Non-asymptotic entanglement distillation arXiv:1706.06221 Kun Fang - PowerPoint PPT Presentation

Non-asymptotic entanglement distillation arXiv:1706.06221 Kun Fang Joint work with Xin Wang, Marco Tomamichel, Runyao Duan Centre for Quantum Software and Information U niversity of T echnology S ydney

Entanglement distillation [Bennett, DiVincenzo, Smolin, Wootters, 1996] ρ ⊗ n � � ⊗ m | 00 � + | 11 � AB √ 2 Π Alice Bob Alice Bob | 00 � + | 11 � √ 2 � m � � � � � � − φ ⊗ m � ρ ⊗ n � � E D ρ AB : � sup n : lim n →∞ inf � Π � 0 . � 1 AB Π ∈ Ω � ������������������������� �� ������������������������� � error Asymptotically, the number of copies of Bell state we can get from per given state ρ . Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

Entanglement distillation [Bennett, DiVincenzo, Smolin, Wootters, 1996] | 00 � + | 11 � √ 2 � m � � � � − φ ⊗ m � � � ρ ⊗ n � � E D ρ AB : � sup n : lim n →∞ inf � Π � 0 . � 1 AB Π ∈ Ω � ������������������������� �� ������������������������� � error ⊚ Theoretically, fundamental and interesting. ⊚ But not easy to calculate in general. ⊚ From practical point of view, lim n →∞ is not possible. How to do estimation when we only have finite copies of state? Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

Concrete example ρ AB � 0 . 7 · | v 1 �� v 1 | + 0 . 3 · | v 2 �� v 2 | , | v 1 � � 1 (| 00 � + | 11 �) , | v 2 � � 1 (| 01 � + | 10 �) . √ √ 2 2 Question: How many copies of Bell state we can get at most from 222 copies of the state ρ (within the error tolerance 0.01) ? Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

One-shot entanglement distillation � � ⊗ m | 00 � + | 11 � √ 2 ρ AB Π Alice Bob Alice Bob ⊚ Fidelity of distillation [Rains, 2001]: � ρ AB , m � Π ∈ Ω F � Π � � , φ ⊗ m � , where φ � | 00 � + | 11 � F Ω : � max ρ AB . √ 2 ⊚ One-shot distillable entanglement: � � : � max � � ρ AB , m � ≤ ε � E ( 1 ) ρ AB m : 1 − F Ω . Ω ,ε ⊚ Asymptotic rate: � � 1 � � n E ( 1 ) ρ ⊗ n E Ω ρ AB � lim ε → 0 lim . Ω ,ε AB n →∞ Ω ∈ {1-LOCC, LOCC, SEP, PPT} Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

A hierarchy of operation classes Local operations and LOCC classical communication A ←→ B Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

A hierarchy of operation classes LOCC Local operations and classical communication A −→ B 1-LOCC Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

A hierarchy of operation classes � � J Π � Π A 1 B 1 → A 2 B 2 φ A 1 B 1 : A ′ 1 B ′ 1 SEP J Π separable ( A ′ 1 A 2 : B ′ 1 B 2 ) LOCC 1-LOCC Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

A hierarchy of operation classes � � PPT J Π � Π A 1 B 1 → A 2 B 2 φ A 1 B 1 : A ′ 1 B ′ 1 T B ′ SEP 1 B 2 J ≥ 0 Π LOCC 1-LOCC Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

One-shot SDP characterization For any state ρ AB and error tolerance ε ∈ ( 0 , 1 ) , � � E ( 1 ) ρ AB � − log min η PPT ,ε s.t. 0 ≤ M ≤ 1 , Tr ρ M ≥ 1 − ε, − η 1 ≤ M T B ≤ η 1 . Efficiently computable Main ingredient of this proof: Symmetry of maximally entangled state φ , i.e., φ is invariant under U ⊗ U . Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

One-shot SDP characterization For any state ρ AB and error tolerance ε ∈ ( 0 , 1 ) , � � E ( 1 ) ρ AB � − log min η PPT ,ε s.t. 0 ≤ M ≤ 1 , Tr ρ M ≥ 1 − ε, − η 1 ≤ M T B ≤ η 1 . Efficiently computable � � Are we done? How about large number of copies E ( 1 ) ρ ⊗ n ? PPT ,ε AB Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

Quantum hypothesis testing ? ρ ∈ { ρ 1 , ρ 2 } Null: ρ � ρ 1 Alternative: ρ � ρ 2 Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

Quantum hypothesis testing ? ρ ∈ { ρ 1 , ρ 2 } Null: ρ � ρ 1 Alternative: ρ � ρ 2 i � 1 , accept ρ � ρ 1 ρ { M 1 , M 2 } i i � 2 , accept ρ � ρ 2 Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

Quantum hypothesis testing ? ρ ∈ { ρ 1 , ρ 2 } Null: ρ � ρ 1 Alternative: ρ � ρ 2 i � 1 , accept ρ � ρ 1 ρ { M 1 , M 2 } i i � 2 , accept ρ � ρ 2 ρ 1 { M 1 , M 2 } 2 Type-I error: Tr M 2 ρ 1 Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

Quantum hypothesis testing ? ρ ∈ { ρ 1 , ρ 2 } Null: ρ � ρ 1 Alternative: ρ � ρ 2 i � 1 , accept ρ � ρ 1 ρ { M 1 , M 2 } i i � 2 , accept ρ � ρ 2 ρ 1 { M 1 , M 2 } 2 Type-I error: Tr M 2 ρ 1 ρ 2 { M 1 , M 2 } Type-II error: Tr M 1 ρ 2 1 Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

Quantum hypothesis testing ρ 1 { M 1 , M 2 } 2 Type-I error: Tr M 2 ρ 1 ρ 2 { M 1 , M 2 } Type-II error: Tr M 1 ρ 2 1 � � D ε ρ 1 || ρ 2 : � − log min Tr M 1 ρ 2 Type-II error H s.t. Tr M 2 ρ 1 ≤ ε, Type-I error M 1 , M 2 ≥ 0 , M 1 + M 2 � 1 . Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

One-shot Hypothesis testing characterization Build a connection , � � � ρ AB � G � E ( 1 ) D ε ρ AB min . � PPT ,ε H � G TB � 1 ≤ 1 Distillation Hypothesis testing Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

One-shot Hypothesis testing characterization Build a connection , Hermitian � � � ρ AB � G � E ( 1 ) D ε ρ AB min . � PPT ,ε H � G TB � 1 ≤ 1 Distillation Hypothesis testing Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

One-shot Hypothesis testing characterization Build a connection , Hermitian � � � ρ AB � G � E ( 1 ) D ε ρ AB min . � PPT ,ε H � G TB � 1 ≤ 1 Distillation Hypothesis testing ρ G "Distance measure" → D ε H Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

One-shot Hypothesis testing characterization Build a connection , Hermitian � � � ρ AB � G � E ( 1 ) D ε ρ AB min . � PPT ,ε H � G TB � 1 ≤ 1 Distillation Hypothesis testing ρ G "Distance measure" → D ε H Main ingredient of this proof: Norm duality between � · � 1 and � · � ∞ . Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

One-shot Hypothesis testing characterization Build a connection , � � � ρ AB � G � E ( 1 ) D ε ρ AB min . � PPT ,ε H � G TB � 1 ≤ 1 Distillation Hypothesis testing Two Applications: ⊚ Recover the Rains bound. ⊚ Second-order estimation. Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

Recover the Rains bound � ρ � � ρ � G � E ( 1 ) D ε min . � PPT ,ε H � G TB � 1 ≤ 1 ⊚ Rains bound [Rains, 2001; Audenaert, Moor, Vollbrecht, Werner, 2002] R � ρ � D � ρ � σ � � ρ � ≤ R � ρ � min E PPT , . � σ ≥ 0 , � σ TB � 1 ≤ 1 Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

Recover the Rains bound � ρ � � ρ � G � E ( 1 ) D ε min . � PPT ,ε H � G TB � 1 ≤ 1 ⊚ Rains bound [Rains, 2001; Audenaert, Moor, Vollbrecht, Werner, 2002] R � ρ � D � ρ � σ � � ρ � ≤ R � ρ � min E PPT , . � σ ≥ 0 , � σ TB � 1 ≤ 1 � ρ ⊗ n � � ρ ⊗ n � G � 1 � 1 n E ( 1 ) D ε min � G TBn � PPT ,ε H n 1 ≤ 1 Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan

Recover the Rains bound � ρ � � ρ � G � E ( 1 ) D ε min . � PPT ,ε H � G TB � 1 ≤ 1 ⊚ Rains bound [Rains, 2001; Audenaert, Moor, Vollbrecht, Werner, 2002] R � ρ � D � ρ � σ � � ρ � ≤ R � ρ � min E PPT , . � σ ≥ 0 , � σ TB � 1 ≤ 1 � ρ ⊗ n � � ρ ⊗ n � G � � ρ ⊗ n � σ ⊗ n � 1 � 1 ≤ 1 n E ( 1 ) D ε n D ε min � G TBn � PPT ,ε H H n 1 ≤ 1 Non-asymptotic entanglement distillation (1706.06221) | K. Fang , X. Wang, M. Tomamichel, R. Duan