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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 7th, 2015

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Open system dynamics

Schrödinger picture: Quantum states evolve under completely positive trace preserving (CPTP) maps E✝ : T ♣HSq Ñ T ♣HSq.

✝♣ q ✏ ♣

❜ q♣ ♣ ❜ ⑤ ②① ⑤q

✝q ✝

♣ q Ñ ♣ q

✝♣ q ✏ ♣

❜ q♣ ♣ ❜ ⑤ ②① ⑤q

✝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 ♣HSq Ñ T ♣HSq. By Stinespring’s dilation theorem: E✝♣ρq ✏ ♣id ❜ trEq♣U♣ρ ❜ ⑤ψE②①ψE⑤qU✝q.

✝

♣ q Ñ ♣ q

✝♣ q ✏ ♣

❜ q♣ ♣ ❜ ⑤ ②① ⑤q

✝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 ♣HSq Ñ T ♣HSq. By Stinespring’s dilation theorem: E✝♣ρq ✏ ♣id ❜ trEq♣U♣ρ ❜ ⑤ψE②①ψE⑤qU✝q. Complementary channel: Ec

✝ : T ♣HSq Ñ T ♣HEq:

Ec

✝♣ρq ✏ ♣trS ❜ idq♣U♣ρ ❜ ⑤ψE②①ψE⑤qU✝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 ♣HSq Ñ T ♣HSq. By Stinespring’s dilation theorem: E✝♣ρq ✏ ♣id ❜ trEq♣U♣ρ ❜ ⑤ψE②①ψE⑤qU✝q. Complementary channel: Ec

✝ : T ♣HSq Ñ T ♣HEq:

Ec

✝♣ρq ✏ ♣trS ❜ idq♣U♣ρ ❜ ⑤ψE②①ψE⑤qU✝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 ♣HSq Ñ T ♣HSq. By Stinespring’s dilation theorem: E✝♣ρq ✏ ♣id ❜ trEq♣U♣ρ ❜ ⑤ψE②①ψE⑤qU✝q. Complementary channel: Ec

✝ : T ♣HSq Ñ T ♣HEq:

Ec

✝♣ρq ✏ ♣trS ❜ idq♣U♣ρ ❜ ⑤ψE②①ψE⑤qU✝q.

Note: Complement is defined up to partial isometry. E✝ and Ec

✝ have dual properties

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Correctable subsystems [Kribs–Laflamme–Poulin ’05 ]

If HS ✏ ♣HA ❜ HBq, then B is a correctable subsystem for E✝ : T ♣HSq Ñ T ♣HSq if ❉ a CPTP map R✝ : T ♣HSq Ñ T ♣HSq such that R✝ ✆ E✝ ✏ F✝ ❜ idB for some CPTP map F✝ : T ♣HAq Ñ T ♣HSq.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Correctable subsystems [Kribs–Laflamme–Poulin ’05 ]

If HS ✏ ♣HA ❜ HBq, then B is a correctable subsystem for E✝ : T ♣HSq Ñ T ♣HSq if ❉ a CPTP map R✝ : T ♣HSq Ñ T ♣HSq such that R✝ ✆ E✝ ✏ F✝ ❜ idB for some CPTP map F✝ : T ♣HAq Ñ T ♣HSq. IDEA: Information stored in B is recoverable after the channel.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

ε-Correctable subsystems

If HS ✏ ♣HA ❜ HBq, then B is an ε-correctable subsystem for E✝ : T ♣HSq Ñ T ♣HSq if ❉ a CPTP map R✝ : T ♣HSq Ñ T ♣HSq such that R✝ ✆ E✝ ✁ F✝ ❜ idBcb ➔ ε for some CPTP map F✝ : T ♣HAq Ñ T ♣HSq. 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 HS ✏ ♣HA ❜ HBq, then B is a private subsystem for E✝ : T ♣HSq Ñ T ♣HSq if ❉ a CPTP map F✝ : T ♣HAq Ñ T ♣HSq such that E✝ ✏ F✝ ❜ trB.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subsystems [Bartlett–Rudolph–Spekkens ’04 ]

If HS ✏ ♣HA ❜ HBq, then B is a private subsystem for E✝ : T ♣HSq Ñ T ♣HSq if ❉ a CPTP map F✝ : T ♣HAq Ñ T ♣HSq such that E✝ ✏ F✝ ❜ trB. IDEA: Information stored in B completely decoheres.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

ε-Private subsystems

If HS ✏ ♣HA ❜ HBq, then B is an ε-private subsystem for E✝ : T ♣HSq Ñ T ♣HSq if ❉ a CPTP map F✝ : T ♣HAq Ñ T ♣HSq such that E✝ ✁ F✝ ❜ trBcb ➔ ε. IDEA: Information stored in B ε-decoheres.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity theorem

Theorem (Kretschmann–Kribs–Spekkens ’08) Let HS ✏ ♣HA ❜ HBq be finite-dimensional and E✝ : T ♣HSq Ñ T ♣HSq be CPTP. Then

✝ ô

❄

✝ ✝ ô

❄

✝

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity theorem

Theorem (Kretschmann–Kribs–Spekkens ’08) Let HS ✏ ♣HA ❜ HBq be finite-dimensional and E✝ : T ♣HSq Ñ T ♣HSq be CPTP. Then B is ε-correctable for E✝ ô B is 2❄ε-private for any Ec

✝. ✝ ô

❄

✝

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity theorem

Theorem (Kretschmann–Kribs–Spekkens ’08) Let HS ✏ ♣HA ❜ HBq be finite-dimensional and E✝ : T ♣HSq Ñ T ♣HSq be CPTP. Then B is ε-correctable for E✝ ô B is 2❄ε-private for any Ec

✝.

B is ε-private for E✝ ô B is 2❄ε-correctable for any Ec

✝.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Heisenberg Picture

Observables evolve under normal unital completely positive (NUCP) maps: E : B♣HSq Ñ B♣HSq. ✏ ♣ ❜ 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♣HSq Ñ B♣HSq. If HS ✏ ♣HA ❜ HBq, then B is correctable for E✝ : T ♣HSq Ñ T ♣HSq iff ❉ a NUCP map R : B♣HBq Ñ B♣HSq such that E ✆ R♣bq ✏ ♣1 ❜ bq for all b P B♣HBq.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Correctable subalgebras [Bény–Kempf–Kribs ’07 ]

A von Neumann subalgebra N ❸ B♣HSq is ε-correctable for E : B♣HSq Ñ B♣HSq if ❉ a NUCP map R : N Ñ B♣HSq such that E ✆ R ✁ idNcb ➔ ε. ✏ ❜ ♣ q ✕

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Correctable subalgebras [Bény–Kempf–Kribs ’07 ]

A von Neumann subalgebra N ❸ B♣HSq is ε-correctable for E : B♣HSq Ñ B♣HSq if ❉ a NUCP map R : N Ñ B♣HSq such that E ✆ R ✁ idNcb ➔ ε. Note: If N is a type I factor, then N ✏ 1 ❜ B♣Hq. ✕

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Correctable subalgebras [Bény–Kempf–Kribs ’07 ]

A von Neumann subalgebra N ❸ B♣HSq is ε-correctable for E : B♣HSq Ñ B♣HSq if ❉ a NUCP map R : N Ñ B♣HSq such that E ✆ R ✁ idNcb ➔ ε. Note: If N is a type I factor, then N ✏ 1 ❜ B♣Hq. Correctable subsystems ✕ Correctable type I factors

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Duality picture

Correctable subalgebras ???? ➈ Correctable subsystems Ø Private subsystems

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Quantum channels

Heisenberg: Observables on the output HS evolve to observables

• n the input HS.

E : B♣HSq Ñ B♣HSq ❸ ♣ q ❸ ♣ q ❸ ♣ q Ñ ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Quantum channels

Heisenberg: Observables on the output HS evolve to observables

• n the input HS.

E : B♣HSq Ñ B♣HSq Subset S ❸ B♣HSq observables ❸ ♣ q ❸ ♣ q Ñ ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Quantum channels

Heisenberg: Observables on the output HS evolve to observables

• n the input HS.

E : B♣HSq Ñ B♣HSq Subset S ❸ B♣HSq observables whose spectral projections lie in M ❸ B♣HSq. ❸ ♣ q Ñ ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Quantum channels

Heisenberg: Observables on the output HS evolve to observables

• n the input HS.

E : B♣HSq Ñ B♣HSq Subset S ❸ B♣HSq observables whose spectral projections lie in M ❸ B♣HSq. Definition Let M ❸ B♣HSq be a von Neumann algebra. A quantum channel is a NUCP map E : M Ñ B♣HSq.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subsystems

If HS ✏ ♣HA ❜ HBq, then B is private for E✝ : T ♣HSq Ñ T ♣HSq iff E✝ ✏ F✝ ❜ trB, F✝ : T ♣HAq Ñ T ♣HSq. ① ♣ q ❜ ② ✏ ① ♣ q ❜ ❜ ② ♣ ♣ qq ❸ ♣ ♣ q ❜ q ✏ ♣ ❜ ♣ qq✶

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subsystems

If HS ✏ ♣HA ❜ HBq, then B is private for E✝ : T ♣HSq Ñ T ♣HSq iff E✝ ✏ F✝ ❜ trB, F✝ : T ♣HAq Ñ T ♣HSq. ①E♣xq, ρA ❜ ρB② ✏ ①F♣xq ❜ 1, ρA ❜ ρB② ♣ ♣ qq ❸ ♣ ♣ q ❜ q ✏ ♣ ❜ ♣ qq✶

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subsystems

If HS ✏ ♣HA ❜ HBq, then B is private for E✝ : T ♣HSq Ñ T ♣HSq iff E✝ ✏ F✝ ❜ trB, F✝ : T ♣HAq Ñ T ♣HSq. ①E♣xq, ρA ❜ ρB② ✏ ①F♣xq ❜ 1, ρA ❜ ρB② so that E♣B♣HSqq ❸ ♣B♣HAq ❜ 1q ✏ ♣1 ❜ B♣HBqq✶.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subalgebras

Definition (C.–Kribs–Levene–Todorov ’14) A von Neumann subalgebra N ❸ B♣HSq is private for E : M Ñ B♣HSq if E♣Mq ❸ N✶. → ❉ Ñ ♣ q ✁ ➔

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subalgebras

Definition (C.–Kribs–Levene–Todorov ’14) A von Neumann subalgebra N ❸ B♣HSq is private for E : M Ñ B♣HSq if E♣Mq ❸ N✶. Given ε → 0, we say that N is ε-private for E if ❉ a quantum channel F : M Ñ B♣HSq such that E ✁ Fcb ➔ ε and N is private for F.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subalgebras

Examples: Any normal conditional expectation E : B♣Hq Ñ N✶, e.g., if π : G Ñ B♣Hq is unitary rep. a compact group, E♣xq ✏ ➺

G

π♣sqxπ♣sq✝ds maps onto π♣Gq✶. ♣ ♣ q q ✏ ① ♣ q ② ✏ ① ♣ q ⑤

✶②

♣ q ✏ ❳

✶

✕

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subalgebras

Examples: Any normal conditional expectation E : B♣Hq Ñ N✶, e.g., if π : G Ñ B♣Hq is unitary rep. a compact group, E♣xq ✏ ➺

G

π♣sqxπ♣sq✝ds maps onto π♣Gq✶. Since E♣E♣xq, ρq ✏ ①E♣xq, ρ② ✏ ①E♣xq, ρ⑤N✶② ♣ q ✏ ❳

✶

✕

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subalgebras

Examples: Any normal conditional expectation E : B♣Hq Ñ N✶, e.g., if π : G Ñ B♣Hq is unitary rep. a compact group, E♣xq ✏ ➺

G

π♣sqxπ♣sq✝ds maps onto π♣Gq✶. Since E♣E♣xq, ρq ✏ ①E♣xq, ρ② ✏ ①E♣xq, ρ⑤N✶② IDEA: The only information contained in N that survives is Z♣Nq ✏ N ❳ N✶. ✕

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subalgebras

Examples: Any normal conditional expectation E : B♣Hq Ñ N✶, e.g., if π : G Ñ B♣Hq is unitary rep. a compact group, E♣xq ✏ ➺

G

π♣sqxπ♣sq✝ds maps onto π♣Gq✶. Since E♣E♣xq, ρq ✏ ①E♣xq, ρ② ✏ ①E♣xq, ρ⑤N✶② IDEA: The only information contained in N that survives is Z♣Nq ✏ N ❳ N✶. If N is a factor, then all information is lost. ✕

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Private subalgebras

Examples: Any normal conditional expectation E : B♣Hq Ñ N✶, e.g., if π : G Ñ B♣Hq is unitary rep. a compact group, E♣xq ✏ ➺

G

π♣sqxπ♣sq✝ds maps onto π♣Gq✶. Since E♣E♣xq, ρq ✏ ①E♣xq, ρ② ✏ ①E♣xq, ρ⑤N✶② IDEA: The only information contained in N that survives is Z♣Nq ✏ N ❳ N✶. If N is a factor, then all information is lost. Private subsystems ✕ Private factors of type I

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Duality picture

Correctable subalgebras ?? Private subalgebras ➈ ➈ Correctable subsystems Ø Private subsystems

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Definition (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel. Given a Stinespring representation ♣π, V , Hq of E, we define the complementary channel to be the NUCP map Ec : π♣Mq✶ Ñ B♣HSq given by Ec♣Xq ✏ V ✝XV , X P π♣Mq✶. ✒ ♣ q✶ ✏ ♣ q ♣ q ✏ ❜ ♣ q✶ ✏ ❜ ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Definition (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel. Given a Stinespring representation ♣π, V , Hq of E, we define the complementary channel to be the NUCP map Ec : π♣Mq✶ Ñ B♣HSq given by Ec♣Xq ✏ V ✝XV , X P π♣Mq✶. Again, defined up to partial isometries. ✒ ♣ q✶ ✏ ♣ q ♣ q ✏ ❜ ♣ q✶ ✏ ❜ ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Definition (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel. Given a Stinespring representation ♣π, V , Hq of E, we define the complementary channel to be the NUCP map Ec : π♣Mq✶ Ñ B♣HSq given by Ec♣Xq ✏ V ✝XV , X P π♣Mq✶. Again, defined up to partial isometries. In this case, environment ✒ π♣Mq✶. ✏ ♣ q ♣ q ✏ ❜ ♣ q✶ ✏ ❜ ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Definition (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel. Given a Stinespring representation ♣π, V , Hq of E, we define the complementary channel to be the NUCP map Ec : π♣Mq✶ Ñ B♣HSq given by Ec♣Xq ✏ V ✝XV , X P π♣Mq✶. Again, defined up to partial isometries. In this case, environment ✒ π♣Mq✶. If M ✏ B♣HSq, then π♣xq ✏ x ❜ 1E and π♣Mq✶ ✏ 1 ❜ B♣HEq.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Theorem (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel, and N ❸ B♣HSq be a von Neumann subalgebra. Then ô ❄ ô ❄ ✁ ❛

• ❛

↕ ✁

✽ ↕

❛ ✁ Ñ ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Theorem (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel, and N ❸ B♣HSq be a von Neumann subalgebra. Then N is ε-private for E ô N is 2❄ε-correctable for any Ec. ô ❄ ✁ ❛

• ❛

↕ ✁

✽ ↕

❛ ✁ Ñ ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Theorem (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel, and N ❸ B♣HSq be a von Neumann subalgebra. Then N is ε-private for E ô N is 2❄ε-correctable for any Ec. N is ε-correctable for E ô N is 8❄ε-private for any Ec. ✁ ❛

• ❛

↕ ✁

✽ ↕

❛ ✁ Ñ ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Theorem (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel, and N ❸ B♣HSq be a von Neumann subalgebra. Then N is ε-private for E ô N is 2❄ε-correctable for any Ec. N is ε-correctable for E ô N is 8❄ε-private for any Ec. Main Tools: Arveson’s commutant lifting and the continuity of Stinespring rep (Kretschmann–Schlingemann–Werner ’08): Φ1 ✁ Φ2cb ❛ Φ1cb ❛ Φ2cb ↕ inf

V1,V2V1 ✁ V2✽ ↕

❛ Φ1 ✁ Φ2cb for CP maps Φ1, Φ2 : A Ñ B♣Hq.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Corollary (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel. Then a von Neumann subalgebra N ❸ B♣HSq is correctable for E iff ❉ a normal faithful ✝-homomorphism π : N Ñ M such that yE♣xq ✏ E♣π♣yqxq, and E♣xqy ✏ E♣xπ♣yqq for all y P N and x P M ✝

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Corollary (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel. Then a von Neumann subalgebra N ❸ B♣HSq is correctable for E iff ❉ a normal faithful ✝-homomorphism π : N Ñ M such that yE♣xq ✏ E♣π♣yqxq, and E♣xqy ✏ E♣xπ♣yqq for all y P N and x P M (Johnston–Kribs ’11). ✝

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Complementarity

Corollary (C.–Kribs–Levene–Todorov ’14) Let E : M Ñ B♣HSq be a quantum channel. Then a von Neumann subalgebra N ❸ B♣HSq is correctable for E iff ❉ a normal faithful ✝-homomorphism π : N Ñ M such that yE♣xq ✏ E♣π♣yqxq, and E♣xqy ✏ E♣xπ♣yqq for all y P N and x P M (Johnston–Kribs ’11). In particular, the recovery operation R may always to taken to be a ✝-homomorphism (Bény–Kempf–Kribs ’09).

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Examples: Gaussian Channels

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Weyl representation

Let R2n represent the phase space of a system of n-bosonic modes. ✏ ♣ ☎ ☎ ☎ q ✏ ♣ q ✏ ♣ q Ñ ♣ ♣ qq ♣ q ✏

① ② ♣ q

♣ q ✏ ♣ q ✏

① ②

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Weyl representation

Let R2n represent the phase space of a system of n-bosonic modes. View vectors as z ✏ ♣x1, y1, x2, y2, ☎ ☎ ☎ , xn, ynq, where x ✏ ♣x1, ..., xnq and y ✏ ♣y1, ..., ynq are the canonical coordinates

• f the n-modes.

Ñ ♣ ♣ qq ♣ q ✏

① ② ♣ q

♣ q ✏ ♣ q ✏

① ②

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Weyl representation

Let R2n represent the phase space of a system of n-bosonic modes. View vectors as z ✏ ♣x1, y1, x2, y2, ☎ ☎ ☎ , xn, ynq, where x ✏ ♣x1, ..., xnq and y ✏ ♣y1, ..., ynq are the canonical coordinates

• f the n-modes.

Define U, V : Rn Ñ B♣L2♣Rnqq by Vxψ♣sq ✏ ei①x,s②ψ♣sq and Uyψ♣sq ✏ ψ♣s yq. ✏

① ②

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Weyl representation

Let R2n represent the phase space of a system of n-bosonic modes. View vectors as z ✏ ♣x1, y1, x2, y2, ☎ ☎ ☎ , xn, ynq, where x ✏ ♣x1, ..., xnq and y ✏ ♣y1, ..., ynq are the canonical coordinates

• f the n-modes.

Define U, V : Rn Ñ B♣L2♣Rnqq by Vxψ♣sq ✏ ei①x,s②ψ♣sq and Uyψ♣sq ✏ ψ♣s yq. These satisfy the Weyl (CCR): UyVx ✏ ei①x,y②VxUy.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Weyl representation

Composing, we obtain W : R2n Ñ B♣L2♣Rnqq given by W ♣zq ✏ e

i 2 ①x,y②VxUy.

♣

✶q ✏ ♣

✶q

♣ q ♣ ✶q ♣

✶q ✏ ➦ ✏ ♣ ✶ ✁ ✶

q ✏ ❵ ✏ ✂ ✁ ✡

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Weyl representation

Composing, we obtain W : R2n Ñ B♣L2♣Rnqq given by W ♣zq ✏ e

i 2 ①x,y②VxUy.

This satisfies Weyl–Segal form of the CCR: W ♣z z✶q ✏ e

i 2 ∆♣z,z✶qW ♣zqW ♣z✶q,

where ∆♣z, z✶q ✏ ➦n

i✏1♣xiy✶ i ✁ x✶ i yiq is the symplectic form on

R2n, represented by the matrix ∆ ✏ ❵n

i✏1

✂ 0 1 ✁1 ✡ .

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic transformations

Let Sp♣2n, Rq ✏ tT P GL♣2n, Rq ⑤ ∆♣Tz, Tz✶q ✏ ∆♣z, z✶q✉. ❅ P ♣ q P ♣ ♣ qq ♣ q ✏

✝

♣ q P ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic transformations

Let Sp♣2n, Rq ✏ tT P GL♣2n, Rq ⑤ ∆♣Tz, Tz✶q ✏ ∆♣z, z✶q✉. Phase space transformations that preserve the CCR ❅ P ♣ q P ♣ ♣ qq ♣ q ✏

✝

♣ q P ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic transformations

Let Sp♣2n, Rq ✏ tT P GL♣2n, Rq ⑤ ∆♣Tz, Tz✶q ✏ ∆♣z, z✶q✉. Phase space transformations that preserve the CCR Stone-von Neumann: ❅ T P Sp♣2n, Rq there exists a unitary UT P B♣L2♣Rnqq such that W ♣Tzq ✏ U✝

TW ♣zqUT,

z P R2n. ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic transformations

Let Sp♣2n, Rq ✏ tT P GL♣2n, Rq ⑤ ∆♣Tz, Tz✶q ✏ ∆♣z, z✶q✉. Phase space transformations that preserve the CCR Stone-von Neumann: ❅ T P Sp♣2n, Rq there exists a unitary UT P B♣L2♣Rnqq such that W ♣Tzq ✏ U✝

TW ♣zqUT,

z P R2n. Metaplectic representation of Sp♣2n, Rq.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Characteristic Functions

For ρ P T ♣L2♣Rnqq, we let ϕρ♣zq :✏ tr♣ρW ♣zqq, for z P R2n. ✏ ♣ q ➺ ♣ q ♣✁ q P ♣ ♣ qq ♣ q ✏ ✂ ① ② ✁ ♣ q ✡ P

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Characteristic Functions

For ρ P T ♣L2♣Rnqq, we let ϕρ♣zq :✏ tr♣ρW ♣zqq, for z P R2n. Characteristic function ϕρ determines ρ via: ρ ✏ 1 ♣2πqn ➺

R2n ϕρ♣zqW ♣✁zqd2nz.

P ♣ ♣ qq ♣ q ✏ ✂ ① ② ✁ ♣ q ✡ P

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Characteristic Functions

For ρ P T ♣L2♣Rnqq, we let ϕρ♣zq :✏ tr♣ρW ♣zqq, for z P R2n. Characteristic function ϕρ determines ρ via: ρ ✏ 1 ♣2πqn ➺

R2n ϕρ♣zqW ♣✁zqd2nz.

A state ρ P T ♣L2♣Rnqq is Gaussian if ϕρ is of the form ϕρ♣zq ✏ exp ✂ i①m, z② ✁ 1 2α♣z, zq ✡ where m P R2n and α is a symmetric bilinear form.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Linear bosonic channels

E : B♣L2♣Rnqq Ñ B♣L2♣Rnqq is linear bosonic if E✝♣ρq ✏ ♣id ❜ trEq♣UT♣ρ ❜ ρEqU✝

Tq,

ρ P T ♣L2♣Rnqq, where ρE P T ♣L2♣Rlqq and UT P B♣L2♣R♣nlqqq representing a symplectic matrix T P Sp♣2♣m lq, Rq of the form T ✏ ✂ K L KE LE ✡ where K : Rn Ñ Rn, L : Rl Ñ Rn, KE : Rn Ñ Rl and LE : Rl Ñ Rl.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

E✝♣ρq ✏ ♣id ❜ trEq♣UT♣ρ ❜ ρEqU✝

Tq

Since Wnl♣Tzq ✏ U✝

TWnl♣zqUT, and

T ✏ ✂ K L KE LE ✡ we get E♣Wn♣zqq ✏ ˆ f ♣zqWn♣Kzq, where ˆ f ♣zq ✏ ϕρE ♣KEzq, z P R2n. ✏

✶

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

E✝♣ρq ✏ ♣id ❜ trEq♣UT♣ρ ❜ ρEqU✝

Tq

Since Wnl♣Tzq ✏ U✝

TWnl♣zqUT, and

T ✏ ✂ K L KE LE ✡ we get E♣Wn♣zqq ✏ ˆ f ♣zqWn♣Kzq, where ˆ f ♣zq ✏ ϕρE ♣KEzq, z P R2n. If ˆ f ✏ ϕρ✶

E for a Gaussian state, then E is a Gaussian channel.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Example: If f ♣zq ✏

1 ♣2παqn④2 e✁ z2

2α , then

E♣xq ✏ ➺

R2n Wn♣zq✝xWn♣zqf ♣zqdz

is a Gaussian channel. ♣ ♣ qq ✏ ♣ q ♣ q ♣ q ✏ ➺

♣ q ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Example: If f ♣zq ✏

1 ♣2παqn④2 e✁ z2

2α , then

E♣xq ✏ ➺

R2n Wn♣zq✝xWn♣zqf ♣zqdz

is a Gaussian channel. It follows that E♣Wn♣zqq ✏ ˆ f ♣zqWn♣zq, ˆ f ♣zq ✏ ➺

R2n ei∆♣w,zqf ♣wqdw.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Example: If f ♣zq ✏

1 ♣2παqn④2 e✁ z2

2α , then

E♣xq ✏ ➺

R2n Wn♣zq✝xWn♣zqf ♣zqdz

is a Gaussian channel. It follows that E♣Wn♣zqq ✏ ˆ f ♣zqWn♣zq, ˆ f ♣zq ✏ ➺

R2n ei∆♣w,zqf ♣wqdw.

In the lab: model classical Gaussian noise in optical fibres.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Example: If f ♣zq ✏

1 ♣2παqn④2 e✁ z2

2α , then

E♣xq ✏ ➺

R2n Wn♣zq✝xWn♣zqf ♣zqdz

is a Gaussian channel. It follows that E♣Wn♣zqq ✏ ˆ f ♣zqWn♣zq, ˆ f ♣zq ✏ ➺

R2n ei∆♣w,zqf ♣wqdw.

In the lab: model classical Gaussian noise in optical fibres. IDEA: Use complementarity to produce explicit private subalgebras for E.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Example: If n ✏ 1, K ✏ 1 0

0 0

✟ and ˆ f ♣zq ✏ e✁ α

2 ♣x2y2q.

E♣W ♣zqq ✏ ˆ f ♣zqW ♣Kzq ✏ ˆ f ♣zqVx P L✽♣Rq✶.

✽♣ q ✽♣ q ✝♣ q ✏ ♣

❜ q♣ ♣ ❜ q

✝ q

✏ ☎ ✝ ✝ ✆ ✁ ✁ ☞ ✍ ✍ ✌

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Example: If n ✏ 1, K ✏ 1 0

0 0

✟ and ˆ f ♣zq ✏ e✁ α

2 ♣x2y2q.

E♣W ♣zqq ✏ ˆ f ♣zqW ♣Kzq ✏ ˆ f ♣zqVx P L✽♣Rq✶. Thus, L✽♣Rq is private for E.

✽♣ q ✝♣ q ✏ ♣

❜ q♣ ♣ ❜ q

✝ q

✏ ☎ ✝ ✝ ✆ ✁ ✁ ☞ ✍ ✍ ✌

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Example: If n ✏ 1, K ✏ 1 0

0 0

✟ and ˆ f ♣zq ✏ e✁ α

2 ♣x2y2q.

E♣W ♣zqq ✏ ˆ f ♣zqW ♣Kzq ✏ ˆ f ♣zqVx P L✽♣Rq✶. Thus, L✽♣Rq is private for E. By complementarity, L✽♣Rq is correctable for any Ec.

✝♣ q ✏ ♣

❜ q♣ ♣ ❜ q

✝ q

✏ ☎ ✝ ✝ ✆ ✁ ✁ ☞ ✍ ✍ ✌

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Example: If n ✏ 1, K ✏ 1 0

0 0

✟ and ˆ f ♣zq ✏ e✁ α

2 ♣x2y2q.

E♣W ♣zqq ✏ ˆ f ♣zqW ♣Kzq ✏ ˆ f ♣zqVx P L✽♣Rq✶. Thus, L✽♣Rq is private for E. By complementarity, L✽♣Rq is correctable for any Ec. E✝♣ρq ✏ ♣id ❜ trEq♣UT♣ρ ❜ ρEqU✝

Tq

with T ✏ ☎ ✝ ✝ ✆ 1 ✁1 ✁1 1 1 1 ☞ ✍ ✍ ✌.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Then Ec : B♣L2♣Rqq¯ ❜B♣L2♣Rqq Ñ B♣L2♣Rqq is Ec♣W ♣zq ❜ W ♣z✶qq ✏ tr♣⑤ψ②①ψ⑤W ♣♣0, yqq ❜ W ♣z✶qqW ♣✁zq, where ♣id ❜ trq♣⑤ψ②①ψ⑤q ✏ ρE.

✽♣ q Ñ

♣ ♣ q ❜ ♣ qq ♣ ♣ qq ✏ ♣♣✁ qq ❜ ✆ ✏

✽♣ q

✏ ☎ ✝ ✝ ✆ ✁ ✁ ☞ ✍ ✍ ✌

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Then Ec : B♣L2♣Rqq¯ ❜B♣L2♣Rqq Ñ B♣L2♣Rqq is Ec♣W ♣zq ❜ W ♣z✶qq ✏ tr♣⑤ψ②①ψ⑤W ♣♣0, yqq ❜ W ♣z✶qqW ♣✁zq, where ♣id ❜ trq♣⑤ψ②①ψ⑤q ✏ ρE. R : L✽♣Rq Ñ B♣L2♣Rq ❜ L2♣Rqq R♣W ♣x, 0qq ✏ W ♣♣✁x, 0qq ❜ 1 satisfies Ec ✆ R ✏ idL✽♣Rq. ✏ ☎ ✝ ✝ ✆ ✁ ✁ ☞ ✍ ✍ ✌

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Gaussian channels

Then Ec : B♣L2♣Rqq¯ ❜B♣L2♣Rqq Ñ B♣L2♣Rqq is Ec♣W ♣zq ❜ W ♣z✶qq ✏ tr♣⑤ψ②①ψ⑤W ♣♣0, yqq ❜ W ♣z✶qqW ♣✁zq, where ♣id ❜ trq♣⑤ψ②①ψ⑤q ✏ ρE. R : L✽♣Rq Ñ B♣L2♣Rq ❜ L2♣Rqq R♣W ♣x, 0qq ✏ W ♣♣✁x, 0qq ❜ 1 satisfies Ec ✆ R ✏ idL✽♣Rq. T ✏ ☎ ✝ ✝ ✆ 1 ✁1 ✁1 1 1 1 ☞ ✍ ✍ ✌.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic duality

T is not unique. Any matrix ✂K ✝ I2 ✝ ✡ P Sp♣4, Rq will do. Ñ ✏

• ✂

✝ ✝ ✡ ♣ q

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic duality

T is not unique. Any matrix ✂K ✝ I2 ✝ ✡ P Sp♣4, Rq will do. In general, if A, B : Rn Ñ Rn satisfy ∆ ✏ At∆A Bt∆B then ✂A ✝ B ✝ ✡ can be completed to Sp♣2n, Rq.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic duality

When rA, Bs ✏ 0, canonical choice: ✂A ✁B∆ B A∆ ✡ P Sp♣2n, Rq, where A∆ ✏ ∆✁1At∆ and B∆ ✏ ∆✁1Bt∆ are the symplectic adjoints of A and B.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic duality

When rA, Bs ✏ 0, canonical choice: ✂A ✁B∆ B A∆ ✡ P Sp♣2n, Rq, where A∆ ✏ ∆✁1At∆ and B∆ ✏ ∆✁1Bt∆ are the symplectic adjoints of A and B. Complementarity & symplectic duality?

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic duality

Given linear bosonic E : B♣L2♣Rnqq Ñ B♣L2♣Rnqq E♣Wn♣zqq ✏ ˆ f ♣zqWn♣Kzq with K : Rn Ñ Rn, ✏ ♣ q ➔ ♣ ♣ ♣ qqq ❸ ♣ q✷ r ♣ q ♣ ✶qs ✏ ô ♣

✶q ✏

♣ q ❸ ♣ q✶ ✏ t P ⑤ ♣ q ✏ ❅ P ✉

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic duality

Given linear bosonic E : B♣L2♣Rnqq Ñ B♣L2♣Rnqq E♣Wn♣zqq ✏ ˆ f ♣zqWn♣Kzq with K : Rn Ñ Rn, if S ✏ K♣Rnq ➔ Rn, then E♣B♣L2♣Rnqqq ❸ Wn♣Sq✷. r ♣ q ♣ ✶qs ✏ ô ♣

✶q ✏

♣ q ❸ ♣ q✶ ✏ t P ⑤ ♣ q ✏ ❅ P ✉

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic duality

Given linear bosonic E : B♣L2♣Rnqq Ñ B♣L2♣Rnqq E♣Wn♣zqq ✏ ˆ f ♣zqWn♣Kzq with K : Rn Ñ Rn, if S ✏ K♣Rnq ➔ Rn, then E♣B♣L2♣Rnqqq ❸ Wn♣Sq✷. Since rWn♣zq, Wn♣z✶qs ✏ 0 ô ∆n♣z, z✶q ✏ 0, we have Wn♣S∆q ❸ Wn♣Sq✶ where S∆ :✏ tz P Rn ⑤ ∆n♣z, wq ✏ 0 ❅w P S✉ is the symplectic complement of S.

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic duality

Thus, E♣B♣L2♣Rnqqq ❸ Wn♣S∆q✶ and Wn♣S∆q✷ is private for E. ♣ q✷ ✏ ♣ q✶

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic duality

Thus, E♣B♣L2♣Rnqqq ❸ Wn♣S∆q✶ and Wn♣S∆q✷ is private for E. Question: is Wn♣S∆q✷ ✏ Wn♣Sq✶ ?

Introduction Private Subalgebras & Complementarity Examples: Gaussian Channels

Symplectic duality

Thus, E♣B♣L2♣Rnqqq ❸ Wn♣S∆q✶ and Wn♣S∆q✷ is private for E. Question: is Wn♣S∆q✷ ✏ Wn♣Sq✶ ?

Thank You!