Spin chains and channels Spin chains and channels with memorywith memory

Martin Plenio Martin Plenio (a)(a) & Shashank & Shashank Virmani Virmani (a,b)(a,b)

quant-ph/0702059, to appear prlquant-ph/0702059, to appear prl

(a)(a) Institute for Mathematical Sciences & Dept. of Physics, Imperial College London.Institute for Mathematical Sciences & Dept. of Physics, Imperial College London.

(b)(b)Dept. of Physics, University of Hertfordshire.Dept. of Physics, University of Hertfordshire.

OutlineOutline Introduction to Channel Capacities.Introduction to Channel Capacities.

Motivation – correlations in error.Motivation – correlations in error.

Connections to many-body physics.Connections to many-body physics.

Validity of assumptions.Validity of assumptions.

Conclusions.Conclusions.

Quantum Channel Quantum Channel CapacitiesCapacities

Alice wants to send qubits to Bob, via a noisy Alice wants to send qubits to Bob, via a noisy channelchannel, e.g. a photon polarisation via an noisy , e.g. a photon polarisation via an noisy optical fibre.optical fibre.

Quantum Error Correction Codes can be used to Quantum Error Correction Codes can be used to reduce error (see Gottesman lectures).reduce error (see Gottesman lectures).

But this comes at a cost – each But this comes at a cost – each logicallogical qubit is qubit is encoded in a larger number of encoded in a larger number of physicalphysical qubits. qubits.

The The Communication rate, R,Communication rate, R, of a code is: of a code is:

qubits physical

qubits logicalRate

Quantum Channel Quantum Channel CapacitiesCapacities

Quantum channel capacities are concerned with Quantum channel capacities are concerned with transmission of transmission of largelarge amounts of quantum data. amounts of quantum data.

If you use a channel If you use a channel εε manymany times, there is a times, there is a maximalmaximal rate Q( rate Q(εε) for which a code can be chosen ) for which a code can be chosen such that errors vanish. such that errors vanish.

This maximal rate is called the This maximal rate is called the quantum channel quantum channel capacitycapacity..

If you try to communicate at a rate R > Q, then If you try to communicate at a rate R > Q, then you will suffer you will suffer errorserrors..

Communication at rates R < Q can be made Communication at rates R < Q can be made essentially essentially error freeerror free by choosing a clever code. by choosing a clever code.

Quantum Channel Quantum Channel CapacitiesCapacities

Q(Q(εε) is the maximal rate at which quantum bits ) is the maximal rate at which quantum bits can be sent essentially error free over many can be sent essentially error free over many uses of a quantum channel uses of a quantum channel εε..

So how do we compute Q(So how do we compute Q(εε) ) Unfortunately it is very difficult !Unfortunately it is very difficult !

Quantum Channel Quantum Channel CapacitiesCapacities

Q(Q(εε) is the maximal rate at which quantum bits ) is the maximal rate at which quantum bits can be sent essentially error free over many can be sent essentially error free over many uses of a quantum channel uses of a quantum channel εε..

So how do we compute Q(So how do we compute Q(εε) ) Unfortunately it is very difficult !Unfortunately it is very difficult !

In fact it is very very difficult….In fact it is very very difficult….

So how do we figure out So how do we figure out Q(Q(εε) ) ??

....

ofon purificati a is : where

))(I())((maxlim)(

n

nnn

SSQ

See e.g. Barnum et. al. ’98, Devetak ’05.See e.g. Barnum et. al. ’98, Devetak ’05.

The best known formula for The best known formula for Q(Q(εε) ) for UNCORRELATED channels is:for UNCORRELATED channels is:

Independence vs. Independence vs. CorrelationsCorrelations

IndependentIndependent error model: each transmission affected error model: each transmission affected by noise independently of the othersby noise independently of the others

Independence vs. Independence vs. CorrelationsCorrelations

However realistic errors can often exhibit However realistic errors can often exhibit correlations correlations ::

E.g. scratches on a CD affect adjacent information pieces, birefringence in optical E.g. scratches on a CD affect adjacent information pieces, birefringence in optical fibres (Banaszek experiments 04)fibres (Banaszek experiments 04)

IndependentIndependent error model: each transmission affected error model: each transmission affected by noise independently of the othersby noise independently of the others

Correlated Errors. Correlated Errors. Independent errors: channel acts on Independent errors: channel acts on n n

qubits as qubits as

)()( 111 ... nnn

Correlated Errors. Correlated Errors. Independent errors: channel acts on Independent errors: channel acts on n n

qubits as qubits as

)()( 111 ... nnn

FamilyFamily of channels { of channels {nn} – for each } – for each number of qubits number of qubits n n ::

)()( 111 ... nnn

So how do correlations in noise affect our So how do correlations in noise affect our ability to communicate ?ability to communicate ?

Motivating ExampleMotivating Example Consider an independent Pauli error channel:Consider an independent Pauli error channel:

).....())

...] [ ...] [ *kji

zy,x,0,ikji

kpp(jp(i.)p(i,j,k...

.)p(i,j,k...

Motivating ExampleMotivating Example

),()()1()|

).....|)|)

jijQip(j

jp(kip(jQ(i.)p(i,j,k...

Channel considered in Macchiavello & Palma ’02:Channel considered in Macchiavello & Palma ’02:

Consider an independent Pauli error channel:Consider an independent Pauli error channel:

).....())

...] [ ...] [ *kji

zy,x,0,ikji

kpp(jp(i.)p(i,j,k...

.)p(i,j,k...

Macchiavello-Palma Macchiavello-Palma channel:channel:

μ

Holevo

μ0

kink in curve

Productstates

Max. entangledstates

Perfect

Also see e.g. Macchiavello et. al. ’04; Karpov et. al. ’06; Banaszek et. al. ’04Also see e.g. Macchiavello et. al. ’04; Karpov et. al. ’06; Banaszek et. al. ’04

Hmmm……Statistical Physics?Hmmm……Statistical Physics? Non-analyticity in large Non-analyticity in large nn, , thermodynamicthermodynamic, limit ? , limit ? Expressions involving Expressions involving entropy entropy ??

That sounds just like Many-body physics!!That sounds just like Many-body physics!!

Hmmm……Statistical Physics?Hmmm……Statistical Physics? Non-analyticity in large Non-analyticity in large nn, , thermodynamicthermodynamic, limit ? , limit ? Expressions involving Expressions involving entropy entropy ??

That sounds just like Many-body physics!!That sounds just like Many-body physics!!

Consider a many-body inspired model for correlated noise:Consider a many-body inspired model for correlated noise:

Unitary Interaction

Transmitted Qubits

Environment Qubits in correlated thermal state

Capacity for Capacity for correlatedcorrelated errorserrors

For our many body models we will compute:For our many body models we will compute:

In general this expression is too difficult to calculate.In general this expression is too difficult to calculate.

But for specific types of channel it can be simplifiedBut for specific types of channel it can be simplified

.... :generalin

ofon purificati a is : where

))(I())((maxlim})({

n

nnn

n SSQ

This will NOT be the capacity in general, but for “sensible” models it will be the capacityThis will NOT be the capacity in general, but for “sensible” models it will be the capacity

Pick a simple interaction!Pick a simple interaction! Simple model:Simple model:

- Consider 2 level systems in environment – - Consider 2 level systems in environment – either either classical or quantum particles classical or quantum particles

- Let interaction be CNOT, environment controls- Let interaction be CNOT, environment controls

Pick a simple interaction!Pick a simple interaction! Simple model:Simple model:

- Consider 2 level systems in environment – - Consider 2 level systems in environment – either either classical or quantum particles classical or quantum particles

- Let interaction be CNOT, environment controls- Let interaction be CNOT, environment controls

Such interaction gives some pleasant properties:Such interaction gives some pleasant properties:

- Essentially probabilistic application of Id or X- Essentially probabilistic application of Id or X- truncated Quantum Cap = Distillable ent.- truncated Quantum Cap = Distillable ent.- Answer given by Hashing bound.- Answer given by Hashing bound.

see Bennett et. al. ’96, Devetak & Winter ’04.see Bennett et. al. ’96, Devetak & Winter ’04.

For such channels:For such channels:

n

)systembody -many(lim1

HQ

n

For classical environments For classical environments HH is just the entropy. is just the entropy. Thermodynamic property!!Thermodynamic property!!

For quantum For quantum H H is the entropy of computational basis is the entropy of computational basis diagonal.diagonal.

This is very convenient! There are years of interesting This is very convenient! There are years of interesting examples, at least for classical environment.examples, at least for classical environment.

Quantum example: Rank-1 Quantum example: Rank-1 MPSMPS

Matrix Product States (e.g. work of Cirac, Verstraete Matrix Product States (e.g. work of Cirac, Verstraete et. al.) are interesting many-body states with efficient et. al.) are interesting many-body states with efficient classical description. classical description.

Convenient result: If matrices are rank-1, Convenient result: If matrices are rank-1, H H reduces to reduces to entropy of a classical Ising chain.entropy of a classical Ising chain.

E.g. ground state of following Hamiltonian (Wolf et. al. arxiv ’05):

2122

12 )1()1()1(2 iiiiii

i

ZXZgXgZZg

Wolf et. al. MPS cont.Wolf et. al. MPS cont.

gg=0

1

I

Diverging gradient

- Slight Cheat : left-right symmetry as channel identical for g, -g

Quantum Ising (Numerics)Quantum Ising (Numerics)

The Assumptions.The Assumptions.

For correlated errors this is NOT the capacity in general.

))(I())((max1

lim})({ nn

nnC SS

nI

Is this the capacity for all many-body environments?

Certainly the Hamiltonian must satisfy some constraints.

What are they ?

We have calculated is actually the coherent information:

Cheat’s guide to correlated Cheat’s guide to correlated codingcoding

Consider the whole system over many uses:

large LIVE blocks, l spins each block

small SPACER blocks, s spins each block

Cheat’s guide to correlated Cheat’s guide to correlated codingcoding

Consider the whole system over many uses:

large LIVE blocks, l spins each block

small SPACER blocks, s spins each block

If correlations in the environment decay sufficiently, reduced state of LIVE blocks will be approximately a product

See e.g. Kretschmann & Werner ‘05

Cheat’s guide to correlated Cheat’s guide to correlated coding IIcoding II

So if correlations decay sufficiently fast, can apply known results on uncorrelated errors.

How fast is sufficiently fast ? Sufficient conditions are:

)exp(||)(|| 1.. 1321FsCvl Ev

LLLLL v

We also require a similar condition, demonstrating that the bulk properties are sufficiently independent of boundary conditions.

These conditions can be proven for MPS and certain bosonic systems.

Conclusions and Further Conclusions and Further work:work: Results from many-body theory can give interesting insight into the coherent information of correlated channels.Results from many-body theory can give interesting insight into the coherent information of correlated channels.

What about more complicated interactions? Methods give LOWER bounds to capacity for all random unitary channels.What about more complicated interactions? Methods give LOWER bounds to capacity for all random unitary channels.

For which many body systems can decay be proven?For which many body systems can decay be proven?

How about other capacities of quantum channels?How about other capacities of quantum channels?

A step towards physically motivated models of correlated error. 2d, 3d…..?A step towards physically motivated models of correlated error. 2d, 3d…..?

Is there a more direct connection to quantum coding.Is there a more direct connection to quantum coding.

Funding by the following is gratefully acknowledged:Funding by the following is gratefully acknowledged: QIP-IRC & EPSRCQIP-IRC & EPSRC Royal Commission for Exhibition of 1851Royal Commission for Exhibition of 1851 The Royal Society UKThe Royal Society UK QUPRODIS & European UnionQUPRODIS & European Union The Leverhulme trustThe Leverhulme trust