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 Information Theory 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 data compression: 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) 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! An Example original image (uncompressed), size 108.5 KB lossless compression (PNG); size 60.1 KB lossy compression; 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 transferB to Bob ? Asymptotic setting initially finally n nnState n B n BE splitting U N How can we transferB 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! What if H () QN ( ) ? Suppose Alice & Bob share entanglement, i.e., shared entanglement E N D Theorem: it is possible to transmit the source over the channel N, up to some distortion D if & only if ea 1 RDq () Ic (,) N 2 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 qeqq qq E|B Resource Inequality : 1 1 1 q IRB: minIRB : e IBE: ; 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 Eqq () Resource Inequality : 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.