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
ψRA φRA B
A (n) n (n) A H E D H
B
(n) (k) E encodes and D decodes nS qubits of any ψRA with vanishing error as n 7→ ∞ S is an achievable rate for protecting entanglement
3 Protection of entanglement can be quantified
R R H H
ψRA φRA B
A (n) n (n) A H E D H
B
(n) (k) E encodes and D decodes nS qubits of any ψRA with vanishing error as n 7→ ∞ 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 | X Shannon entropy: H(X ) = p(x) log p(x) − x∈X 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) − x∈X 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 |
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 | X Shannon entropy: H(X ) = p(x) log p(x) − x∈X 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 | X Shannon entropy: H(X ) = p(x) log p(x) − x∈X 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 | X Shannon entropy: H(X ) = p(x) log p(x) − x∈X 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→∞ 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→∞ 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→∞ 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→∞ 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→∞ 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→∞ 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→∞ 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→∞ Q B
7 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→∞ Q B Question : Given a channel is Q( ) > 0? B B
7 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→∞ 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
7 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→∞ 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
7 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→∞ 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
7 Answer : Some examples, but much is unknown
Non-additivity introduces some challenges Quantum Capacity ( ) = lim (1)( ⊗k )/k. Q B k7→∞ 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
7 Non-additivity introduces some challenges Quantum Capacity ( ) = lim (1)( ⊗k )/k. Q B k7→∞ 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
7 Develop new theoretical tools to understand quantum communication •
Gain conceptual insight about quantum communication •
Leverage non-additivity for quantum error correction •
Non-additivity brings new opportunities
8 Gain conceptual insight about quantum communication •
Leverage non-additivity for quantum error correction •
Non-additivity brings new opportunities
Develop new theoretical tools to understand quantum communication •
8 Leverage non-additivity for quantum error correction •
Non-additivity brings new opportunities
Develop new theoretical tools to understand quantum communication •
Gain conceptual insight about quantum communication •
8 Non-additivity brings new opportunities
Develop new theoretical tools to understand quantum communication •
Gain conceptual insight about quantum communication •
Leverage non-additivity for quantum error correction •
8 Non-additivity brings new opportunities
Develop new theoretical tools to understand quantum communication •
Gain conceptual insight about quantum communication •
Leverage non-additivity for quantum error correction •
8 Basic Idea: ρ() has entropy
P S() := Tr(ρ log ρ) = λi log λi − − i
Suppose: λ1 = , λ2 = 1 − S() has an log-singularity
dS()/d = O(log ) for small
Using log-singularities, we develop
A method to check Q(1) > 0 A method to check if Q(1) non-additive
New theoretical tool: log-singularity in the von-Neumann entropy
9 Suppose: λ1 = , λ2 = 1 − S() has an log-singularity
dS()/d = O(log ) for small
Using log-singularities, we develop
A method to check Q(1) > 0 A method to check if Q(1) non-additive
New theoretical tool: log-singularity in the von-Neumann entropy
Basic Idea: ρ() has entropy
P S() := Tr(ρ log ρ) = λi log λi − − i
9 Using log-singularities, we develop
A method to check Q(1) > 0 A method to check if Q(1) non-additive
New theoretical tool: log-singularity in the von-Neumann entropy
Basic Idea: ρ() has entropy
P S() := Tr(ρ log ρ) = λi log λi − − i
Suppose: λ1 = , λ2 = 1 − S() has an log-singularity
dS()/d = O(log ) for small
9 New theoretical tool: log-singularity in the von-Neumann entropy
Basic Idea: ρ() has entropy
P S() := Tr(ρ log ρ) = λi log λi − − i
Suppose: λ1 = , λ2 = 1 − S() has an log-singularity
dS()/d = O(log ) for small
Using log-singularities, we develop
A method to check Q(1) > 0 A method to check if Q(1) non-additive
9 (1) Surprisingly ( 1) = 0, p 1/2 but ( g ) > 0 p, λ > 0 Q C ≤ Q C
Application 1: incomplete erasure g (ρ) = (1 λ) 1(ρ) λρ C − C ⊕ 1 qubit amplitude damping with probability 1 p C −
10 Application 1: incomplete erasure g (ρ) = (1 λ) 1(ρ) λρ C − C ⊕ 1 qubit amplitude damping with probability 1 p C −
(1) Surprisingly ( 1) = 0, p 1/2 but ( g ) > 0 p, λ > 0 Q C ≤ Q C 10 (1) Corollary 1: If da > 1 and db > da(dc 1) then ( ) > 0. − Q B
(1) Corollary 2: If is a qubit channel (da = db = 2) then ( ) > 0 whenever dc > 2. B Q C
Application 2: General results about positivity of (1) Q
Quantum Channel Pair
da ρ db B
C
dc
(1) Theorem: If dc < db and maps some pure state to an output of rank dc , then Q ( ) > 0. B B
11 Application 2: General results about positivity of (1) Q
Quantum Channel Pair
da ρ db B
C
dc
(1) Theorem: If dc < db and maps some pure state to an output of rank dc , then Q ( ) > 0. B B
(1) Corollary 1: If da > 1 and db > da(dc 1) then ( ) > 0. − Q B
(1) Corollary 2: If is a qubit channel (da = db = 2) then ( ) > 0 whenever dc > 2. B Q C 11 A simple zero capacity channel boosts error correction rates
New type of non-additivity using log-singularity
(1)( 0) > (1)( ) + (1)( 0) Q B ⊗ B Q B Q B
: Qubit amplitude damping channel with ( ) = 0 B Q B 0 : Simple qutrit channel B
V.Siddhu, arXiv:2003.10367 (2020)
12 New type of non-additivity using log-singularity
(1)( 0) > (1)( ) + (1)( 0) Q B ⊗ B Q B Q B
: Qubit amplitude damping channel with ( ) = 0 B Q B 0 : Simple qutrit channel B A simple zero capacity channel boosts error correction rates
V.Siddhu, arXiv:2003.10367 (2020)
12 Belief: Allowing leakage of quantum information to a channel’s environment is detrimental for its ability to preserve entanglement.
Result: Quantum capacity can be boosted by allowing leakage of quantum information to a channel’s environment.
Non-additivity can give rise to conceptual surprises
V.Siddhu, arXiv:2011.15116 (2020)
13 Result: Quantum capacity can be boosted by allowing leakage of quantum information to a channel’s environment.
Non-additivity can give rise to conceptual surprises
Belief: Allowing leakage of quantum information to a channel’s environment is detrimental for its ability to preserve entanglement.
V.Siddhu, arXiv:2011.15116 (2020)
13 Non-additivity can give rise to conceptual surprises
Belief: Allowing leakage of quantum information to a channel’s environment is detrimental for its ability to preserve entanglement.
Result: Quantum capacity can be boosted by allowing leakage of quantum information to a channel’s environment.
V.Siddhu, arXiv:2011.15116 (2020)
13 What is the leakage procedure?
Quantum Channel Pair ρ B
C
Trash
14 What is the leakage procedure?
Quantum Channel Pair With Leakage
ρ ρ l B B
l C C
Trash More Trash
15 Allowing qutrit channel to leak takes l B B 7→ B
16 Theoretical tool: log-singularities in the von-Neumann entropy • arXiv:2003.10367
Conceptual surprise: allowing leakage of information can boost quantum communication • arXiv:2011.15116
Message: Study of quantum capacities/non-additivity is a fundamental and fertile area for • developing new theoretical and conceptual tools
Contribution of this talk
17 Conceptual surprise: allowing leakage of information can boost quantum communication • arXiv:2011.15116
Message: Study of quantum capacities/non-additivity is a fundamental and fertile area for • developing new theoretical and conceptual tools
Contribution of this talk
Theoretical tool: log-singularities in the von-Neumann entropy • arXiv:2003.10367
17 Message: Study of quantum capacities/non-additivity is a fundamental and fertile area for • developing new theoretical and conceptual tools
Contribution of this talk
Theoretical tool: log-singularities in the von-Neumann entropy • arXiv:2003.10367
Conceptual surprise: allowing leakage of information can boost quantum communication • arXiv:2011.15116
17 Contribution of this talk
Theoretical tool: log-singularities in the von-Neumann entropy • arXiv:2003.10367
Conceptual surprise: allowing leakage of information can boost quantum communication • arXiv:2011.15116
Message: Study of quantum capacities/non-additivity is a fundamental and fertile area for • developing new theoretical and conceptual tools
17 THANK YOU
18