1, … , 𝑜 𝑇 ≤ 𝑜 2 ⟹ 𝜍 𝑇 = Tr 𝑇 𝑑 𝜔 𝜔 ∝ 𝟚 2 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Existence of AME states  Still fundamental questions open. [1] A. Higuchi, A. and Sudbery, Phys. Lett. A 273,213 (2000). [2] A. J. Scott, Phys. Rev. A, 69, 052330 (2004). [3] F. Huber, O. Gühne, and J. Siewert, Phys. Rev. Lett. 118, 200502 (2017). 3 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Existence of AME states  Still fundamental questions open. For qubits, ( 𝑟 = 2 ): 𝑜 = 2, 3 • 𝐵 𝐶 𝜚 + = 00 + 11 𝜍 𝑗 = 𝟚 ∀𝑗 𝐷 𝜍 𝑗 = 𝟚 𝐻𝐼𝑎 = 000 + 111 ∀𝑗 𝐵 𝐶 [1] A. Higuchi, A. and Sudbery, Phys. Lett. A 273,213 (2000). [2] A. J. Scott, Phys. Rev. A, 69, 052330 (2004). [3] F. Huber, O. Gühne, and J. Siewert, Phys. Rev. Lett. 118, 200502 (2017). 3 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Existence of AME states  Still fundamental questions open. For qubits, ( 𝑟 = 2 ): 𝑜 = 2, 3, 4, 5, 6, 7, 8, 9, … • [1,2,3] 𝐵 𝐶 𝜚 + = 00 + 11 𝜍 𝑗 = 𝟚 ∀𝑗 𝐷 𝜍 𝑗 = 𝟚 𝐻𝐼𝑎 = 000 + 111 ∀𝑗 𝐵 𝐶 [1] A. Higuchi, A. and Sudbery, Phys. Lett. A 273,213 (2000). [2] A. J. Scott, Phys. Rev. A, 69, 052330 (2004). [3] F. Huber, O. Gühne, and J. Siewert, Phys. Rev. Lett. 118, 200502 (2017). 3 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Existence of AME states  Still fundamental questions open. For qubits, ( 𝑟 = 2 ): 𝑜 = 2, 3, 4, 5, 6, 7, 8, 9, … • [1,2,3] By increasing the local dimension 𝑟 , we can find AME state  For qutrits, ( 𝑟 = 3 ): • 𝐵 𝐶 𝐷 𝐸 2 𝜍 𝑗𝑘 = 𝟚 𝐵𝑁𝐹(4,3) = 𝑗, 𝑘, 𝑗 + 𝑘, 𝑗 + 2𝑘 ∀𝑗, 𝑘 𝑗,𝑘=0 modulo(3) 3 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

𝑙 -uniform states  Since AME states may not always exist, one can loosen the criteria for maximal mixedness, 1 2 3 4 𝑜 𝜔 = . . . 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 ⟹ Tr 𝑇 𝑑 𝜔 𝜔 ∝ 𝟚 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 ⟹ Tr 𝑇 𝑑 𝜔 𝜔 ∝ 𝟚 𝑜 Obviously, an AME state is a 𝑙 =  2 -uniform state. 4 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Why are 𝑙 -uniform states interesting?  Natural generalization of EPR and GHZ states  Resource for multipartite parallel teleportation and quantum secret sharing [1] 𝐵 𝐶 𝐵 𝐶 𝜚 + 𝐵𝐶 = 00 + 11 𝜒 = 𝑏 0 + 𝑐 1  𝑙 -uniform states are a type of quantum error correcting codes having the maximal distance allowed by the Singleton bound (optimal codes) [2,3] [1] W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.K. Lo, Phys. Rev. A, 86, 052335 (2012). [2] A.J. Scott, Phys. Rev. A 69, 052330 (2004). [3] M. Grassl and M Rötteler, IEEE Int. Symp. Inf. Teory (ISTT), 1108 (2015) . 5 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Why are AME states interesting? AME states of minimal support Classical error Quantum error correcting codes correcting codes Perfect tensors [1]  Holographic models implementing the AdS/CFT correspondence [2] [1] Z. R., C. Gogolin, A. Riera, A. Acín, J. Phys. A, 51, 7 (2018) [2] F. Patawski, B. Yoshida, D. Harlow, and J. Preskill, HEP, 06, 149 (2015). 6 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Content of this talk 𝑙 -uniform states of Classical error 𝜔 1. correcting codes minimal support quantum part Classical part Constructing 𝑙 -uniform state of 2. non-minimal support: ⋮ ⋮ All terms of a state of complete orthonormal 𝜚 min support basis LU equivalent 𝜔 𝜚 … Graph states: 3. … 7 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

𝑙 -uniform states of minimal support 𝑙 -uniform states of Classical error correcting codes minimal support 8 Constructing 𝑙 -uniform states of non-minimal support - study the graph states 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. 𝑟−1 𝜔 = 𝑘 1 ,…,𝑘 𝑜 𝑘 1 , … , 𝑘 𝑜 𝑑 𝑘 1 ,…,𝑘 𝑜 =0 𝑟 𝑙 ≤ # terms ≤ 𝑟 𝑜  # terms = 𝑟 𝑙 : states with this number of terms or local unitary equivalent to this state are called 𝑙 -uniform of minimal support.  # terms > 𝑟 𝑙 : states with this number of terms are 𝑙 -uniform of non-minimal support. 9 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Classical error correcting codes Channel Alice Bob Word 0 1 F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1. 10 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Classical error correcting codes error Channel Alice Bob Word 1 − 𝑞 0 0 𝑞 1 1 SEND RECEIVE F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1. 10 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Classical error correcting codes error Channel Alice Bob Encoding Word (Codewords) 0 000 111 1 F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1. 10 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Classical error correcting codes error Channel Alice Bob Encoding Error Word (Codewords) 0 000 010 111 1 101 F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1. 10 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Classical error correcting codes error Channel Alice Bob Encoding Error Correction Decoding Word (Codewords) 0 000 000 0 010 111 1 101 111 1 F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1. 10 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Classical error correcting codes error Channel Alice Bob Encoding Error Correction Decoding Word (Codewords) 0 000 000 0 010 111 1 101 111 1 𝑙 = 1 𝑛 = (𝑦 1 , … , 𝑦 𝑙 ) F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1. 10 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Classical error correcting codes error Channel Alice Bob Encoding Error Correction Decoding Word (Codewords) 0 000 000 0 010 111 1 101 111 1 𝑜 = 3 𝑙 = 1 𝑛 = (𝑦 1 , … , 𝑦 𝑙 ) 𝑑 = (𝑦 1 , … , 𝑦 𝑙 , 𝑦 𝑙+1 , … , 𝑦 𝑜 ) Message symbols Check symbols F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1. 10 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi

Classical error correcting codes error Channel Alice Bob Encoding Error Correction Decoding Word (Codewords) 0 000 000 0 010 𝑒 𝐼 111 1 101 111 1 𝑜 = 3 𝑙 = 1 𝑒 𝐼 = 2𝑢 + 1 𝑛 = (𝑦 1 , … , 𝑦 𝑙 ) 𝑑 = (𝑦 1 , … , 𝑦 𝑙 , 𝑦 𝑙+1 , … , 𝑦 𝑜 ) Message symbols Check symbols F.J. MacWilliams and N.J.A. Sloane, The theory of error-correction codes (1977) - chapter 1. 10 Constructing 𝑙 -uniform states of non-minimal support - study the graph states Zahra Raissi