Source coding for a mixed source:determination of second order asymptotics

arXiv:1407.6616

Felix Leditzky Nilanjana Datta

27 August 2014QESS, Smolenice

Table of Contents

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Fixed-length visible source coding

I Quantum source: pure-state ensemble {pi , ψi}

I Visible setting: Alice knows identity of signals ψi

→ receives classical information in form of label i

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Fixed-length visible source coding

Encoding

Input

Decoding

I Visible encoder: V : {1, . . . , k} → D(Hc) (can be non-linear!)

I Hc : compressed Hilbert space with M := dimHc < dimH.

I Decoding: D : D(Hc)→ D(H) (CPTP map)

I Goal: Retrieve output state that is close to input state on average

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Fixed-length visible source coding

I Figure of merit: Ensemble average fidelity

F̄ (E,V,D) :=∑

ipi Tr((D ◦ V)(i)ψi )

I For ε ∈ (0, 1), a code (V,D,M) is ε-admissible if

F̄ (E,V,D) ≥ 1− ε.

Definition

ε-error one-shot minimum compression length:

m(1),ε(ρ) := inf{log M | (V,D,M) is an ε-admissible code}

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Mixed source coding

I Mixed source: emits signals from one of two i.i.d. sources:

1 2

1 2

n

n

t

I Source state: ρ(n) = tρ⊗n1 + (1− t)ρ⊗n

2

I After the first signal, the source sticks to the chosen source.

I Simple example of a non-i.i.d. protocol

I For ε ∈ (0, 1) we define mn,ε(ρ1, ρ2, t) := m(1),ε(ρ(n)).

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Mixed source coding

I Goal: determine the second order asymptotics of mn,ε(ρ1, ρ2, t)

I Find a and b in the expansion

mn,ε(ρ1, ρ2, t) = na +√

nb +O(log n).

I Compare: Single source with i.i.d. state ρ(n) = ρ⊗n:

B Schumacher: a = S(ρ)B b given by a Gaussian approximation [Datta, FL]

I Mixed source with non-i.i.d. source state

ρ(n) = tρ⊗n1 + (1− t)ρ⊗n

2

is much harder to evaluate!

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One-shot bounds

I Bounds for the minimum compression length in terms of theinformation spectrum entropy

Hεs(ρ) := inf{γ ∈ R | Tr(ρ− 2−γ1)+ ≥ 1− ε}

I Properties: positive for states, quasi-concavity, etc.

I Source state for a single use of the source: ρ = tρ1 + (1− t)ρ2

Theorem

For ε ∈ (0, 1), we have

m1,ε(ρ1, ρ2, t) ≈ max{

Hε1s (ρ1) ,H

ε2s (ρ2)

}where εi = εi (ε, t).

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Second order asymptotics

I Key tool: Second order asymptotic expansion of Hεs(ρ⊗n):

Hεs(ρ⊗n) = nS(ρ)−

√n s(ρ)Φ−1 (ε) +O(log n)

B s(ρ) := Tr(ρ(log ρ)2)− S(ρ)2 is the quantum information variance.B Φ−1 (ε) = sup{z ∈ R | Φ(z) ≤ ε} is the inverse of the cdf of a normal

random variable.

I Goal: determine expansion

mn,ε(ρ1, ρ2, t) = na +√

nb +O(log n).

I Problem: due to maximum of Hεs(ρi ) appearing in one-shot bounds,

direct application of the second order expansion is not possible.

I Solution: distinction between three cases based on the von Neumannentropies S(ρi ) and mixing parameter t.

I Inspired by classical analysis of mixed sources [Nomura, Han].

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Second order asymptotics

I The three cases: abbreviate Si ≡ S(ρi ), si ≡ s(ρi )

B Case 1: S1 = S2, s1 < s2, ε ∈ (0, 1/2)B Case 2: S1 > S2, t > εB Case 3: S1 > S2, t < ε

Main result

Let ε, t ∈ (0, 1) and ρ1, ρ2 ∈ D(H).

I Case 1: mn,ε(ρ1, ρ2, t) = nS2 −√

n s2Φ−1 (ε) +O(log n)

I Case 2: mn,ε(ρ1, ρ2, t) = nS1 −√

n s1Φ−1(εt

)+O(log n)

I Case 3: mn,ε(ρ1, ρ2, t) = nS2 −√

n s2Φ−1(ε−t1−t

)+O(log n)

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References & Acknowledgements

References:

I FL, Nilanjana Datta, “Source coding for a mixed source:determination of second order asymptotics,” arXiv:1407.6616[quant-ph], 2014.

I Nilanjana Datta, FL, “Second-order asymptotics for source coding,dense coding and pure-state entanglement conversions,”arXiv:1403.2543 [quant-ph], 2014.

I Marco Tomamichel, Masahito Hayashi, “A hierarchy of informationquantities for finite block length analysis of quantum tasks, IEEETrans. on Inf. Th. 59, no. 11, 76937710, 2013.

Special thanks to Francesco Buscemi, Nilanjana Datta, Will Matthews andDavid Reeb for valuable feedback.

Thank you very much for your attention!

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Second order asymptotics

I How do we prove this?

I Key observation: Growth of Hεs(ρ⊗n

i ) is mainly governed by firstorder term Si in asymptotic expansion

Hεs(ρ⊗n

i ) = nSi −√

n siΦ−1 (ε) +O(log n).

I Example: Case 2

B By assumption, S1 > S2 and t > ε.B For sufficiently large n, this implies that

Hε1s

(ρ⊗n1

)> H

ε2s

(ρ⊗n2

)for any ε1, ε2 ∈ (0, 1).

B Hence, the first source ρ1 dominates in the second order asymptotics ofthe minimum compression length.

I Similar assertions hold in Cases 1 and 3.

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