A Quantum Multiparty Packing Lemma and the Relay Channel Dawei Ding - - PowerPoint PPT Presentation

A Quantum Multiparty Packing Lemma and the Relay Channel

Dawei Ding Stanford University Joint with Hrant Gharibyan, Patrick Hayden, and Michael Walter

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Outline

Introduction Relay channel Relay channel definition Multihop bound Quantum multiparty packing lemma Packing lemma statement Conclusion

Dawei Ding (Stanford University) |

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Introduction

Message packing

Input Output

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Introduction

Message packing

Encoded Packed Raw Decoded

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Introduction

Communication as message packing

Taken from Mark Wilde’s From Classical to Quantum Shannon Theory

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Introduction

Encoding messages into quantum systems

◮ Classical-quantum black box: Input is classical, output is quantum

Dawei Ding (Stanford University) |

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Introduction

Encoding messages into quantum systems

◮ Classical-quantum black box: Input is classical, output is quantum

Theorem (Holevo-Schumacher-Westmoreland theorem)

The classical capacity of a quantum channel NA→B with separable encodings is C(N) = max

{pX ,ρ(x)

A }

I(X; B)ρ where ρXB ≡

• x

pX(x) |x x|X ⊗ ρ(x)

B .

Dawei Ding (Stanford University) |

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Introduction

Encoding messages into quantum systems

◮ Network information theory: multiple senders and/or multiple

receivers

Dawei Ding (Stanford University) |

6

Introduction

Encoding messages into quantum systems

◮ Network information theory: multiple senders and/or multiple

receivers

◮ Classical-quantum channels: Multiple classical inputs, quantum

• utput

Dawei Ding (Stanford University) |

6

Introduction

Encoding messages into quantum systems

◮ Network information theory: multiple senders and/or multiple

receivers

◮ Classical-quantum channels: Multiple classical inputs, quantum

• utput

◮ Mostly open for quantum channels, especially one-shot Dawei Ding (Stanford University) |

6

Introduction

Encoding messages into quantum systems

◮ Network information theory: multiple senders and/or multiple

receivers

◮ Classical-quantum channels: Multiple classical inputs, quantum

• utput

◮ Mostly open for quantum channels, especially one-shot

◮ Multiparty packing: encoding multiple messages Mj into a

quantum system B via multiple classical systems Xv

Dawei Ding (Stanford University) |

6

Introduction

Encoding messages into quantum systems

◮ Network information theory: multiple senders and/or multiple

receivers

◮ Classical-quantum channels: Multiple classical inputs, quantum

• utput

◮ Mostly open for quantum channels, especially one-shot

◮ Multiparty packing: encoding multiple messages Mj into a

quantum system B via multiple classical systems Xv

◮ Multiple senders: multiple access channel, relay channel Dawei Ding (Stanford University) |

6

Introduction

Encoding messages into quantum systems

◮ Network information theory: multiple senders and/or multiple

receivers

◮ Classical-quantum channels: Multiple classical inputs, quantum

• utput

◮ Mostly open for quantum channels, especially one-shot

◮ Multiparty packing: encoding multiple messages Mj into a

quantum system B via multiple classical systems Xv

◮ Multiple senders: multiple access channel, relay channel

◮ Assuming classical-quantum channel, codebook is of the form

{xv(m)}v∈V,m∈M, where M =×

j∈J Mj

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Relay channel

Relay channel definition Sender Receiver Relay

◮ NX1X2→B2B3 : X1 × X2 → HB2 ⊗ HB3, (x1, x2) → ρ(x1x2) B2B3 (SWV, 2012)

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Relay channel

Relay channel definition Sender Receiver Relay

◮ NX1X2→B2B3 : X1 × X2 → HB2 ⊗ HB3, (x1, x2) → ρ(x1x2) B2B3 (SWV, 2012) ◮ Relay’s transmission affects relay’s system, sender’s transmission

affects receiver’s: More general than concatenated channels!

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Relay channel definition

Multihop bound

Quantum generalization of classical protocols

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Relay channel definition

Multihop bound

Quantum generalization of classical protocols

◮ Multihop

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Relay channel definition

Multihop bound

Quantum generalization of classical protocols

◮ Multihop

◮ Treat relay channel as concatenated channel: sender transmits

message to relay, relay transmits decoded message to receiver

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Relay channel definition

Multihop bound

Random codebook C =

b

• j=1

{(x1)j(mj), (x2)j(mj−1)}mj∈Mj,mj−1∈Mj−1

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Relay channel definition

Multihop bound

Random codebook C =

b

• j=1

{(x1)j(mj), (x2)j(mj−1)}mj∈Mj,mj−1∈Mj−1 Code:

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Relay channel

Remarks

◮ Coherent multihop, decode forward, partial decode forward

◮ All straightforward generalizations of classical protocols Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Random codebooks for network communication

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Random codebooks for network communication

◮ Has structure depending on network setting and protocol chosen

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Random codebooks for network communication

◮ Has structure depending on network setting and protocol chosen ◮ How to represent mathematically?

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Random codebooks for network communication

◮ Has structure depending on network setting and protocol chosen ◮ How to represent mathematically?

◮ Muliplex Bayesian network Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Detailed definition:

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Detailed definition:

◮ Let X be a Bayesian network with respect to (DAG) G = (V, E)

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Detailed definition:

◮ Let X be a Bayesian network with respect to (DAG) G = (V, E)

◮ Bayesian network: statistical model that represents a set of random

variables and their conditional dependencies via a DAG

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Detailed definition:

◮ Let X be a Bayesian network with respect to (DAG) G = (V, E)

◮ Bayesian network: statistical model that represents a set of random

variables and their conditional dependencies via a DAG

◮ X composed of Xv for v ∈ V

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Detailed definition:

◮ Let X be a Bayesian network with respect to (DAG) G = (V, E)

◮ Bayesian network: statistical model that represents a set of random

variables and their conditional dependencies via a DAG

◮ X composed of Xv for v ∈ V ◮ For v ∈ V, let pa(v) ≡ {v′ ∈ V|(v′, v) ∈ E}

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Detailed definition:

◮ Let X be a Bayesian network with respect to (DAG) G = (V, E)

◮ Bayesian network: statistical model that represents a set of random

variables and their conditional dependencies via a DAG

◮ X composed of Xv for v ∈ V ◮ For v ∈ V, let pa(v) ≡ {v′ ∈ V|(v′, v) ∈ E} ◮ Message sets: let J be an index set labeling the multiple message

sets

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Detailed definition:

◮ Let X be a Bayesian network with respect to (DAG) G = (V, E)

◮ Bayesian network: statistical model that represents a set of random

variables and their conditional dependencies via a DAG

◮ X composed of Xv for v ∈ V ◮ For v ∈ V, let pa(v) ≡ {v′ ∈ V|(v′, v) ∈ E} ◮ Message sets: let J be an index set labeling the multiple message

sets

◮ Let ind : V → P(J) denote the (indices of) the message sets the

random variable Xv will be generated over

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Detailed definition:

◮ Let X be a Bayesian network with respect to (DAG) G = (V, E)

◮ Bayesian network: statistical model that represents a set of random

variables and their conditional dependencies via a DAG

◮ X composed of Xv for v ∈ V ◮ For v ∈ V, let pa(v) ≡ {v′ ∈ V|(v′, v) ∈ E} ◮ Message sets: let J be an index set labeling the multiple message

sets

◮ Let ind : V → P(J) denote the (indices of) the message sets the

random variable Xv will be generated over

◮ Index inheritance: For v ∈ V, ind(v ′) ⊆ ind(v) for all v ′ ∈ pa(v) Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Definition of multiplex Bayesian network

We call the tuple B = (G, X, M, ind), where M ≡×

j∈J Mj, a

multiplex Bayesian network. Visualization:

◮ Adjoin to G additional vertices Mj for each j ∈ J ◮ Connect with edge Xv to all Mj where j ∈ ind(v)

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Fixing multiplex Bayesian network (G, X, M, ind), generate a random codebook {xv(m)}v∈V,m∈M

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Fixing multiplex Bayesian network (G, X, M, ind), generate a random codebook {xv(m)}v∈V,m∈M

◮ xv corresponds to vertices of G

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Fixing multiplex Bayesian network (G, X, M, ind), generate a random codebook {xv(m)}v∈V,m∈M

◮ xv corresponds to vertices of G ◮ M is message set

Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Fixing multiplex Bayesian network (G, X, M, ind), generate a random codebook {xv(m)}v∈V,m∈M

◮ xv corresponds to vertices of G ◮ M is message set ◮ xv(m) only depends on mj where j ∈ ind(v)

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Quantum multiparty packing lemma

Multiplex Bayesian networks

Algorithm for generating random codebook: for v ∈ V do for mv ∈ Mind(v) do generate xv(mv) according to pXv|Xpa(v)

• ·|xpa(v)(mpa(v))
• for m¯

v ∈ Mind(v) do

xv(mv, m¯

v) = xv(mv)

end for end for end for

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Quantum Multiparty Packing Lemma

Simultaneous Decoding

◮ How to find a decoder?

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Quantum Multiparty Packing Lemma

Simultaneous Decoding

◮ How to find a decoder? ◮ First guess: sequential cancellation decoding

Dawei Ding (Stanford University) |

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Quantum Multiparty Packing Lemma

Simultaneous Decoding

◮ How to find a decoder? ◮ First guess: sequential cancellation decoding

◮ Measurement disturbance Dawei Ding (Stanford University) |

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Quantum Multiparty Packing Lemma

Simultaneous Decoding

◮ How to find a decoder? ◮ First guess: sequential cancellation decoding

◮ Measurement disturbance ◮ Necessity of time sharing Dawei Ding (Stanford University) |

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Quantum Multiparty Packing Lemma

Simultaneous Decoding

◮ How to find a decoder? ◮ First guess: sequential cancellation decoding

◮ Measurement disturbance ◮ Necessity of time sharing

◮ Simultaneous decoder!

Dawei Ding (Stanford University) |

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Quantum Multiparty Packing Lemma

Simultaneous Decoding

◮ How to find a decoder? ◮ First guess: sequential cancellation decoding

◮ Measurement disturbance ◮ Necessity of time sharing

◮ Simultaneous decoder!

◮ Needs quantum joint typicality Dawei Ding (Stanford University) |

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Quantum Multiparty Packing Lemma

Sen’s quantum joint typicality

◮ Quantum joint typicality

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Quantum Multiparty Packing Lemma

Sen’s quantum joint typicality

◮ Quantum joint typicality

◮ Long open question Dawei Ding (Stanford University) |

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Quantum Multiparty Packing Lemma

Sen’s quantum joint typicality

◮ Quantum joint typicality

◮ Long open question

◮ Recent breakthrough: Pranab Sen, arXiv 1806.0727

“A one-shot quantum joint typicality lemma”

◮ [Insert complicated equations here.] Dawei Ding (Stanford University) |

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Quantum Multiparty Packing Lemma

Sen’s quantum joint typicality

◮ Quantum joint typicality

◮ Long open question

◮ Recent breakthrough: Pranab Sen, arXiv 1806.0727

“A one-shot quantum joint typicality lemma”

◮ [Insert complicated equations here.]

Typicality ⇐ ⇒ Packing lemma ⇐ ⇒ Capacity theorem

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Quantum multiparty packing lemma

Packing lemma statement

◮ Let C be codebook generated by multiplex Bayesian network

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Quantum multiparty packing lemma

Packing lemma statement

◮ Let C be codebook generated by multiplex Bayesian network ◮ Let {ρ(x) B }x∈X be family of quantum states

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Quantum multiparty packing lemma

Packing lemma statement

◮ Let C be codebook generated by multiplex Bayesian network ◮ Let {ρ(x) B }x∈X be family of quantum states

◮ Defines the black box Dawei Ding (Stanford University) |

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Quantum multiparty packing lemma

Packing lemma statement

◮ Let C be codebook generated by multiplex Bayesian network ◮ Let {ρ(x) B }x∈X be family of quantum states

◮ Defines the black box

◮ Let D ⊆ J be subset of messages to be decoded, D be guess for

• ther messages

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Quantum multiparty packing lemma

Packing lemma statement

Asymptotic quantum multiparty packing lemma (simplified)

Let B = (G, X, M, ind), Cn = {xn(m)}m∈M, {ρ(x)

B }x∈X , D ⊆ J, and

ε ∈ (0, 1). Then there exists a POVM {Q

(mD|mD) B

}mD∈MD for each mD ∈ MD that, assuming the guess was right, can decode (mD, mD) ∈ M encoded into n

i=1 ρ (xi(mD,mD)) Bi

with vanishing probability of error as n → ∞ if ∀ ∅ = T ⊆ D,

• t∈T

Rt < I(XST ; B|XST )ρ, where Rt ≡ log |Mt|, ST ≡ {v ∈ V | ind(v) ∩ T = ∅} ⊆ V, ρXB ≡

• x∈X

pX(x) |x x|X ⊗ ρ(x)

B .

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Conclusion

Summary

◮ Established a one-shot quantum multiparty packing lemma

Dawei Ding (Stanford University) |

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Conclusion

Summary

◮ Established a one-shot quantum multiparty packing lemma

◮ Multiplex Bayesian networks Dawei Ding (Stanford University) |

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Conclusion

Summary

◮ Established a one-shot quantum multiparty packing lemma

◮ Multiplex Bayesian networks ◮ Cornerstone of classical network information theory is packing

lemma

Dawei Ding (Stanford University) |

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Conclusion

Summary

◮ Established a one-shot quantum multiparty packing lemma

◮ Multiplex Bayesian networks ◮ Cornerstone of classical network information theory is packing

lemma

◮ Allows hitherto impossible direct quantum generalization Dawei Ding (Stanford University) |

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Conclusion

Summary

◮ Established a one-shot quantum multiparty packing lemma

◮ Multiplex Bayesian networks ◮ Cornerstone of classical network information theory is packing

lemma

◮ Allows hitherto impossible direct quantum generalization

◮ Demonstrated direct quantum generalization for

classical-quantum relay channel

Dawei Ding (Stanford University) |

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Conclusion

Summary

◮ Established a one-shot quantum multiparty packing lemma

◮ Multiplex Bayesian networks ◮ Cornerstone of classical network information theory is packing

lemma

◮ Allows hitherto impossible direct quantum generalization

◮ Demonstrated direct quantum generalization for

classical-quantum relay channel

◮ Multihop ◮ Cohere multihop ◮ Decode forward ◮ Partial decode forward Dawei Ding (Stanford University) |

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Conclusion

Open questions

Open questions and future work:

Dawei Ding (Stanford University) |

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Conclusion

Open questions

Open questions and future work:

◮ Apply packing lemma to other settings?

Dawei Ding (Stanford University) |

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Conclusion

Open questions

Open questions and future work:

◮ Apply packing lemma to other settings? ◮ Other notions of joint typicality?

Dawei Ding (Stanford University) |

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Conclusion

Open questions

Open questions and future work:

◮ Apply packing lemma to other settings? ◮ Other notions of joint typicality? ◮ Rate region simplification?

Dawei Ding (Stanford University) |

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Conclusion

Open questions

Open questions and future work:

◮ Apply packing lemma to other settings? ◮ Other notions of joint typicality? ◮ Rate region simplification? ◮ More general packing lemma?

Dawei Ding (Stanford University) |

Thank you!

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Backup

Reduction to classical packing lemma

Lemma (Classical packing lemma)

Let (U, X, Y) be a triple of random variables with joint distribution pUXY. For each n, let ( Un, Y n) be a pair of arbitrarily distributed random sequences and { X n(m)} a family of at most 2nR random sequences such that each X n(m) is conditionally independent of Y n given Un. Further assume that each X n(m) is distributed as ⊗n

i=1pX|U= Ui given

• Un. Then, there exists δ(ε) that tends to zero as ε → 0 such that

lim

n→∞ Pr((

Un, X n(m), Y n) ∈ T (n)

ε

for some m) = 0 if R < I(X; Y|U) − δ(ε), where T (n)

ε

is the set of ε-typical strings of length n with respect to pUXY. Usual channel coding follows from U = ∅ and Y n ∼ p⊗n

Y .

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Backup

Reduction to classical packing lemma

i.i.d. version of classical packing lemma is a corollary of our quantum multiparty packing lemma.

◮ Run algorithm n times, codebook generated is

Un, X n(m)

◮ Set of quantum states

• ρ(u,x)
• Y

≡

• y∈Y pY|UX(

y|u, x) | y y|

Y

• u∈U,x∈X

◮ “Typicality test” POVM

• Q(m)
• Y n
• m∈M

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Backup

Sen’s quantum joint typicality lemma (baby version)

Let B1 . . . Bk be a k-partite quantum system with each Bi isomorphic to a Hilbert space H. Let ρB1...Bk be a quantum state in B1 . . . Bk. For a subset S ⊆ [k], let BS denote the systems {Bs : s ∈ S}. Let ρBS denote the marginal state on BS obtained by tracing out the systems in ¯ S := [k] \ S from ρB1...Bk . Let 0 < ǫ < 1. Let K be a Hilbert space of dimension dK There exist a state τK⊗[k] independent of ρB[k], a state (ρ′)B′

[k], and a POVM element Π′

B′

[k] on B′

1 . . . B′ k where

B′

i ∼

= Bi ⊗ K, with the following properties:

• 1. (ρ′)B′

[k] − ρB[k] ⊗ τK⊗[k]1 ≤ f(k, ε);

• 2. tr[(Π′)B′

[k](ρ′)B′ [k]] ≥ 1 − g(k, ε);

• 3. For every set S, {} = S ⊂ [k],

tr[(Π′)B′

[k]((ρ′)B′ S⊗(ρ′)B′ ¯ S)] ≤ 2−Dǫ H(ρB[k]ρBS ⊗ρB¯ S ).+h(k, dH, dK)

From Pranab Sen, “A one-shot quantum joint typicality lemma”

Dawei Ding (Stanford University) |