Distillation of quantum coherence ( 1711.10512 & 1804.09500 ) Kun - PowerPoint PPT Presentation

Distillation of quantum coherence ( 1711.10512 & 1804.09500 ) Kun Fang 1 RMS:QI workshop 2018, JILA Based on joint works with Gerardo Adesso 2 , Ludovico Lami 2 , Bartosz Regula 2 , Xin Wang 1 1 Centre for Quantum Software and Information University of Technology Sydney 2 School of Mathematical Sciences University of Nottingham

⊚ Coherence theory background ⊚ Deterministic setting ⊚ Probabilistic setting ⊚ Summary and discussions

Distillation of quantum coherence Quantum coherence 1711.10512 & 1804.09500 Resource theory: ⊚ Free states, e.g. separable states; 1 � m ⊚ Resource states, e.g. entangled states like | Φ m � � i � 1 | ii � ; √ m ⊚ Free operations, e.g. LOCC, SEP, SEPP, PPT... A special case of resource theory: ⊚ Free states: incoherent states I : � � ρ ≥ 0 : Tr ρ � 1 , ρ � ∆ � ρ � � ; 1 � m ⊚ Resource states: coherent state like | Ψ m � � i � 1 | i � . √ m ⊚ Free operatioins, e.g. SIO, IO, DIO, MIO. Quantum coherence as a resource: ⊚ Implement the Deutsch-Jozsa algorithm [Hillery, 2016] ; ⊚ Quantum state merging [Streltsov et al., 2016] ; ⊚ Quantum channel simulation [Díaz et al., 2018] ; ⊚ ...

Distillation of quantum coherence Free operations 1711.10512 & 1804.09500 MIO Semidefinite conditions for MIO and DIO: DIO IO ⊚ MIO: E (| i �� i |) � ∆ ( E (| i �� i |)) for all i . SIO ⊚ DIO: MIO and ∆ � E � | i �� j | �� � 0 for i � j . ⊚ Maximally incoherent operations (MIO): E ( I ) ⊆ I ; ⊚ Dephasing-covariant incoherent operations (DIO): [ E , ∆ ] � 0 ; ⊚ Incoherent operations (IO): E i ρ E † i ∈ I for all ρ ∈ I ; Kraus operators { E i } such that Tr E i ρ E † i ⊚ Strictly incoherent operations (SIO): both E i and E † i are incoherent. More about quantum coherence theory refer to [Streltsov, Adesso, Plenio, 2017] and quantum resource theory refer to [Chitambar and Gour, 2018] ...

⊚ Coherence theory background ⊚ Deterministic setting ⊚ Probabilistic setting ⊚ Summary and discussions

Distillation of quantum coherence Deterministic setting 1711.10512 & 1804.09500 Target Resource state state Π ρ Ψ m The fidelity of coherence distillation under the class of operations Ω is defined by � ρ, m � Π ∈ Ω Tr Π � ρ � F Ω : � max Ψ m . (1) The one-shot ε -error distillable coherence under the class of operation Ω is defined as � ρ � : � log max � m ∈ N � � ρ, m � ≥ 1 − ε � C ( 1 ) ,ε F Ω . (2) d , Ω The asymptotic distillable coherence can be given as � ρ � � ρ ⊗ n � 1 n C ( 1 ) ,ε C d , Ω � lim ε → 0 lim . (3) d , Ω n →∞ � ρ � Similarly we can define the coherence cost of a quantum state C c , Ω .

Distillation of quantum coherence Reversibility 1711.10512 & 1804.09500 � ρ � � ρ � � ρ � � ρ � C d , DIO ≤ C d , MIO ≤ C c , MIO ≤ C c , DIO

Distillation of quantum coherence Reversibility 1711.10512 & 1804.09500 � ρ � σ ∈ I D � ρ � σ � � D � ρ � ∆ � ρ �� C r : � min � ρ � [Winter and Yang, 2016] C r � ρ � � ρ � � ρ � � ρ � C d , DIO ≤ C d , MIO ≤ C c , MIO ≤ C c , DIO

Distillation of quantum coherence Reversibility 1711.10512 & 1804.09500 � ρ � σ ∈ I D � ρ � σ � � D � ρ � ∆ � ρ �� C r : � min � ρ � [Winter and Yang, 2016] [Zhao et al., 2017] C r � ρ � � ρ � � ρ � � ρ � C d , DIO ≤ C d , MIO ≤ C c , MIO ≤ C c , DIO

Distillation of quantum coherence Reversibility 1711.10512 & 1804.09500 � ρ � σ ∈ I D � ρ � σ � � D � ρ � ∆ � ρ �� C r : � min � ρ � [Winter and Yang, 2016] [Zhao et al., 2017] C r � ρ � � ρ � � ρ � � ρ � C d , DIO ≤ C d , MIO ≤ C c , MIO ≤ C c , DIO [Chitambar, 2017] � ρ � C r

Distillation of quantum coherence Reversibility 1711.10512 & 1804.09500 � ρ � σ ∈ I D � ρ � σ � � D � ρ � ∆ � ρ �� C r : � min � ρ � [Winter and Yang, 2016] [Zhao et al., 2017] C r � ρ � � ρ � � ρ � � ρ � C d , DIO � C d , MIO � C c , MIO � C c , DIO [Chitambar, 2017] � ρ � C r

Distillation of quantum coherence Reversibility 1711.10512 & 1804.09500 � ρ � σ ∈ I D � ρ � σ � � D � ρ � ∆ � ρ �� C r : � min � ρ � [Winter and Yang, 2016] [Zhao et al., 2017] C r � ρ � � ρ � � ρ � � ρ � C d , DIO � C d , MIO � C c , MIO � C c , DIO [Chitambar, 2017] � ρ � C r Reversibility for entanglement theory [Brandão and Plenio, 2010] and other resource theory [Brandão and Gour, 2015] only known under resource (asymptotically) non-generating maps . The case of coherence theory set a difference from the others.

Distillation of quantum coherence Reversibility 1711.10512 & 1804.09500 � ρ � σ ∈ I D � ρ � σ � � D � ρ � ∆ � ρ �� C r : � min � ρ � [Winter and Yang, 2016] [Zhao et al., 2017] C r � ρ � � ρ � � ρ � � ρ � C d , DIO � C d , MIO � C c , MIO � C c , DIO [Chitambar, 2017] � ρ � C r Reversibility for entanglement theory [Brandão and Plenio, 2010] and other resource theory [Brandão and Gour, 2015] only known under resource (asymptotically) non-generating maps . The case of coherence theory set a difference from the others.

 Distillation of quantum coherence SDP characterizations 1711.10512 & 1804.09500 Theorem For any state ρ and operation class Ω ∈ { MIO , DIO } , the fidelity of coherence distillation and the one-shot distillable coherence can both be written as the following SDPs: � ρ, m � � 0 ≤ G ≤ 1 , ∆ ( G ) � 1 � Tr G ρ � � � F Ω � max m 1 , (4) � � � ρ � η � C ( 1 ) ,ε � � Tr G ρ ≥ 1 − ε, 0 ≤ G ≤ 1 , ∆ ( G ) � η 1 � − log min . (5) d , Ω Proof ingredients: symmetry of Ψ m and semidefinite conditions for MIO. Then we observe that the optimal operation MIO admits the structure of DIO.

Distillation of quantum coherence SDP characterizations 1711.10512 & 1804.09500 Theorem For any state ρ and operation class Ω ∈ { MIO , DIO } , the fidelity of coherence distillation and the one-shot distillable coherence can both be written as the following SDPs: � ρ, m � � 0 ≤ G ≤ 1 , ∆ ( G ) � 1 � Tr G ρ � � � F Ω � max m 1 , (4) � � � ρ � η � C ( 1 ) ,ε � � Tr G ρ ≥ 1 − ε, 0 ≤ G ≤ 1 , ∆ ( G ) � η 1 � − log min . (5) d , Ω Proof ingredients: symmetry of Ψ m and semidefinite conditions for MIO. Then we observe that the optimal operation MIO admits the structure of DIO. Denote the set of diagonal Hermitian G operators with unit trace, ( ρ � G ) ε D H  J � � G � Tr G � 1 , ∆ ( G ) � G � . � ρ � � ρ � G � C ( 1 ) ,ε G ∈ J D ε Then � min . H d , Ω

Distillation of quantum coherence SDP characterizations 1711.10512 & 1804.09500 Theorem For any state ρ and operation class Ω ∈ { MIO , DIO } , the fidelity of coherence distillation and the one-shot distillable coherence can both be written as the following SDPs: � ρ, m � � 0 ≤ G ≤ 1 , ∆ ( G ) � 1 � Tr G ρ � � � F Ω � max m 1 , (4) � � � ρ � η � C ( 1 ) ,ε � Tr G ρ ≥ 1 − ε, 0 ≤ G ≤ 1 , ∆ ( G ) � η 1 � � − log min . (5) d , Ω Proof ingredients: symmetry of Ψ m and semidefinite conditions for MIO. Then we observe that the optimal operation MIO admits the structure of DIO. Denote the set of diagonal Hermitian G operators with unit trace, ( ρ � G ) ε D H  J � � G � Tr G � 1 , ∆ ( G ) � G � . � ρ � � ρ � G � C ( 1 ) ,ε G ∈ J D ε Then � min . H d , Ω Remark: Similar characterizations independently found by Winter’s group.

Distillation of quantum coherence Pure state 1711.10512 & 1804.09500 For the case of pure states, we go beyond MIO and DIO. Theorem For any pure state | ψ � , we have � ψ, m � � ψ, m � � ψ, m � � ψ, m � F SIO � F IO � F DIO � F MIO , � ψ � � ψ � � ψ � � ψ � C ( 1 ) ,ε � C ( 1 ) ,ε � C ( 1 ) ,ε � C ( 1 ) ,ε . d , SIO d , IO d , DIO d , MIO � ψ, m � � ψ, m � Sketch of proof: F SIO � F MIO ⊚ Introduce a intermediate quantity 1 m �| ψ �� 2 [ m ] which MIO admits max � m 1 � Tr ψ W : 0 ≤ W ≤ 1 , ∆ ( W ) ≤ 1 ; DIO IO � ψ, m � ≤ 1 m �| ψ �� 2 ⊚ Compare SDPs and have F MIO [ m ] ; SIO SIO ⊚ Construct | η � such that λ ψ ≺ λ η ( ψ − − − → η ) and F � � � ψ, m � � 1 m �| ψ �� 2 ≥ 1 m �| ψ �� 2 η, Ψ m [ m ] , thus F SIO [ m ] . More details refer to arXiv: 1711.10512.

⊚ Coherence theory background ⊚ Deterministic setting ⊚ Probabilistic setting ⊚ Summary and discussions