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

Transcription

1 Quantum rate distortion, reverse Shannon theorems, and source-channel separation ilanjana Datta, Min-Hsiu Hsieh, Mark Wilde (1) University of Cambridge,U.K. (2) McGill University, Montreal, Canada

2 Classical Information Theory Shannon s Source CodingTheorem Compression of information emitted by a source Shannon s oisy Channel Coding Theorem Transmission of information through a noisy channel

3 Shannon s Coding Theorems Source Coding Theorem memoryless stochastically independent signals X ~ p X ; x X The fundamental limit of data compression: = Shannon entropy of the source: H ( X ) Channel Coding Theorem X ~ p X input noisy channel memoryless Y output The fundamental limit on the rate of reliable information transmission: = Capacity of the channel C( ) max I( X : Y) p x mutual information

4 Lossless Data Compression s original signal from source E compressed signal s ' s D s ' decompressed signal (exact replica!) Shannon s Source CodingTheorem corresponds to asymptotically lossless data compression multiple ( n) uses of the channel signals: sequences p ( ) e n x1, x2..., xn 0 as n 0 (average probability of error in recovering original signal)

5 Lossless Compression oftentoostringent a condition for cases of multimedia data; e.g. audio, video 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!

6 An Example original image (uncompressed), size KB lossless compression (PG); size 60.1 KB lossy compression; 4.82 KB lossy compression; 1.14 KB

7 a data compression scheme : instead D: Lossy Data Compression recovered data original data D recovered data original data 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

8 Classical Rate Distortion Theory For a given memoryless information source ; If the maximum allowed distortion is The minimum rate of data compression: R( D) rate distortion function min ( a) D I( X : Y) (Shannon) X ~ p X ; 0 D1, H( X) mutual information ( a) : p YX stochastic maps ; Ε( d( X, Y )) D average distortion Y : d( x, y) : output of a stochastic map distortion measure e.g. pyx : X Y 2 d( x, y) x y

9 Our Aim To obtain rate distortion functions in the quantum realm

10 Storage setting: Scenario for Quantum Rate Distortion for a fixed 0 D 1, H A a memoryless quantum information source

11 Storage setting: Scenario for Quantum Rate Distortion for a fixed 0 D 1 n uses of it n E n compressed state in c H n D recovered signal Rate of data compression achievable if q lim d (, D E ) n R if n R D R Quantum Rate Distortion function dim H n 2 D ( ): inf{ : achievable} c nr (min. rate of compression under given distortion)

12 Quantum Rate Distortion Function Barnum proved: R D min, q ( ) Ic (1) coherent information : CPTP d(, ) D RA R. A. I S R B c. R. B RB, ( ) ; d(, ) 1 F (, ) 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!!

13 Storage setting: Equivalent Scenarios for Quantum Rate Distortion n uses of the source n E nrq ( D ) qubits D Communication setting: n uses of the source E nrq ( D ) noiseless qubit channels D

14 Another Scenario for Quantum Rate Distortion Communication setting: (entanglement assisted) shared entanglement n uses of the source E ea nrq ( D) noiseless qubit channels D R ea q ( D) min. number of noiseless qubit channels needed per use of the source in the presence of entanglement for a given distortion D entanglement-assisted quantum rate distortion function

15 Result - I Entanglement-assisted quantum rate distortion function: R ( D) min : CPTP ea q d (, ) D 1 : 2 I R B (2) quantum mutual information RA Proof: R. A. ea q (i) Prove : R ( D) RHS of (2). R. B RB using entropic inequalities (ii) Prove: This bound is achievable, i.e., " " holds

16 The rate Sketch of proof: R 1 min : 2 : CPTP d(, ) D IR B is achievable, Source H A ; Its purification: RA Let the minimizing CPTP map Let U : A BE a Stinespring isometry of RA R. A. (id U ) R E R RBE B entanglement noiseless qubit channel

17 RA R. A. R (id U ) 1 min I R: B 2 To prove: R : CPTP d(, ) D E R RBE B entanglement noiseless qubit channel is achievable ote: B satisfies d(, ) : ( ) D Suffices to prove: 1 : B 2 I R B uses of a noiseless qubit channel per use of the source can be sent to Bob by Alice through

18 How can we transfer B to Bob? initially n U n Asymptotic setting n BE State splitting finally n n B

19 How can we transfer B to Bob? RBE R q qubits from Alice to Bob R RBE E B + LO E B entanglement q 1 2 I R: B : CPTP d(, ) D achievable rate! 1 min : 2 IR B

20 n (i) local preparation of U (ii) State splitting with the help of shared entanglement such that n n BE n B simulating the (output of the) quantum channel n when the input is (using shared entanglement) n a special case of another protocol -- channel simulation Quantum Reverse Shannon Theorem (QRST) achievability of R ea q ( D)

21 Result - II Unassisted Quantum rate distortion function: lim 1 n Rq ( D ) ( n) n min : CPTP n ( n) d(, ) D ( n ) n E ( ) P regularized entanglement of purification For any AB ABR BR : : CPTP E min H (id ) 0 P BR R B R BR where, for any state, H ( ): Von eumann entropy of

22 Unassisted Quantum rate distortion function: R D ( n) n lim min E ( ) P q ( ) n Result - II 1 n : CPTP ( n) d(, ) D Channel simulation (QRST) in the absence of shared entanglement achievability of this rate To simulate output of n when input is n qubits sent at a rate E P ( ) RB B n ( ) B

23 Source Channel Separation Theorems Shannon: connects the two pillars of Classical Info. Theory Is it possible to transmit a classical information source reliably over a classical channel? Yes -- if & only if H ( X) C( ) ( ) ( X ) Implication: data compression code channel code decoding a 2-stage encoding & decoding with the best data compression + error correction codes is optimal!

24 Quantum Source Channel Separation Theorems Theorem : It possible to transmit a quantum information source ( ) Result - III reliably over a quantum channel ( ) if & only if von eumann entropy H ( ) Q ( ) quantum capacity Implication: data compression code channel code decoding a 2-stage encoding & decoding with the best data compression + quantum error correction codes is optimal!

25 H ( ) Q ( ) What if? Suppose Alice & Bob share entanglement, i.e., shared entanglement E D Theorem: it is possible to transmit the source over the channel, if & only if 1 ea q up to some distortion R ( D) I (, ) 2 c D

26 R ea q ( D): Summary entanglement-assisted quantum rate distortion function in terms of the quantum mutual information Rq ( D ): unassisted quantum rate distortion function in terms of regularised entanglement of purification Quantum Source Channel Separation Theorems

27 The following are some extra slides which I have removed/modified to make the talk 20 minute long.

28 Shannon s Coding Theorems Source Coding Theorem memoryless stochastically independent signals X ~ p X ; x X The fundamental limit of data compression: = Shannon entropy of the source: H ( X ) Channel Coding Theorem X ~ p X input noisy channel memoryless Y output The fundamental limit on the rate of reliable information transmission: = Capacity of the channel C( ) max I( X : Y) p x mutual information

29 Rate Distortion Function RD ( ) min I( X: Y) p YX : E( d ( XY, )) D For D 0, Y X R( D) H( X) D R( D) H( X) For 0,

30 Barnum s Conjecture Quantum Rate Distortion function R ( ) q D : CPTP d (, ) D min, Ic Motivation : Rate Distortion Function RD ( ) min I( X: Y) pyx ( y x): E( d ( xy, )) D mutual information similarity in the classical case Capacity of a noisy channel C( ) max I( X : Y) X ~ px p x p Y X Y Quantum capacity of a 1 n n Q ( ) lim max I, noisy quantum channel c n coherent information

31 State Splitting RBE R LO & QC from Alice to Bob R RBE E B E B entanglement BE Resource Inequality : q qq e qq 1 e IB: E ; 2 q 1 2 I R: B : CPTP d(, ) D achievable rate! 1 min : 2 E B IR B

32 Unassisted Quantum rate distortion function: R D ( n) n lim min E ( ) P q ( ) n Result - II 1 n : CPTP ( n) d(, ) D Channel simulation (QRST) in the absence of shared entanglement achievability of this rate To simulate output of n when input is Resource Inequality : E ( ) q q E P ( ) qubits RB P RB B ( ) B n

33 Unassisted Quantum rate distortion function: R D ( n) n lim min E ( ) P q ( ) n 1 n : CPTP ( n) d(, ) D Why the double regularization? Storage setting: n uses of the source n E D whereas in QRST: simulates n D C n " " To prove need to prove the converse done using entropic inequalities