Quantum rate distortion, reverse Shannon theorems, and source-channel separation

Nilanjana Datta, Min-Hsiu Hsieh, Mark Wilde

(1) University of Cambridge,U.K. (2) McGill University, Montreal, Canada Classical

Shannon’s Source Shannon’s Noisy Channel CodingTheorem Coding Theorem

Compression of information Transmission of information emitted by a source through a noisy channel Shannon’s Coding Theorems

 Source Coding Theorem  Channel Coding Theorem stochastically X ~ p independent X noisy Y memoryless signals channel N input output memoryless X ~ pX ; x X The fundamental limit The fundamental limit of on the rate of reliable : information transmission:

= Shannon entropy of the = Capacity of the channel source: CN() max(:) IX Y H ()X px mutual information Lossless Data Compression

s s ' E compressed D original signal decompressed signal from source signal

ss'  (exact replica!)

Shannon’s Source CodingTheorem

corresponds to asymptotically lossless data compression

 multiple () n uses of the channel

 signals: sequences xx12, ...., xn

pne () 0 as n  0

 (average probability of error in recovering original signal) –oftentoostringent a condition

 for cases of multimedia data; e.g. audio, and still images

 when storage space is insufficient

 Why ?

Typically, a substantial amount of data can be discarded before the information is sufficiently degraded to be noticeable! An Example

original image (uncompressed), size 108.5 KB

lossless compression (PNG); size 60.1 KB

; 4.82 KB

lossy compression; 1.14 KB Lossy Data Compression

 a data compression scheme :

 recovered data  original data instead D  recovered data  original data D: the allowed distortion

Rate Distortion Theory  the theory of Lossy Data Compression

tradeoff  rate of compression the allowed distortion

fundamental limit on the asymptotic rate of data compression for a given maximum distortion Classical Rate Distortion Theory (Shannon)

 For a given memoryless information source ; X ~ pX  If the maximum allowed distortion is DD ; 0 1,

The minimum rate of data compression:  min I(:)XY  HX() R()D ()a rate distortion function mutual information ()a : p YX| ;((,))Ε dXY  D stochastic maps average distortion

Y : output of a stochastic map pXYYX| :  2 dxy(, ):distortion measure e.g. dxy(, ) x y Our Aim

To obtain rate distortion functions in the quantum realm Scenario for Quantum Rate Distortion for a fixed 01 D   Storage setting:

a ,HA memoryless quantum information source Scenario for Quantum Rate Distortion for a fixed 01 D   Storage setting: n   n n uses of it E D compressed recovered c signal state in Hn

Rate of data compression if cnR  R dim Hn  2 achievable –if limd ( n ,DE )  D n

RDq (): inf{R : achievable} Quantum Rate Distortion function (min. rate of compression under given distortion) Quantum Rate Distortion Function

Barnum proved: R ()D  min Ic , N  ……(1) q N: CPTP dN(, )D coherent information

R . . R RA    RB A . N . B  dN(, ) 1 F (, N ) INSRBc ,(|);  e entanglement fidelity

 He conjectured: "" holds in (1) problem!!

 the coherent information can be negative BUT  The rate of a data compression scheme is non-negative !! Equivalent Scenarios for Quantum Rate Distortion

 Storage setting:

n uses of E D the source nRq () D qubits  n

 Communication setting:

nRq () D n uses of E D the source noiseless qubit channels Another Scenario for Quantum Rate Distortion

 Communication setting: (entanglement assisted)

shared entanglement

ea nRq () D E D n uses of the source noiseless qubit channels

ea min. number of noiseless qubit channels needed RDq () per use of the source in the presence of entanglement for a given distortion D

entanglement-assisted quantum rate distortion function Result - I

 Entanglement-assisted quantum rate distortion function: 1 RDea () min I RB: …………(2) q N: CPTP 2  d (, N )D quantum mutual information

R . . R RA    RB A . N . B   Proof: ea using entropic (i) Prove : RD() RHS of (2) q inequalities

(ii) Prove: This bound is achievable, i.e., "" holds 1  The rate R  minIRB : is achievable N: CPTP 2  dN(, )D Sketch of proof:

RA  Source Its purification: ,;HA  

 Let N  the minimizing CPTP map a Stinespring isometry of N  Let UABEN : 

R . R RBE RA  entanglement  

A (idR U N ) . E B noiseless  qubit channel R . R RBE RA  entanglement  

A (idR U N ) . E B noiseless  qubit channel 1 To prove: R  minIRB : is achievable N: CPTP 2  dN(, )D

 Note: B  :() N  satisfies d(, N ) D Suffices to prove: 1  can be sent to Bob by Alice through I RB: B 2  uses of a noiseless qubit channel per use of the source  How can we transferB to Bob ?

Asymptotic setting  initially  finally n nnState     n  B  n BE splitting U N  How can we transferB to Bob ? R R qubits from q RBE Alice to Bob   RBE + LO EB E B

entanglement

1 1 q  IRB:  minIRB :  N: CPTP 2  2 dN(, )D achievable rate !   nn   (i) local preparation of  n BE U N (ii) State splitting with the help n of shared entanglement such that B

n simulating the (output of the) quantum channel N  n  when the input is (using shared entanglement)

 a special case of another protocol -- channel simulation – Quantum Reverse Shannon Theorem (QRST)

achievability of ea  RDq () Result - II

 Unassisted Quantum rate distortion function: 1 min ()nn ()n EN ()  RDq () lim N : CPTP P n n dN(,nn() )D regularized entanglement of purification

 For any AB ABR   BR

E : PBR   minH  (idBRBR )   0 R :CPTP

 where, Von Neumann entropy of for any state , H ():  Result - II

 Unassisted Quantum rate distortion function: 1 ()nn RD() lim minENP  ( ) q nN n : CPTP dN(, ()n )D

Channel simulation (QRST) in the achievability of absence of shared entanglement  this rate

n n To simulate output of N when input is 

qubits sent at a rate n  B E  () P RB B  N() Source Channel Separation Theorems

 Shannon: connects the two pillars of Classical Info. Theory

 Is it possible to transmit a classical information source ()X reliably over a classical channel ()N ?

 Yes -- if & only if H ()XCN ()

 Implication:

data compression channel decoding code code N

 a 2-stage encoding & decoding with the best data compression + error correction codes is optimal! Result - III

Quantum Source Channel Separation Theorems

Theorem : It possible to transmit a quantum information source () reliably over a quantum channel()N if & only if von Neumann entropy H ()  QN ( ) quantum capacity

 Implication:

data compression channel decoding code code N

 a 2-stage encoding & decoding with the best data compression + quantum error correction codes is optimal!  D

D source , i.e., the

 c N QN 2 1 transmit  

 up to some distortion () ( ) , () (,) H N ea q share entanglement shared entanglement RD I N Bob E & What if ? it is possible to channel Alice over the if & only if Theorem:

  Suppose  Summary

ea  entanglement-assisted quantum rate distortion RDq (): function

 in terms of the quantum mutual information

RD(): unassisted quantum rate distortion function q  in terms of regularised entanglement of purification

 Quantum Source Channel Separation Theorems  The following are some extra slides which I have removed/modified to make the talk 20 minute long. Shannon’s Coding Theorems

 Source Coding Theorem  Channel Coding Theorem stochastically X ~ p independent X noisy Y memoryless signals channel N input output memoryless X ~ pX ; x X The fundamental limit The fundamental limit of on the rate of reliable data compression: information transmission:

= Shannon entropy of the = Capacity of the channel source: CN() max(:) IX Y H ()X px mutual information Rate Distortion Function

RD() min( I X :) Y pYX| : E(d (XY , ))D

 For D  0, R()DHX () YX

 For D  0, R()DHX () Barnum’s Conjecture

Quantum Rate Distortion RD() minIc  , N  function q N: CPTP d (, N )D

 Motivation : similarity in the classical case

Rate Distortion Function Capacity of a noisy channel CN() max(:) IX Y RD() min( I X :) Y p pyxYX| (|): x E(d (xy , ))D Xp~ X Y N mutual information p YX|

Quantum capacity of a 1 n QN() limmax Ic  , N noisy quantum channel n n  coherent information State Splitting R R LO & QC from RBE Alice to Bob   RBE

EB  E B

entanglement

BE  E|B Resource Inequality :  qeqq qq   1 1 1 q  IRB: e  IBE: ;  minIRB :  2 N: CPTP 2  2 dN(, )D achievable rate ! Result - II

 Unassisted Quantum rate distortion function: 1 ()nn RD() lim minENP  ( ) q nN n : CPTP dN(, ()n )D

Channel simulation (QRST) in the achievability of absence of shared entanglement  this rate

n n To simulate output of N when input is   Resource Inequality : Eqq()     P RB B E  () P RB   N() qubits B  Unassisted Quantum rate distortion function: 1 ()nn RD() lim minENP  ( ) q nN n : CPTP dN(, ()n )D

 Why the double regularization?

 Storage setting:  n n uses of E D the source

n n DC  N  whereas in QRST: simulates N

 To prove"" need to prove the converse

 done using entropic inequalities