Quantum Communication and Cryptography · Quantum Communication and Cryptography Introduction Outline Overview recent work in quantum information theory. (in the strict sense) 1 Resource

Quantum Communication and Cryptography


Joseph M. Renes

Theoretical Quantum Physics, Institut für Angewandte PhysikTechnische Universität Darmstadt

IQING 5 Innsbruck 2007 April 12

J. M. Renes IQING 5


Introduction

Overview

Alice Bobquantum channel

classical channel

shared quantum states

shared classical randomness

Communication between distant partiesSend messagesDistributed computationCrytographic useCreate entanglement

Major point: Interchangeability of resourcesnoisy channel −→ noiseless channel; noiseless + randomness −→ noisylow-fielity EPR pairs + classical communication −→ maximal entanglement. . .

J. M. Renes IQING 5


Introduction

Outline

Overview recent work in quantum information theory. (in the strict sense)

1 Resource framework: how the myriad of protocols fit together.Expressed as inequalities, for instance teleportation:[qq] + 2[c → c] ≤ [q → q]Quantum family tree: Fully-Quantum Slepian-Wolf

2 Connection between Entanglement and Secret Keysstatic/dynamic & quantum/classical:

entanglement generation & distillationsecret key generation & distillation

protocols have essentially the same proof3 Erasure is Fundamental

Closer look at quantum capacityDecoupling from the environment is enoughFor coding, just pick a random subspace

J. M. Renes IQING 5


Resource Framework

Unit Resources and Simple Protocols

Unit Resources

Ideal cbit channel [c → c][q → q] Ideal qbit channel

Private cbit channel (c → c)

Shared randomness [cc] [qq] Shared entanglement

Simple Protocols

Teleportation [qq] + 2[c → c] ≥ [q → q]

Dense Coding [q → q] + [qq] ≥ 2[c → c]

Entanglement Distribution [q → q] ≥ [qq]

Private Communication [qq] + [c → c] ≥ (c → c)[q → q] ≥ (c → c)

I. Devetak, A. W. Harrow, and A. Winter, quant-ph/0512015

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Resource Framework

General Resources and Standard Protocols

General Resources

noisy channel 〈N A→B〉 shared state 〈ϕAB〉

Standard Protocols

Schumacher compression S(ρA)[q → q] ≥ 〈idA→B〉HSW theorem 〈N T→B〉 ≥ I(X: B)[c → c]

Quantum capacity 〈N T→B〉 ≥ I(A〉B)[q → q]

Entanglement distillation 〈ρAB〉+ I(A:E)[c → c] ≥ I(A〉B)[qq]

J. M. Renes IQING 5


Resource Framework


General Resources


Standard Protocols




Transmit ρA by first compressing it, then using a noiseless channel

J. M. Renes IQING 5


Resource Framework


General Resources


Standard Protocols




Transmit classical message over a quantum channel

Start with the state ρXT =∑

x px |x〉〈x|X ⊗ ρT

x

Send T to obtain σXB = N T→B(ρXT)

I(X: B) is the Holevo quantity χ = S(∑

x pxσBx ) −

∑x pxS(σB

x ).

J. M. Renes IQING 5


Resource Framework


General Resources


Standard Protocols




Encode quantum messages for transmission over a noisy channel

Input ensemble state ρT has purification |ψ〉〈ψ|TA

Channel outputs σAB = N T→B(ρTA)

I(A〉B) = −H(A|B) is the coherent information.

J. M. Renes IQING 5


Resource Framework


General Resources


Standard Protocols




Distill maximal entanglement from ϕAB

use only local operations and classical communication

E comes from purifying ϕAB to |ψ〉〈ψ|ABE

J. M. Renes IQING 5


Resource Framework

Quantum Family Tree

FQSWFQRS

♀

EntanglementDistillation

Noisy Dense Coding

Noisy Teleportation

♂Entanglement-

assisted Capacity

Quantum Capacity

State Merging

Broadcast

QMAC

Environment-assistedCapacity Fully-Quantum Slepian-Wolf

Given: state |ϕ〉ABR and noiselessquantum communication

Goal: distill EPR pairs andtransfer ϕA completely to Bob

R A

B

|ϕ〉

R A

BB

|Φ〉|ϕ〉

〈WS→AB : ϕS〉+ 12 I(A:R)[q → q] ≥

12 I(A:B)[qq] + 〈idS→B

: ϕS〉.A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter,quant-ph/0606225

J. M. Renes IQING 5


Entanglement and Secret Keys

Four Protocols

r = I(A〉B) = H(B) − H(AB)I. Devetak and A. Winter, PRL 93 080501 (2004).

Entanglement Distillation

Convert (ρAB)⊗n into nr ebitsΦ⊗nr using 1-LOCC

Secret Key Distillation

Convert (ρAB)⊗n into nr secretkey bits κ⊗nr using 1-LOPC

Entanglement Generation

Use n copies of the channel Nto generate nr bits of sharedentanglement, (ΦAB)⊗nr

Secret Key Generation

Use n copies of the channel Nto generate nr bits of sharedsecret key, (κABE)⊗nr

Rows: Static vs. DynamicColumns: Classical vs. Quantum

J. M. Renes IQING 5



Four Protocols










A T E N D B

Specify Encoding for Alice and Decoding for Bob.Same as quantum channel capacity

J. M. Renes IQING 5



Four Protocols










T E N M

Specify Encoding for Alice and Measurement for Bob.κABE = 1

2∑

k |kk〉〈kk|AB ⊗ ρE

J. M. Renes IQING 5



Four Protocols










ρAB is purified to |ψ〉ABE =∑

x√

px |x〉A ⊗ |ψk〉BE

regard |ψk〉BE as N ◦ E(|ϕk〉T)

rate bound is known as the Hashing inequality

J. M. Renes IQING 5



Four Protocols










Starting point is ρAB =∑

k pk |k〉〈k|A ⊗ψBk , the decohered version.

Use only one-way local operations and public communication

J. M. Renes IQING 5



Four Protocols










J. M. Renes IQING 5



Typical Sequences and Subspaces

In the asymptotic case we use the power of typical sequences and sets.

Consider coin with Prheads = p:

After n → ∞ tosses, only the set of sequences for which#heads ≈ np has nonnegligible probability

Moreover, sequences occur with essentially equal probability

Roughly 2nH2(p) such sequences, for binary entropy H2

Apply the same to density matrices by working in the eigenbasis:

ρ has eigenvalues {p,1−p}

à ρ⊗n has support on subspace of dimension 2nH2(p) = 2nS(ρ)

à ρ⊗n is essentially a projector onto this subspace

J. M. Renes IQING 5



Secret Key Distillation: Information Reconciliation

Start with n copies of ψABE =∑

k pk |k〉〈k|A ⊗ψBEk

First correct errors: Bob doesn’t know k exactly. ⇒ HSW Theorem!

Alice k strings Bob state support

Bob cannot reliably distinguish the ψBk . Can distinguish elements of a random subset!

The ψBk are essentially projectors of dimension ≈ 2nH(B| K)

Average state ψB lives on a space of size ≈ 2nH(B)

à Intuitively we can pack in 2nI(K:B) disjoint ψBk

Hence, Alice projects onto a random subset of this size, and tells Bob which one.

J. M. Renes IQING 5



Secret Key Distillation: Information Reconciliation

Start with n copies of ψABE =∑

k pk |k〉〈k|A ⊗ψBEk

First correct errors: Bob doesn’t know k exactly. ⇒ HSW Theorem!

Alice k strings Bob state support

Bob cannot reliably distinguish the ψBk . Can distinguish elements of a random subset!

The ψBk are essentially projectors of dimension ≈ 2nH(B| K)

Average state ψB lives on a space of size ≈ 2nH(B)

à Intuitively we can pack in 2nI(K:B) disjoint ψBk

Hence, Alice projects onto a random subset of this size, and tells Bob which one.

J. M. Renes IQING 5



Secret Key Distillation: Privacy Amplification

Now perform privacy amplification to remove Eve’s information.

Alice k strings Eve state support

ψEk partially distinguishable; Eve can learn some information about the string k.

Average state ψE is supported on a subspace of size ≈ 2nH(E)

Each ψEk is a projector of dimension ≈ 2nH(E| K)

à Intuitively, only ≈ 2nI(K:E) states are needed to cover the support of the average

Hence, if we label every state in the “cover” with the same secret key letter, Eve willhave no information about it.

J. M. Renes IQING 5



Secret Key Distillation: Privacy Amplification

Now perform privacy amplification to remove Eve’s information.

Alice k strings Eve state support

ψEk partially distinguishable; Eve can learn some information about the string k.

Average state ψE is supported on a subspace of size ≈ 2nH(E)

Each ψEk is a projector of dimension ≈ 2nH(E| K)

à Intuitively, only ≈ 2nI(K:E) states are needed to cover the support of the average

Hence, if we label every state in the “cover” with the same secret key letter, Eve willhave no information about it.

J. M. Renes IQING 5



Putting it all together

+ How do we know these sorts of HSW and PA codes really exist?

Invoke the usual trick due to Shannon: Show that random codingworks with high probability! The codewords km,s are such that:

Bob can distinguish the ψBm,s

Coarse-graining over s leaves Eve with indistinguishable ψEm

Rate is r = I(A〉B) = I(K : B) − I(K : E)Moreover, Alice only has to communicate nH(K|B) bits to Bob.

+ For secret key generation, just feed a code into to the channel.

+ Entanglement case: make states & operations coherentStart with |Ψ〉ABE

=∑

k√

pk |k〉A ⊗ |ψk〉BE in the static casecoherently project onto one of the random HSW subsets. . .convert static to dynamic as before

J. M. Renes IQING 5



Private State Distillation

Can instead construct PA codes based on the uncertainty principle!

1 From ρAA ′BB ′, measure ZA and ZB to create a key

Key is private iff Bob+A ′ can predict XA mmt (think entanglement)Defines the set of private states Horodecki3 and Oppenheim, PRL 94, 160502 (2005)

UP: Bob’s knowledge of XA constrains Eve’s knowledge of ZA.

2 If ρBB ′x gives only partial information. . .

Use HSW theorem again!Alice picks a random subset, erasing info Bob doesn’t have

3 Watch out!XA measurement is hypothetical, isn’t actually performedFigure out what key would have been. . .If actual key distillation and XA measurement commute, okNoisy preprocessing examined in: J. M. Renes and G. Smith, PRL 98, 020502 (2007).General shield (but using only linear functions) considered in:J. M. Renes and J.-C. Boileau, quant-ph/0702187.

J. M. Renes IQING 5


Erasure is Fundamental

Destruction is a relatively indiscriminate goal

From FQSW:“In contrast to most proofs in information theory, instead of showing how to establish perfectcorrelation of some kind between the sender and the receiver, our proof proceeds by showing thatthe protocol destroys all correlation between the sender and a reference system. Since destructionis a relatively indiscriminate goal, the resulting proof is correspondingly simple.”

Quantum capacity strategy: P. Hayden, M. Horodecki, J. Yard, and A. Winter, quant-ph/0702005

View problem as sending entanglement with high fidelityPick a random subspace at the encoderDecoupling from the environment ensures a decoder

Decoupling implies (good) decoding

Given a channel N T→R, purify it to a unitary VT→REN by using an additional

system E. Now put half a maximally entangled state |Φ〉AT through thechannel, producing |ψ〉ARE

= VT→REN |Φ〉AT.

Then there exists a decoding map DR→B such that

F(|Φ〉AB ,1A ⊗D ◦ N (ΦAT)

)≥ 1 −

∥∥ψAE − 1A/dA ⊗ψE∥∥

1

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Erasure is Fundamental

One-shot Version to Memoryless Channels

Use a random subspace of the appropriate size.

Suppose |ψ〉ARE comes out of the channel such that ψA ismaximally-mixed. (Put in an entangled state. . . )

Now project onto a subspace A using ΠA→A, obtaining ψAEΠ .

By first applying a unitary we can use Π to project onto any subspacewe like: ψAE

U ∝ (ΠU ⊗ 1E)ψAE(U†Π⊗ 1

E)

Averaging over all choices of Π yields∫U(A)

dU∥∥∥ψAE

U − 1A/dA ⊗ψ

EU

∥∥∥1≤

√|A| |E| Tr[(ψAE)2]

à RHS small ⇒ channel output decoupled ⇒ decoding operation

à Encoding operation is UᵀΠ, since original input maximally-entangled.

à For the case of block inputs to a memoryless channel, we can evaluatethe rhs by using typical sequences and subspaces. . .

J. M. Renes IQING 5

Quantum Communication and Cryptography · Quantum Communication and Cryptography Introduction Outline Overview recent work in quantum information theory. (in the strict sense) 1 Resource

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