Leaking information to gain entanglement via log-singularities Vikesh Siddhu Feb 1, 2021, QIP 2021 JILA, CU Boulder Formerly at Carnegie Mellon University arXiv:2003.10367 (log-singularity) arXiv:2011.15116 (Leakage) Theoretical interest and practical need to protecting entanglement from noise • When, at what rate, and by what means can noisy entanglement be distilled back? Fertile ground for exchange of research ideas • Use and make contributions to Shannon Theory, Error-correction, Optimization theory,... What care about it? Entanglement is a fundamental non-classical phenomenon • Would be nice to understand and harness in experiments and real-world applications 1 Fertile ground for exchange of research ideas • Use and make contributions to Shannon Theory, Error-correction, Optimization theory,... What care about it? Entanglement is a fundamental non-classical phenomenon • Would be nice to understand and harness in experiments and real-world applications Theoretical interest and practical need to protecting entanglement from noise • When, at what rate, and by what means can noisy entanglement be distilled back? 1 What care about it? Entanglement is a fundamental non-classical phenomenon • Would be nice to understand and harness in experiments and real-world applications Theoretical interest and practical need to protecting entanglement from noise • When, at what rate, and by what means can noisy entanglement be distilled back? Fertile ground for exchange of research ideas • Use and make contributions to Shannon Theory, Error-correction, Optimization theory,... 1 New ideas that help find state-of-the-art rates for protecting entanglement from noise • Theoretical tool: Singularity in the von-Neumann entropy gives rise to positivity and • non-additivity in rates for protecting entanglement Conceptual surprise: Allowing a noisy channel to leak quantum information to its • environment can boosts its ability to protect entanglement. What we found 2 Theoretical tool: Singularity in the von-Neumann entropy gives rise to positivity and • non-additivity in rates for protecting entanglement Conceptual surprise: Allowing a noisy channel to leak quantum information to its • environment can boosts its ability to protect entanglement. What we found New ideas that help find state-of-the-art rates for protecting entanglement from noise • 2 Conceptual surprise: Allowing a noisy channel to leak quantum information to its • environment can boosts its ability to protect entanglement. What we found New ideas that help find state-of-the-art rates for protecting entanglement from noise • Theoretical tool: Singularity in the von-Neumann entropy gives rise to positivity and • non-additivity in rates for protecting entanglement 2 What we found New ideas that help find state-of-the-art rates for protecting entanglement from noise • Theoretical tool: Singularity in the von-Neumann entropy gives rise to positivity and • non-additivity in rates for protecting entanglement Conceptual surprise: Allowing a noisy channel to leak quantum information to its • environment can boosts its ability to protect entanglement. 2 Asymptotic Quantum Capacity Q( ) = max S B Protection of entanglement can be quantified R R H H RA φRA B A (n) n (n) A H E D H B (n) (k) E encodes and D decodes nS qubits of any RA with vanishing error as n 7! 1 S is an achievable rate for protecting entanglement 3 Protection of entanglement can be quantified R R H H RA φRA B A (n) n (n) A H E D H B (n) (k) E encodes and D decodes nS qubits of any RA with vanishing error as n 7! 1 S is an achievable rate for protecting entanglement Asymptotic Quantum Capacity Q( ) = max S B 3 Sending quantum information and performing quantum error-correction An achievable lower bound on rates for quantum key distribution • Achievable rates and quantum capacity are a fundamental quantities Information theoretic rates for • Preserving entanglement (read distilling EPR pairs) 4 An achievable lower bound on rates for quantum key distribution • Achievable rates and quantum capacity are a fundamental quantities Information theoretic rates for • Preserving entanglement (read distilling EPR pairs) Sending quantum information and performing quantum error-correction 4 Achievable rates and quantum capacity are a fundamental quantities Information theoretic rates for • Preserving entanglement (read distilling EPR pairs) Sending quantum information and performing quantum error-correction An achievable lower bound on rates for quantum key distribution • 4 1. Classical Channel - : i.e. p(y x) N X 7! Y j X Shannon entropy: H(X ) = p(x) log p(x) − x2X Mutual Information: I (X ; Y ) = H(X ) + H(Y ) H(X ; Y ) − Channel Mutual Information: C (1)( ) = max I (X ; Y ) N p(x) 2. Quantum Channel - : a b B 7! von-Neumann entropy: S(a) = Tr(ρa log ρa) − Coherent Information: ∆( ; ρa) = S(b) S(c) B − (1) Channel Coherent Information: Q ( ) = max ∆( ; ρa) B ρa B Shannon like recipe for finding an achievable rate 5 X Shannon entropy: H(X ) = p(x) log p(x) − x2X Mutual Information: I (X ; Y ) = H(X ) + H(Y ) H(X ; Y ) − Channel Mutual Information: C (1)( ) = max I (X ; Y ) N p(x) 2. Quantum Channel - : a b B 7! von-Neumann entropy: S(a) = Tr(ρa log ρa) − Coherent Information: ∆( ; ρa) = S(b) S(c) B − (1) Channel Coherent Information: Q ( ) = max ∆( ; ρa) B ρa B Shannon like recipe for finding an achievable rate 1. Classical Channel - : i.e. p(y x) N X 7! Y j 5 2. Quantum Channel - : a b B 7! von-Neumann entropy: S(a) = Tr(ρa log ρa) − Coherent Information: ∆( ; ρa) = S(b) S(c) B − (1) Channel Coherent Information: Q ( ) = max ∆( ; ρa) B ρa B Shannon like recipe for finding an achievable rate 1. Classical Channel - : i.e. p(y x) N X 7! Y j X Shannon entropy: H(X ) = p(x) log p(x) − x2X Mutual Information: I (X ; Y ) = H(X ) + H(Y ) H(X ; Y ) − Channel Mutual Information: C (1)( ) = max I (X ; Y ) N p(x) 5 von-Neumann entropy: S(a) = Tr(ρa log ρa) − Coherent Information: ∆( ; ρa) = S(b) S(c) B − (1) Channel Coherent Information: Q ( ) = max ∆( ; ρa) B ρa B Shannon like recipe for finding an achievable rate 1. Classical Channel - : i.e. p(y x) N X 7! Y j X Shannon entropy: H(X ) = p(x) log p(x) − x2X Mutual Information: I (X ; Y ) = H(X ) + H(Y ) H(X ; Y ) − Channel Mutual Information: C (1)( ) = max I (X ; Y ) N p(x) 2. Quantum Channel - : a b B 7! 5 Shannon like recipe for finding an achievable rate 1. Classical Channel - : i.e. p(y x) N X 7! Y j X Shannon entropy: H(X ) = p(x) log p(x) − x2X Mutual Information: I (X ; Y ) = H(X ) + H(Y ) H(X ; Y ) − Channel Mutual Information: C (1)( ) = max I (X ; Y ) N p(x) 2. Quantum Channel - : a b B 7! von-Neumann entropy: S(a) = Tr(ρa log ρa) − Coherent Information: ∆( ; ρa) = S(b) S(c) B − (1) Channel Coherent Information: Q ( ) = max ∆( ; ρa) B ρa B 5 1. Classical channel Additivity: C (1)( 0) = C (1)( ) + C (1)( 0) N × N N N Channel Capacity: C( ) = lim C (1)( ×k )=k = C (1)( ) N k7!1 N N Joint Channel Capacity: C( 0) = C( ) + C( 0) N × N N N 2. Quantum channel Non-Additivity: Q(1)( 0) Q(1)( ) + Q(1)( 0) B ⊗ B ≥ B B Quantum Capacity: Q( ) = lim Q(1)( ⊗k )=k Q(1)( ) B k7!1 B ≥ B Joint Quantum Capacity: Q( 0) Q( ) + Q( 0) B ⊗ B ≥ B B The inequalities can be strict Shannon like recipe for finding channel capacity 6 2. Quantum channel Non-Additivity: Q(1)( 0) Q(1)( ) + Q(1)( 0) B ⊗ B ≥ B B Quantum Capacity: Q( ) = lim Q(1)( ⊗k )=k Q(1)( ) B k7!1 B ≥ B Joint Quantum Capacity: Q( 0) Q( ) + Q( 0) B ⊗ B ≥ B B The inequalities can be strict Shannon like recipe for finding channel capacity 1. Classical channel Additivity: C (1)( 0) = C (1)( ) + C (1)( 0) N × N N N Channel Capacity: C( ) = lim C (1)( ×k )=k = C (1)( ) N k7!1 N N Joint Channel Capacity: C( 0) = C( ) + C( 0) N × N N N 6 Shannon like recipe for finding channel capacity 1. Classical channel Additivity: C (1)( 0) = C (1)( ) + C (1)( 0) N × N N N Channel Capacity: C( ) = lim C (1)( ×k )=k = C (1)( ) N k7!1 N N Joint Channel Capacity: C( 0) = C( ) + C( 0) N × N N N 2. Quantum channel Non-Additivity: Q(1)( 0) Q(1)( ) + Q(1)( 0) B ⊗ B ≥ B B Quantum Capacity: Q( ) = lim Q(1)( ⊗k )=k Q(1)( ) B k7!1 B ≥ B Joint Quantum Capacity: Q( 0) Q( ) + Q( 0) B ⊗ B ≥ B B The inequalities can be strict 6 Quantum Capacity ( ) = lim (1)( ⊗k )=k: Q B k7!1 Q B Question : Given a channel is Q( ) > 0? B B Answer : Hard to say, for all n > 1 there is some s.t. B (1)( ⊗n) = 0 but (1)( ⊗n+1) > 0 Q B Q B Question : If ( ) = 0 then is useless? Q B B Answer : Not really, one can have ( ) = ( 0) = 0 but ( 0) > 0 Q B Q B Q B ⊗ B Question : When and why are (1) and non-additive? Q Q Answer : Some examples, but much is unknown Non-additivity introduces some challenges 7 Question : Given a channel is Q( ) > 0? B B Answer : Hard to say, for all n > 1 there is some s.t. B (1)( ⊗n) = 0 but (1)( ⊗n+1) > 0 Q B Q B Question : If ( ) = 0 then is useless? Q B B Answer : Not really, one can have ( ) = ( 0) = 0 but ( 0) > 0 Q B Q B Q B ⊗ B Question : When and why are (1) and non-additive? Q Q Answer : Some examples, but much is unknown Non-additivity introduces some challenges Quantum Capacity ( ) = lim (1)( ⊗k )=k: Q B k7!1 Q B 7 Answer : Hard to say, for all n > 1 there is some s.t.