A unified framework for complementarity in quantum information - PowerPoint PPT Presentation

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels A unified framework for complementarity in quantum information Jason Crann with D. Kribs, R. Levene and I. Todorov. Carleton University and Université Lille 1 Recent Developments in Quantum Groups Operator Algebras and Applications February 7 th , 2015

✝ q ✝ ♣ q ✏ ♣ ❜ q♣ ♣ ❜ ⑤ ②① ⑤q ♣ q Ñ ♣ q ✝ ✝ q ✝ ♣ q ✏ ♣ ❜ q♣ ♣ ❜ ⑤ ②① ⑤q ✝ ✝ Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Open system dynamics Schrödinger picture : Quantum states evolve under completely positive trace preserving (CPTP) maps E ✝ : T ♣ H S q Ñ T ♣ H S q .

♣ q Ñ ♣ q ✝ ✝ q ✝ ♣ q ✏ ♣ ❜ q♣ ♣ ❜ ⑤ ②① ⑤q ✝ ✝ Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Open system dynamics Schrödinger picture : Quantum states evolve under completely positive trace preserving (CPTP) maps E ✝ : T ♣ H S q Ñ T ♣ H S q . By Stinespring’s dilation theorem: E ✝ ♣ ρ q ✏ ♣ id ❜ tr E q♣ U ♣ ρ ❜ ⑤ ψ E ②① ψ E ⑤q U ✝ q .

✝ ✝ Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Open system dynamics Schrödinger picture : Quantum states evolve under completely positive trace preserving (CPTP) maps E ✝ : T ♣ H S q Ñ T ♣ H S q . By Stinespring’s dilation theorem: E ✝ ♣ ρ q ✏ ♣ id ❜ tr E q♣ U ♣ ρ ❜ ⑤ ψ E ②① ψ E ⑤q U ✝ q . Complementary channel : E c ✝ : T ♣ H S q Ñ T ♣ H E q : E c ✝ ♣ ρ q ✏ ♣ tr S ❜ id q♣ U ♣ ρ ❜ ⑤ ψ E ②① ψ E ⑤q U ✝ q .

✝ ✝ Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Open system dynamics Schrödinger picture : Quantum states evolve under completely positive trace preserving (CPTP) maps E ✝ : T ♣ H S q Ñ T ♣ H S q . By Stinespring’s dilation theorem: E ✝ ♣ ρ q ✏ ♣ id ❜ tr E q♣ U ♣ ρ ❜ ⑤ ψ E ②① ψ E ⑤q U ✝ q . Complementary channel : E c ✝ : T ♣ H S q Ñ T ♣ H E q : E c ✝ ♣ ρ q ✏ ♣ tr S ❜ id q♣ U ♣ ρ ❜ ⑤ ψ E ②① ψ E ⑤q U ✝ q . Note : Complement is defined up to partial isometry.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Open system dynamics Schrödinger picture : Quantum states evolve under completely positive trace preserving (CPTP) maps E ✝ : T ♣ H S q Ñ T ♣ H S q . By Stinespring’s dilation theorem: E ✝ ♣ ρ q ✏ ♣ id ❜ tr E q♣ U ♣ ρ ❜ ⑤ ψ E ②① ψ E ⑤q U ✝ q . Complementary channel : E c ✝ : T ♣ H S q Ñ T ♣ H E q : E c ✝ ♣ ρ q ✏ ♣ tr S ❜ id q♣ U ♣ ρ ❜ ⑤ ψ E ②① ψ E ⑤q U ✝ q . Note : Complement is defined up to partial isometry. E ✝ and E c ✝ have dual properties

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Correctable subsystems [ Kribs–Laflamme–Poulin ’05 ] If H S ✏ ♣ H A ❜ H B q , then B is a correctable subsystem for E ✝ : T ♣ H S q Ñ T ♣ H S q if ❉ a CPTP map R ✝ : T ♣ H S q Ñ T ♣ H S q such that R ✝ ✆ E ✝ ✏ F ✝ ❜ id B for some CPTP map F ✝ : T ♣ H A q Ñ T ♣ H S q .

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Correctable subsystems [ Kribs–Laflamme–Poulin ’05 ] If H S ✏ ♣ H A ❜ H B q , then B is a correctable subsystem for E ✝ : T ♣ H S q Ñ T ♣ H S q if ❉ a CPTP map R ✝ : T ♣ H S q Ñ T ♣ H S q such that R ✝ ✆ E ✝ ✏ F ✝ ❜ id B for some CPTP map F ✝ : T ♣ H A q Ñ T ♣ H S q . IDEA : Information stored in B is recoverable after the channel.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels ε -Correctable subsystems If H S ✏ ♣ H A ❜ H B q , then B is an ε -correctable subsystem for E ✝ : T ♣ H S q Ñ T ♣ H S q if ❉ a CPTP map R ✝ : T ♣ H S q Ñ T ♣ H S q such that �R ✝ ✆ E ✝ ✁ F ✝ ❜ id B � cb ➔ ε for some CPTP map F ✝ : T ♣ H A q Ñ T ♣ H S q . IDEA : Information stored in B is ε -recoverable after the channel.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Private subsystems [ Bartlett–Rudolph–Spekkens ’04 ] If H S ✏ ♣ H A ❜ H B q , then B is a private subsystem for E ✝ : T ♣ H S q Ñ T ♣ H S q if ❉ a CPTP map F ✝ : T ♣ H A q Ñ T ♣ H S q such that E ✝ ✏ F ✝ ❜ tr B .

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Private subsystems [ Bartlett–Rudolph–Spekkens ’04 ] If H S ✏ ♣ H A ❜ H B q , then B is a private subsystem for E ✝ : T ♣ H S q Ñ T ♣ H S q if ❉ a CPTP map F ✝ : T ♣ H A q Ñ T ♣ H S q such that E ✝ ✏ F ✝ ❜ tr B . IDEA : Information stored in B completely decoheres.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels ε -Private subsystems If H S ✏ ♣ H A ❜ H B q , then B is an ε -private subsystem for E ✝ : T ♣ H S q Ñ T ♣ H S q if ❉ a CPTP map F ✝ : T ♣ H A q Ñ T ♣ H S q such that �E ✝ ✁ F ✝ ❜ tr B � cb ➔ ε. IDEA : Information stored in B ε -decoheres.

❄ ✝ ô ✝ ❄ ✝ ô ✝ Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Complementarity theorem Theorem (Kretschmann–Kribs–Spekkens ’08) Let H S ✏ ♣ H A ❜ H B q be finite-dimensional and E ✝ : T ♣ H S q Ñ T ♣ H S q be CPTP. Then

❄ ✝ ô ✝ Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Complementarity theorem Theorem (Kretschmann–Kribs–Spekkens ’08) Let H S ✏ ♣ H A ❜ H B q be finite-dimensional and E ✝ : T ♣ H S q Ñ T ♣ H S q be CPTP. Then B is ε -correctable for E ✝ ô B is 2 ❄ ε -private for any E c ✝ .

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Complementarity theorem Theorem (Kretschmann–Kribs–Spekkens ’08) Let H S ✏ ♣ H A ❜ H B q be finite-dimensional and E ✝ : T ♣ H S q Ñ T ♣ H S q be CPTP. Then B is ε -correctable for E ✝ ô B is 2 ❄ ε -private for any E c ✝ . B is ε -private for E ✝ ô B is 2 ❄ ε -correctable for any E c ✝ .

✏ ♣ ❜ q ♣ q Ñ ♣ q ✝ ❉ ♣ q Ñ ♣ q ✆ ♣ q ✏ ♣ ❜ q P ♣ q Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Heisenberg Picture Observables evolve under normal unital completely positive (NUCP) maps: E : B ♣ H S q Ñ B ♣ H S q .

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Heisenberg Picture Observables evolve under normal unital completely positive (NUCP) maps: E : B ♣ H S q Ñ B ♣ H S q . If H S ✏ ♣ H A ❜ H B q , then B is correctable for E ✝ : T ♣ H S q Ñ T ♣ H S q iff ❉ a NUCP map R : B ♣ H B q Ñ B ♣ H S q such that E ✆ R ♣ b q ✏ ♣ 1 ❜ b q for all b P B ♣ H B q .

✏ ❜ ♣ q ✕ Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Correctable subalgebras [ Bény–Kempf–Kribs ’07 ] A von Neumann subalgebra N ❸ B ♣ H S q is ε -correctable for E : B ♣ H S q Ñ B ♣ H S q if ❉ a NUCP map R : N Ñ B ♣ H S q such that �E ✆ R ✁ id N � cb ➔ ε.

✕ Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Correctable subalgebras [ Bény–Kempf–Kribs ’07 ] A von Neumann subalgebra N ❸ B ♣ H S q is ε -correctable for E : B ♣ H S q Ñ B ♣ H S q if ❉ a NUCP map R : N Ñ B ♣ H S q such that �E ✆ R ✁ id N � cb ➔ ε. Note : If N is a type I factor, then N ✏ 1 ❜ B ♣ H q .

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Correctable subalgebras [ Bény–Kempf–Kribs ’07 ] A von Neumann subalgebra N ❸ B ♣ H S q is ε -correctable for E : B ♣ H S q Ñ B ♣ H S q if ❉ a NUCP map R : N Ñ B ♣ H S q such that �E ✆ R ✁ id N � cb ➔ ε. Note : If N is a type I factor, then N ✏ 1 ❜ B ♣ H q . Correctable subsystems ✕ Correctable type I factors

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Duality picture Correctable subalgebras ???? ➈ Correctable subsystems Ø Private subsystems

❸ ♣ q ❸ ♣ q ❸ ♣ q Ñ ♣ q Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Quantum channels Heisenberg : Observables on the output H S evolve to observables on the input H S . E : B ♣ H S q Ñ B ♣ H S q

❸ ♣ q ❸ ♣ q Ñ ♣ q Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Quantum channels Heisenberg : Observables on the output H S evolve to observables on the input H S . E : B ♣ H S q Ñ B ♣ H S q Subset S ❸ B ♣ H S q observables

❸ ♣ q Ñ ♣ q Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels Quantum channels Heisenberg : Observables on the output H S evolve to observables on the input H S . E : B ♣ H S q Ñ B ♣ H S q Subset S ❸ B ♣ H S q observables whose spectral projections lie in M ❸ B ♣ H S q .