COMPRESSION FOR QUANTUM POPULATION CODING

Transcription

1 COMPRESSION FOR QUANTUM POPULATION CODING Ge Bai, The University of Hong Kong Collaborative work with: Yuxiang Yang, Giulio Chiribella, Masahito Hayashi

2 INTRODUCTION Population: A group of identical states

3 ENCODING OF IDENTICAL PREPARED STATES Alice has a device that generates nn identical states Bob wants to use the states Eg. to perform tomography Generator Communication Channel Operations Alice Bob Faithful transmission, minimal memory usage

4 COMPRESSION FOR QUANTUM POPULATION CODING Input: quantum population coding ρρ θθ with unknown θθ Encoder/decoder: quantum channels Faithfulness: output a state almost the same as the input Optimality: minimizing memory size Encoder Memory Decoder

5 RELATED WORKS Memory cost for compressing nn identical 1. Qubits log nn qubits + 1/2 log nn bits [YY, GC, MH; PRL 16] 2. Clock states 1/2 log nn qubits [YY, GC, MH; arxiv: ] 3. General qudits OO log nn [YY, GC, Ebler; PRL 16] 4. Classical population coding 1/2 log nn bits [MH, Vincent Tan; 17]

6 MAIN RESULT Memory cost for compressing

7 PARAMETRIZATION CLASSICAL AND QUANTUM PARAMETERS A state family : θθ = μμ, ξξ Θ characterized by two kinds of parameters: = UU ξξ ρρ μμ UU ξξ Classical parameters μμ R dd 1 : determining the spectrum Quantum parameters ξξ R dd2 dd : determining the eigenbasis ff cc, ff qq : number of free classical/quantum parameters

8 PARAMETRIZATION EXAMPLES Full qudit state family: ff cc = dd 1 and ff qq = dd 2 dd Phase-covariant state family: ff cc = 0 and ff qq = dd 1 = UU θθ ρρ 0 UU θθ UU θθ = ee iiθθ kk kk kk kk

9 MAIN RESULT MEMORY SIZE A state family ρρ θθ : θθ Θ can be faithfully compressed into: 1/2 + δδ lognn bits per free classical parameter 1/2 log nn bits + δδ log nn qubits per free quantum parameter where δδ > 0 can be made arbitrarily small (but not zero) Mixed qubits [YY, GC, MH; PRL 16] Classical population coding [MH, Vincent Tan, 17] 3/2 log nn (qu)bits 1/2 log nn bits

10 MAIN RESULT OPTIMALITY Optimality of memory size: Any compression protocol using 1/2 εε log nn (qu)bits cannot be faithful Necessity of quantum memory: Any compression protocol using only classical memory cannot be faithful unless the family is classical (no free quantum parameter)

11 BUILDING THE PROTOCOL Achieving the minimal memory cost

12 BUILDING BLOCKS QUANTUM LAN Quantum local asymptotic normality (Q-LAN) [Kahn, Guta; CMP 09] In a neighborhood of θθ 0, is asymptotically (nn ) equivalent to a classical-quantum Gaussian state: Quantum channels γγ μμ cccccccccc γγ ξξ qqqqqqqqqq Classical mode γγ μμ cccccccccc : a Gaussian distribution with ff cc variates Quantum mode γγ ξξ qqqqqqqqqq : a multimode (number of modes depending on ff qq ) displaced thermal state

13 BUILDING BLOCKS DISPLACED THERMAL STATE ENCODER Displaced thermal states ρρ αα,ββ = DD αα ρρ ββ DD αα where αα C, ββ R, DD αα is the displacement operator and ρρ ββ is a fixed thermal state (state of a system in equilibrium) 1. Concentrate the displacement ρρ Beam splitters nn 1 αα,ββ ρρ nnαα,ββ ρρ ββ 2. Truncate in photon number basis Truncate δδlognn nnqubits

14 BUILDING BLOCKS LESS QUANTUM MEMORY Compressing ρρ αα,ββ = DD αα ρρ ββ DD αα with less quantum memory 1. Concentrate the displacement ρρ αα,ββ Beam splitters ρρ nnαα,ββ ρρ ββ nn 1 2. Estimate αα by heterodyne measurement 3. Truncate in displaced photon number basis According to estimated αα Estimate αα Truncate αα log nn bits δδlog nn qubits

15 THE COMPRESSION PROTOCOL Localization 1 δδ/2 θθ Neighborhood Θ nn,δδ θθ 0 ff cc +ff qq 2 log nn bits nn 1 δδ/2 Q-LAN Total memory: 1 + δδ ff 2 cc + ff qq log nn (qu)bits Quantum/classical ratio can be arbitrarily small Gaussian state compression γγ μμ cccccccccc Truncation ff cc δδ log nn bits γγ ξξ qqqqqqqqqq Displaced thermal state encoder ff qq δδ log nn qubits

16 OPTIMALITY & ERROR BOUND The information-theoretical limit

17 OPTIMALITY OF MEMORY SIZE Construct a mesh MM on Θ containing n ff/2 εε (ff = ff cc + ff qq ) mutually distinguishable states Θ The following protocol θθ E Memory can faithfully communicate n ff/2 εε different messages if E, DD is faithful DD θθ θθ 1 θθ 2 The memory cost cannot be smaller than log # messages = ff/2 εε log nn OO nn 1/2+αα, αα > 0 ρρ θθ1 distinguishable from 2

18 ERROR BOUND The compression error is upper bounded as εε nn = OO(nn δδ/2 ) + OO nn κκ δδ, where the latter is the error of Q-LAN. κκ(δδ) > 0 for δδ (0,2/9) Faithfulness lim nn εε nn = 0 is guaranteed as long as δδ (0,2/9) The upper bound vanishes more slowly when less quantum memory (smaller δδ) is used

19 QUANTUM MEMORY IS ESSENTIAL The ratio between quantum/classical memory can be arbitrarily small Can we use zero quantum memory? No. purely classical information Lemma: a state family can be perfectly compressed into classical memory if and only if it is classical, i.e. ρρ 1, ρρ 2 = 0 for any ρρ 1, ρρ 2 from the family.

20 QUANTUM MEMORY IS ESSENTIAL Pick states ρρ θθ0, 0, tt contains one free quantum +tt/ nn parameter 0 Q-LAN γγ 0 Q-LAN Independent on nn 0 +tt/ nn γγ tt A protocol for ρρ θθ0, 0 +tt/ nn a protocol for γγ 0, γγ tt nn a protocol for γγ 0, γγ tt with arbitrarily small error nn γγ 0, γγ tt = 0 (which is false)

21 SUMMARY & FUTURE WORKS Generator Alice Faithful Communication Extension to non-product states with symmetry Real population coding eg. states of bosonic systems Bob ~1/2 log nn for each free parameter Arbitrarily small, but non-zero quantum memory Non-zero quantum memory how small can it be? Operations

22 AUTHORS OF THIS WORK Yuxiang Yang Masahito Hayashi Ge Bai Giulio Chiribella

23 AUTHORS OF THIS WORK Yuxiang Yang Masahito Hayashi Giulio Chiribella Ge Bai

24 THANKS! Full version of this paper: arxiv