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Entanglement and secret-key-agreement capacities of...
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Entanglement and secret-key-agreement capacities ofbipartite quantum interactions and read-only memory devices
Siddhartha Das 1* Stefan Bauml 2 Mark M. Wilde 1
1Louisiana State University, USA
2Delft University of Technology, Netherlands & NTT Japan
arXiv:1712.00827
8th International Conference on Quantum Cryptography, Shanghai, China
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 1 / 16
Bipartite quantum interactions
Bipartite unitary interactions are the most elementary many-body interactions. Due tounavoidable interaction with environment, study of bipartite noisy interactions is pertinent.
E'A'
B'
EA
B
Figure: Systems of interest A′ and B′ interacting inpresence of the bath E ′.
U is unitary transformationcorresponding to underlyinginteraction Hamiltonian H amongA′,B′,E ′.Before action of interactionHamiltonian H: ωA′B′ ⊗ τE ′ , wherebath E ′ is in some fixed state anduncorrelated to A′B′.After action of H:
ρABE := UA′B′E ′→ABE (ωA′B′ ⊗ τE ′).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 2 / 16
Bipartite quantum interactions
Bipartite unitary interactions are the most elementary many-body interactions. Due tounavoidable interaction with environment, study of bipartite noisy interactions is pertinent.
E'A'
B'
EA
B
Figure: Systems of interest A′ and B′ interacting inpresence of the bath E ′.
U is unitary transformationcorresponding to underlyinginteraction Hamiltonian H amongA′,B′,E ′.Before action of interactionHamiltonian H: ωA′B′ ⊗ τE ′ , wherebath E ′ is in some fixed state anduncorrelated to A′B′.After action of H:
ρABE := UA′B′E ′→ABE (ωA′B′ ⊗ τE ′).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 2 / 16
Bipartite quantum interactions
Bipartite unitary interactions are the most elementary many-body interactions. Due tounavoidable interaction with environment, study of bipartite noisy interactions is pertinent.
E'A'
B'
EA
B
Figure: Systems of interest A′ and B′ interacting inpresence of the bath E ′.
U is unitary transformationcorresponding to underlyinginteraction Hamiltonian H amongA′,B′,E ′.Before action of interactionHamiltonian H: ωA′B′ ⊗ τE ′ , wherebath E ′ is in some fixed state anduncorrelated to A′B′.After action of H:
ρABE := UA′B′E ′→ABE (ωA′B′ ⊗ τE ′).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 2 / 16
Bipartite quantum interactions
Bipartite unitary interactions are the most elementary many-body interactions. Due tounavoidable interaction with environment, study of bipartite noisy interactions is pertinent.
E'A'
B'
EA
B
Figure: Systems of interest A′ and B′ interacting inpresence of the bath E ′.
U is unitary transformationcorresponding to underlyinginteraction Hamiltonian H amongA′,B′,E ′.Before action of interactionHamiltonian H: ωA′B′ ⊗ τE ′ , wherebath E ′ is in some fixed state anduncorrelated to A′B′.After action of H:
ρABE := UA′B′E ′→ABE (ωA′B′ ⊗ τE ′).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 2 / 16
Bidirectional quantum channelA bipartite quantum channel NA′B′→AB is a completely positive, trace-preserving map thattransforms composite system A′B′ to AB.
A'
B'
A
BFigure: Two parties of interest: Alice holds A′,Aand Bob holds B′,B.
When A′,A are held by Alice andB′,B are held by Bob, bipartitechannel N is called bidirectionalchannel.It corresponds to noisy bipartiteinteraction, when bath is inaccessible.For all input state ωA′B′ :
N (ωA′B′) = ρAB, whereρAB := TrE{UA′B′E ′→ABE (ωA′B′ ⊗ τE ′)},
when initial state τE ′ of bath is fixed.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 3 / 16
Bidirectional quantum channelA bipartite quantum channel NA′B′→AB is a completely positive, trace-preserving map thattransforms composite system A′B′ to AB.
A'
B'
A
BFigure: Two parties of interest: Alice holds A′,Aand Bob holds B′,B.
When A′,A are held by Alice andB′,B are held by Bob, bipartitechannel N is called bidirectionalchannel.It corresponds to noisy bipartiteinteraction, when bath is inaccessible.For all input state ωA′B′ :
N (ωA′B′) = ρAB, whereρAB := TrE{UA′B′E ′→ABE (ωA′B′ ⊗ τE ′)},
when initial state τE ′ of bath is fixed.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 3 / 16
Bidirectional quantum channelA bipartite quantum channel NA′B′→AB is a completely positive, trace-preserving map thattransforms composite system A′B′ to AB.
A'
B'
A
BFigure: Two parties of interest: Alice holds A′,Aand Bob holds B′,B.
When A′,A are held by Alice andB′,B are held by Bob, bipartitechannel N is called bidirectionalchannel.It corresponds to noisy bipartiteinteraction, when bath is inaccessible.For all input state ωA′B′ :
N (ωA′B′) = ρAB, whereρAB := TrE{UA′B′E ′→ABE (ωA′B′ ⊗ τE ′)},
when initial state τE ′ of bath is fixed.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 3 / 16
Motivation
Bidirectional channels:Simple model of quantum network with 2 clients, Alice and Bob.Model for quantum gates – CNOT, SWAP, etc.– in noisy intermediate-scale quantum(NISQ) computers.
Entanglement may increase, decrease or not change due to bipartite quantum interactions.
Entanglement distillation: Maximally entangled states are useful resource for severalinformation processing tasks: quantum key distribution, quantum teleportation, etc.
Secret key distillation: Need for secure communication protocols between two parties overnetwork – private reading.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 4 / 16
Motivation
Bidirectional channels:Simple model of quantum network with 2 clients, Alice and Bob.Model for quantum gates – CNOT, SWAP, etc.– in noisy intermediate-scale quantum(NISQ) computers.
Entanglement may increase, decrease or not change due to bipartite quantum interactions.
Entanglement distillation: Maximally entangled states are useful resource for severalinformation processing tasks: quantum key distribution, quantum teleportation, etc.
Secret key distillation: Need for secure communication protocols between two parties overnetwork – private reading.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 4 / 16
Motivation
Bidirectional channels:Simple model of quantum network with 2 clients, Alice and Bob.Model for quantum gates – CNOT, SWAP, etc.– in noisy intermediate-scale quantum(NISQ) computers.
Entanglement may increase, decrease or not change due to bipartite quantum interactions.
Entanglement distillation: Maximally entangled states are useful resource for severalinformation processing tasks: quantum key distribution, quantum teleportation, etc.
Secret key distillation: Need for secure communication protocols between two parties overnetwork – private reading.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 4 / 16
Motivation
Bidirectional channels:Simple model of quantum network with 2 clients, Alice and Bob.Model for quantum gates – CNOT, SWAP, etc.– in noisy intermediate-scale quantum(NISQ) computers.
Entanglement may increase, decrease or not change due to bipartite quantum interactions.
Entanglement distillation: Maximally entangled states are useful resource for severalinformation processing tasks: quantum key distribution, quantum teleportation, etc.
Secret key distillation: Need for secure communication protocols between two parties overnetwork – private reading.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 4 / 16
Motivation
Bidirectional channels:Simple model of quantum network with 2 clients, Alice and Bob.Model for quantum gates – CNOT, SWAP, etc.– in noisy intermediate-scale quantum(NISQ) computers.
Entanglement may increase, decrease or not change due to bipartite quantum interactions.
Entanglement distillation: Maximally entangled states are useful resource for severalinformation processing tasks: quantum key distribution, quantum teleportation, etc.
Secret key distillation: Need for secure communication protocols between two parties overnetwork – private reading.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 4 / 16
Goal
Two different information-processing tasks relevant for bipartite quantum interactions:
1 Entanglement distillation: generation of singlet state from two separated systems.
2 Secret key agreement: generation of maximal classical correlation between two separatedsystems, such that there’s no correlation with the bath.
New secure communication protocol between two parties, called private reading.
Non-asymptotic capacity of a channel N for a task: Maximum rate at which a given task canbe accomplished by allowing the use of N a finite number of times.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 5 / 16
Goal
Two different information-processing tasks relevant for bipartite quantum interactions:
1 Entanglement distillation: generation of singlet state from two separated systems.
2 Secret key agreement: generation of maximal classical correlation between two separatedsystems, such that there’s no correlation with the bath.
New secure communication protocol between two parties, called private reading.
Non-asymptotic capacity of a channel N for a task: Maximum rate at which a given task canbe accomplished by allowing the use of N a finite number of times.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 5 / 16
Goal
Two different information-processing tasks relevant for bipartite quantum interactions:
1 Entanglement distillation: generation of singlet state from two separated systems.
2 Secret key agreement: generation of maximal classical correlation between two separatedsystems, such that there’s no correlation with the bath.
New secure communication protocol between two parties, called private reading.
Non-asymptotic capacity of a channel N for a task: Maximum rate at which a given task canbe accomplished by allowing the use of N a finite number of times.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 5 / 16
Goal
Two different information-processing tasks relevant for bipartite quantum interactions:
1 Entanglement distillation: generation of singlet state from two separated systems.
2 Secret key agreement: generation of maximal classical correlation between two separatedsystems, such that there’s no correlation with the bath.
New secure communication protocol between two parties, called private reading.
Non-asymptotic capacity of a channel N for a task: Maximum rate at which a given task canbe accomplished by allowing the use of N a finite number of times.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 5 / 16
Goal
Two different information-processing tasks relevant for bipartite quantum interactions:
1 Entanglement distillation: generation of singlet state from two separated systems.
2 Secret key agreement: generation of maximal classical correlation between two separatedsystems, such that there’s no correlation with the bath.
New secure communication protocol between two parties, called private reading.
Non-asymptotic capacity of a channel N for a task: Maximum rate at which a given task canbe accomplished by allowing the use of N a finite number of times.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 5 / 16
Secret key generation over bidirectional channel
LOCC LOCC LOCC LOCC LOCC
LOCC-assisted bidirectional secret-key-agreement capacity [P2→2LOCC (NA′B′→AB).]
Bidirectional max-relative entropy of entanglement:
E 2→2max (NA′B′→AB) = sup
ψSAA′⊗ϕB′SB
Emax(SAA; BSB)N (ψ⊗ϕ),
where ψSAA′ , ϕB′SB are pure states, SA ' A′,SB ' B′,Emax(A : B)ρ = minσAB∈SEP Dmax(ρ‖σ), such that Dmax(ρ‖σ) = inf{λ : ρ ≤ 2λσ}.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 6 / 16
Secret key generation over bidirectional channel
LOCC LOCC LOCC LOCC LOCC
LOCC-assisted bidirectional secret-key-agreement capacity [P2→2LOCC (NA′B′→AB).]
Bidirectional max-relative entropy of entanglement:
E 2→2max (NA′B′→AB) = sup
ψSAA′⊗ϕB′SB
Emax(SAA; BSB)N (ψ⊗ϕ),
where ψSAA′ , ϕB′SB are pure states, SA ' A′,SB ' B′,Emax(A : B)ρ = minσAB∈SEP Dmax(ρ‖σ), such that Dmax(ρ‖σ) = inf{λ : ρ ≤ 2λσ}.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 6 / 16
Secret key generation over bidirectional channel
LOCC LOCC LOCC LOCC LOCC
LOCC-assisted bidirectional secret-key-agreement capacity [P2→2LOCC (NA′B′→AB).]
Bidirectional max-relative entropy of entanglement:
E 2→2max (NA′B′→AB) = sup
ψSAA′⊗ϕB′SB
Emax(SAA; BSB)N (ψ⊗ϕ),
where ψSAA′ , ϕB′SB are pure states, SA ' A′,SB ' B′,Emax(A : B)ρ = minσAB∈SEP Dmax(ρ‖σ), such that Dmax(ρ‖σ) = inf{λ : ρ ≤ 2λσ}.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 6 / 16
Secret key generation over bidirectional channel
LOCC LOCC LOCC LOCC LOCC
Theorem (LOCC-assisted secret key agreement)1n log2 M ≤ E 2→2
max (N ) + 1n log2
( 11− ε
).
P2→2LOCC (NA′B′→AB) ≤ E 2→2
max (NA′B′→AB), and this upper bound is in fact a strong conversebound.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 7 / 16
Secret key generation over bidirectional channel
LOCC LOCC LOCC LOCC LOCC
Theorem (LOCC-assisted secret key agreement)1n log2 M ≤ E 2→2
max (N ) + 1n log2
( 11− ε
).
P2→2LOCC (NA′B′→AB) ≤ E 2→2
max (NA′B′→AB), and this upper bound is in fact a strong conversebound.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 7 / 16
Secret key generation over bidirectional channel
LOCC LOCC LOCC LOCC LOCC
Theorem (LOCC-assisted secret key agreement)1n log2 M ≤ E 2→2
max (N ) + 1n log2
( 11− ε
).
P2→2LOCC (NA′B′→AB) ≤ E 2→2
max (NA′B′→AB), and this upper bound is in fact a strong conversebound.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 7 / 16
Proof outline: Secret key generation
LOCC LOCC LOCC LOCC LOCC
Private states [HHHO05,HHHO09]: State γSAKA:KBSB containing log2 K private bits.
Success probability in privacy test: Tr {Πγω} ≥ 1− ε. By [HHHO05,HHHO09],Tr {Πγσ} ≤ 1
K for σ ∈ SEP.
Main observation: E 2→2max is not enhanced by amortization.
Emax(SAA; BSB)σ ≤ Emax(SAA′; B′SB)ρ + E 2→2max (NA′B′→AB),
where σSAABSB = NA′B′→AB(ρSAA′B′SB ).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 8 / 16
Proof outline: Secret key generation
LOCC LOCC LOCC LOCC LOCC
Private states [HHHO05,HHHO09]: State γSAKA:KBSB containing log2 K private bits.
Success probability in privacy test: Tr {Πγω} ≥ 1− ε. By [HHHO05,HHHO09],Tr {Πγσ} ≤ 1
K for σ ∈ SEP.
Main observation: E 2→2max is not enhanced by amortization.
Emax(SAA; BSB)σ ≤ Emax(SAA′; B′SB)ρ + E 2→2max (NA′B′→AB),
where σSAABSB = NA′B′→AB(ρSAA′B′SB ).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 8 / 16
Proof outline: Secret key generation
LOCC LOCC LOCC LOCC LOCC
Private states [HHHO05,HHHO09]: State γSAKA:KBSB containing log2 K private bits.
Success probability in privacy test: Tr {Πγω} ≥ 1− ε. By [HHHO05,HHHO09],Tr {Πγσ} ≤ 1
K for σ ∈ SEP.
Main observation: E 2→2max is not enhanced by amortization.
Emax(SAA; BSB)σ ≤ Emax(SAA′; B′SB)ρ + E 2→2max (NA′B′→AB),
where σSAABSB = NA′B′→AB(ρSAA′B′SB ).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 8 / 16
Proof outline: Secret key generation
LOCC LOCC LOCC LOCC LOCC
Private states [HHHO05,HHHO09]: State γSAKA:KBSB containing log2 K private bits.
Success probability in privacy test: Tr {Πγω} ≥ 1− ε. By [HHHO05,HHHO09],Tr {Πγσ} ≤ 1
K for σ ∈ SEP.
Main observation: E 2→2max is not enhanced by amortization.
Emax(SAA; BSB)σ ≤ Emax(SAA′; B′SB)ρ + E 2→2max (NA′B′→AB),
where σSAABSB = NA′B′→AB(ρSAA′B′SB ).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 8 / 16
Entanglement generation over bidirectional channel
N
LA
LB
B’2 B2A’2
N
LA2
A2
LB2
PPT-PB’1 B1A’1
PPT-PPPT-P N B’ B
APPT-PPPT-P
n
n n
AM
BM
LA1
A1
LB1
A’nn
n
PPT-assisted bidirectional quantum capacity[Q2→2
PPT (NA′B′→AB).]
Bidirectional max-Rains Information[R2→2
max (NA′B′→AB) = log Γ2→2(NA′B′→AB)], where
Γ2→2(NA′B′→AB) = minimize ‖TrAB{VSAABSB + YSAABSB}‖∞subject to VSAABSB ,YSAABSB ≥ 0,
TBSB{VSAABSB − YSAABSB} ≥ JNSAABSB ,
such that SA ' A′, SB ' B′.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 9 / 16
Entanglement generation over bidirectional channel
N
LA
LB
B’2 B2A’2
N
LA2
A2
LB2
PPT-PB’1 B1A’1
PPT-PPPT-P N B’ B
APPT-PPPT-P
n
n n
AM
BM
LA1
A1
LB1
A’nn
n
PPT-assisted bidirectional quantum capacity[Q2→2
PPT (NA′B′→AB).]
Bidirectional max-Rains Information[R2→2
max (NA′B′→AB) = log Γ2→2(NA′B′→AB)], where
Γ2→2(NA′B′→AB) = minimize ‖TrAB{VSAABSB + YSAABSB}‖∞subject to VSAABSB ,YSAABSB ≥ 0,
TBSB{VSAABSB − YSAABSB} ≥ JNSAABSB ,
such that SA ' A′, SB ' B′.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 9 / 16
Entanglement generation over bidirectional channel
N
LA
LB
B’2 B2A’2
N
LA2
A2
LB2
PPT-PB’1 B1A’1
PPT-PPPT-P N B’ B
APPT-PPPT-P
n
n n
AM
BM
LA1
A1
LB1
A’nn
n
PPT-assisted bidirectional quantum capacity[Q2→2
PPT (NA′B′→AB).]
Bidirectional max-Rains Information[R2→2
max (NA′B′→AB) = log Γ2→2(NA′B′→AB)], where
Γ2→2(NA′B′→AB) = minimize ‖TrAB{VSAABSB + YSAABSB}‖∞subject to VSAABSB ,YSAABSB ≥ 0,
TBSB{VSAABSB − YSAABSB} ≥ JNSAABSB ,
such that SA ' A′, SB ' B′.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 9 / 16
Entanglement generation over bidirectional channel
N
LA
LB
B’2 B2A’2
N
LA2
A2
LB2
PPT-PB’1 B1A’1
PPT-PPPT-P N B’ B
APPT-PPPT-P
n
n n
AM
BM
LA1
A1
LB1
A’nn
n
PPT-assisted bidirectional quantum capacity[Q2→2
PPT (NA′B′→AB).]
Bidirectional max-Rains Information[R2→2
max (NA′B′→AB) = log Γ2→2(NA′B′→AB)], where
Γ2→2(NA′B′→AB) = minimize ‖TrAB{VSAABSB + YSAABSB}‖∞subject to VSAABSB ,YSAABSB ≥ 0,
TBSB{VSAABSB − YSAABSB} ≥ JNSAABSB ,
such that SA ' A′, SB ' B′.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 9 / 16
Entanglement generation over bidirectional channel
N
LA
LB
B’2 B2A’2
N
LA2
A2
LB2
PPT-PB’1 B1A’1
PPT-PPPT-P N B’ B
APPT-PPPT-P
n
n n
AM
BM
LA1
A1
LB1
A’nn
n
Theorem (PPT-assisted distillable entanglement generation)1n log2 M ≤ R2→2
max (N ) + 1n log2
( 11− ε
).
Q2→2PPT (NA′B′→AB) ≤ R2→2
max (NA′B′→AB), and this upper bound is in fact a strong conversebound.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 10 / 16
Entanglement generation over bidirectional channel
N
LA
LB
B’2 B2A’2
N
LA2
A2
LB2
PPT-PB’1 B1A’1
PPT-PPPT-P N B’ B
APPT-PPPT-P
n
n n
AM
BM
LA1
A1
LB1
A’nn
n
Theorem (PPT-assisted distillable entanglement generation)1n log2 M ≤ R2→2
max (N ) + 1n log2
( 11− ε
).
Q2→2PPT (NA′B′→AB) ≤ R2→2
max (NA′B′→AB), and this upper bound is in fact a strong conversebound.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 10 / 16
Entanglement generation over bidirectional channel
N
LA
LB
B’2 B2A’2
N
LA2
A2
LB2
PPT-PB’1 B1A’1
PPT-PPPT-P N B’ B
APPT-PPPT-P
n
n n
AM
BM
LA1
A1
LB1
A’nn
n
Theorem (PPT-assisted distillable entanglement generation)1n log2 M ≤ R2→2
max (N ) + 1n log2
( 11− ε
).
Q2→2PPT (NA′B′→AB) ≤ R2→2
max (NA′B′→AB), and this upper bound is in fact a strong conversebound.
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 10 / 16
Application: Private ReadingFirst recall: (Quantum) Reading [BRV00,Pir11] of memory devices.Memory device: message encoded into a sequence of channels from a memory cellSX = {N x
B′→B}x∈X .Alice encodes m ∈M into codewords (x1(m), ..., xn(m)), and sets the device to(N x1(m)
B′→B, ...,Nxn(m)B′→B
).
Bob can enter quantum states and do channel discrimination to learn m. Natural toemploy adaptive strategy [DW17].
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 11 / 16
Application: Private ReadingFirst recall: (Quantum) Reading [BRV00,Pir11] of memory devices.Memory device: message encoded into a sequence of channels from a memory cellSX = {N x
B′→B}x∈X .Alice encodes m ∈M into codewords (x1(m), ..., xn(m)), and sets the device to(N x1(m)
B′→B, ...,Nxn(m)B′→B
).
Bob can enter quantum states and do channel discrimination to learn m. Natural toemploy adaptive strategy [DW17].
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 11 / 16
Application: Private ReadingFirst recall: (Quantum) Reading [BRV00,Pir11] of memory devices.Memory device: message encoded into a sequence of channels from a memory cellSX = {N x
B′→B}x∈X .Alice encodes m ∈M into codewords (x1(m), ..., xn(m)), and sets the device to(N x1(m)
B′→B, ...,Nxn(m)B′→B
).
Bob can enter quantum states and do channel discrimination to learn m. Natural toemploy adaptive strategy [DW17].
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 11 / 16
Application: Private ReadingFirst recall: (Quantum) Reading [BRV00,Pir11] of memory devices.Memory device: message encoded into a sequence of channels from a memory cellSX = {N x
B′→B}x∈X .Alice encodes m ∈M into codewords (x1(m), ..., xn(m)), and sets the device to(N x1(m)
B′→B, ...,Nxn(m)B′→B
).
Bob can enter quantum states and do channel discrimination to learn m. Natural toemploy adaptive strategy [DW17].
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 11 / 16
Application: Private Reading
E3E2 êk
B3B’3B’2B1x1(k)MB’1 B2
A1
x2(k)M
A2
x3(k)M
L L L
E1
B1 B2 B3
Private reading: Eve present when Bob performs the readout: Wiretap memory cellSX = {Mx
B′→BE}x∈X . Special case: Isometric wiretap memory cell S isoX .
The non-adaptive private reading capacity of a wiretap memory cell SX is given by
P readn-a
(MX
)= sup
nmax
pXn ,σLBB′n
1n [I(Xn; LBBn)τ − I(Xn; En)τ ] ,
where τXnLBBnEn :=∑
xn pXn (xn)|xn〉〈xn|Xn ⊗MxnB′n→BnEn (σLBB′n ).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 12 / 16
Application: Private Reading
E3E2 êk
B3B’3B’2B1x1(k)MB’1 B2
A1
x2(k)M
A2
x3(k)M
L L L
E1
B1 B2 B3
Private reading: Eve present when Bob performs the readout: Wiretap memory cellSX = {Mx
B′→BE}x∈X . Special case: Isometric wiretap memory cell S isoX .
The non-adaptive private reading capacity of a wiretap memory cell SX is given by
P readn-a
(MX
)= sup
nmax
pXn ,σLBB′n
1n [I(Xn; LBBn)τ − I(Xn; En)τ ] ,
where τXnLBBnEn :=∑
xn pXn (xn)|xn〉〈xn|Xn ⊗MxnB′n→BnEn (σLBB′n ).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 12 / 16
Application: Private Reading
E3E2 êk
B3B’3B’2B1x1(k)MB’1 B2
A1
x2(k)M
A2
x3(k)M
L L L
E1
B1 B2 B3
Private reading: Eve present when Bob performs the readout: Wiretap memory cellSX = {Mx
B′→BE}x∈X . Special case: Isometric wiretap memory cell S isoX .
The non-adaptive private reading capacity of a wiretap memory cell SX is given by
P readn-a
(SX)
= supn
maxpXn ,σLBB′n
1n [I(Xn; LBBn)τ − I(Xn; En)τ ] ,
where τXnLBBnEn :=∑
xn pXn (xn)|xn〉〈xn|Xn ⊗MxnB′n→BnEn (σLBB′n ).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 13 / 16
Application: Private Reading
E3E2 êk
B3B’3B’2B1x1(k)MB’1 B2
A1
x2(k)M
A2
x3(k)M
L L L
E1
B1 B2 B3
Private reading: Eve present when Bob performs the readout: Wiretap memory cellSX = {Mx
B′→BE}x∈X . Special case: Isometric wiretap memory cell S isoX .
The non-adaptive private reading capacity of a wiretap memory cell SX is given by
P readn-a
(SX)
= supn
maxpXn ,σLBB′n
1n [I(Xn; LBBn)τ − I(Xn; En)τ ] ,
where τXnLBBnEn :=∑
xn pXn (xn)|xn〉〈xn|Xn ⊗MxnB′n→BnEn (σLBB′n ).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 13 / 16
Application: Private Reading
E3E2 êk
B3B’3B’2B1x1(k)MB’1 B2
A1
x2(k)M
A2
x3(k)M
L L L
E1
B1 B2 B3
Private reading: Eve present when Bob performs the readout: Wiretap memory cellSX = {Mx
B′→BE}x∈X . Special case: Isometric wiretap memory cell S isoX .
The non-adaptive private reading capacity of a wiretap memory cell SX is given by
P readn-a
(SX)
= supn
maxpXn ,σLBB′n
1n [I(Xn; LBBn)τ − I(Xn; En)τ ] ,
where τXnLBBnEn :=∑
xn pXn (xn)|xn〉〈xn|Xn ⊗MxnB′n→BnEn (σLBB′n ).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 13 / 16
Application: Private Reading
E3E2 êk
B3B’3B’2B1x1(k)MB’1 B2
A1
x2(k)M
A2
x3(k)M
L L L
E1
B1 B2 B3
Private reading: Eve present when Bob performs the readout: Wiretap memory cellSX = {Mx
B′→BE}x∈X . Special case: Isometric wiretap memory cell S isoX .
The non-adaptive private reading capacity of a wiretap memory cell SX is given by
P readn-a
(SX)
= supn
maxpXn ,σLBB′n
1n [I(Xn; LBBn)τ − I(Xn; En)τ ] ,
where τXnLBBnEn :=∑
xn pXn (xn)|xn〉〈xn|Xn ⊗MxnB′n→BnEn (σLBB′n ).
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 13 / 16
Private reading capacity
The strong converse private reading capacity P read(S isoX ) of an isometric wiretap memory cell
S isoX = {UMx
B′→BE}x∈X is bounded from above as
P read(S isoX ) ≤ E 2→2
max (N SXB′→XB),
whereN SXB′→XB(·) := TrE
{U SXB′→XBE (·)
(U SXB′→XBE
)†},
such thatU SXB′→XBE :=
∑x∈X|x〉〈x |X ⊗ UMx
B′→BE .
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 14 / 16
Conclusion
We derived upper bounds on entanglement generation and secret-key-agreementcapacities over bidirectional channels. Sizes of reference systems are same as size of inputsystems (Open question in [BHLS03]).
Obtain tighter upper bounds for channels obeying certain symmetries, see [DBW17].
Introduced secure protocol for reading of memory devices under scrutiny of aneavesdropper. Both upper and lower bounds for this protocol can be found in [DBW17].
[DBW17] arXiv : 1712.00827
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 15 / 16
Conclusion
We derived upper bounds on entanglement generation and secret-key-agreementcapacities over bidirectional channels. Sizes of reference systems are same as size of inputsystems (Open question in [BHLS03]).
Obtain tighter upper bounds for channels obeying certain symmetries, see [DBW17].
Introduced secure protocol for reading of memory devices under scrutiny of aneavesdropper. Both upper and lower bounds for this protocol can be found in [DBW17].
[DBW17] arXiv : 1712.00827
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 15 / 16
Conclusion
We derived upper bounds on entanglement generation and secret-key-agreementcapacities over bidirectional channels. Sizes of reference systems are same as size of inputsystems (Open question in [BHLS03]).
Obtain tighter upper bounds for channels obeying certain symmetries, see [DBW17].
Introduced secure protocol for reading of memory devices under scrutiny of aneavesdropper. Both upper and lower bounds for this protocol can be found in [DBW17].
[DBW17] arXiv : 1712.00827
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 15 / 16
Conclusion
We derived upper bounds on entanglement generation and secret-key-agreementcapacities over bidirectional channels. Sizes of reference systems are same as size of inputsystems (Open question in [BHLS03]).
Obtain tighter upper bounds for channels obeying certain symmetries, see [DBW17].
Introduced secure protocol for reading of memory devices under scrutiny of aneavesdropper. Both upper and lower bounds for this protocol can be found in [DBW17].
[DBW17] arXiv : 1712.00827
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 15 / 16
SWAP and Collective dephasing
0 0.2 0.4 0.6 0.8 1p
1
1.2
1.4
1.6
1.8
2
R2-
2m
ax
Collective Dephasing
2
2 /3/2
2 /5/3
2 /7/4
Collective dephasing: |00〉 → |00〉,|01〉 → eiφ|01〉, |10〉 → eiφ|10〉,|11〉 → e2iφ|11〉.Swap operator S =
∑ij |ij〉〈ji | and
collective dephasing:
NA′B′→AB(ρ)=pSρS† + (1− p)UφSρS†Uφ†
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 16 / 16
SWAP and Collective dephasing
0 0.2 0.4 0.6 0.8 1p
1
1.2
1.4
1.6
1.8
2
R2-
2m
ax
Collective Dephasing
2
2 /3/2
2 /5/3
2 /7/4
Collective dephasing: |00〉 → |00〉,|01〉 → eiφ|01〉, |10〉 → eiφ|10〉,|11〉 → e2iφ|11〉.Swap operator S =
∑ij |ij〉〈ji | and
collective dephasing:
NA′B′→AB(ρ)=pSρS† + (1− p)UφSρS†Uφ†
Siddhartha Das (QST@LSU) Bipartite quantum interactions QCrypt ’18 16 / 16