This is a 10-minute talk given at a summer school on Quantum Information Theory in Smolenice, Slovakia in August 2014. The topic is the derivation of second order asymptotics for quantum source coding using a mixed source.

1. Source coding for a mixed source:
determination of second order asymptotics
arXiv:1407.6616
Felix Leditzky Nilanjana Datta
27 August 2014
QESS, Smolenice

2. Table of Contents
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 2 / 1

3. Fixed-length visible source coding
Quantum source: pure-state ensemble {pi , ψi }
Visible setting: Alice knows identity of signals ψi
→ receives classical information in form of label i
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 3 / 1

4. Fixed-length visible source coding
Encoding
Input
Decoding
Visible encoder: V : {1, . . . , k} → D(Hc) (can be non-linear!)
Hc: compressed Hilbert space with M := dim Hc < dim H.
Decoding: D : D(Hc) → D(H) (CPTP map)
Goal: Retrieve output state that is close to input state on average
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 4 / 1

5. Fixed-length visible source coding
Figure of merit: Ensemble average fidelity
¯F(E, V, D) :=
i
pi Tr((D ◦ V)(i)ψi )
For ∈ (0, 1), a code (V, D, M) is -admissible if
¯F(E, V, D) ≥ 1 − .
Definition
-error one-shot minimum compression length:
m(1),
(ρ) := inf{log M | (V, D, M) is an -admissible code}
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 5 / 1

6. Table of Contents
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 6 / 1

7. Mixed source coding
Mixed source: emits signals from one of two i.i.d. sources:
1 2
1 2
n
n
t
Source state: ρ(n) = tρ⊗n
1 + (1 − t)ρ⊗n
2
After the first signal, the source sticks to the chosen source.
Simple example of a non-i.i.d. protocol
For ∈ (0, 1) we define mn, (ρ1, ρ2, t) := m(1), ρ(n) .
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 7 / 1

8. Mixed source coding
Goal: determine the second order asymptotics of mn, (ρ1, ρ2, t)
Find a and b in the expansion
mn,
(ρ1, ρ2, t) = na +
√
nb + O(log n).
Compare: Single source with i.i.d. state ρ(n) = ρ⊗n:
Schumacher: a = S(ρ)
b given by a Gaussian approximation [Datta, FL]
Mixed source with non-i.i.d. source state
ρ(n)
= tρ⊗n
1 + (1 − t)ρ⊗n
2
is much harder to evaluate!
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 8 / 1

9. Table of Contents
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 9 / 1

10. One-shot bounds
Bounds for the minimum compression length in terms of the
information spectrum entropy
Hs(ρ) := inf{γ ∈ R | Tr(ρ − 2−γ
1)+ ≥ 1 − }
Properties: positive for states, quasi-concavity, etc.
Source state for a single use of the source: ρ = tρ1 + (1 − t)ρ2
Theorem
For ∈ (0, 1), we have
m1,
(ρ1, ρ2, t) ≈ max H 1
s (ρ1) , H 2
s (ρ2)
where i = i ( , t).
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 10 / 1

11. Table of Contents
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 11 / 1

12. Second order asymptotics
Key tool: Second order asymptotic expansion of Hs(ρ⊗n):
Hs(ρ⊗n
) = nS(ρ) −
√
n s(ρ)Φ−1
( ) + O(log n)
s(ρ) := Tr(ρ(log ρ)2
) − S(ρ)2
is the quantum information variance.
Φ−1
( ) = sup{z ∈ R | Φ(z) ≤ } is the inverse of the cdf of a normal
random variable.
Goal: determine expansion
mn,
(ρ1, ρ2, t) = na +
√
nb + O(log n).
Problem: due to maximum of Hs(ρi ) appearing in one-shot bounds,
direct application of the second order expansion is not possible.
Solution: distinction between three cases based on the von Neumann
entropies S(ρi ) and mixing parameter t.
Inspired by classical analysis of mixed sources [Nomura, Han].
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 12 / 1

13. Second order asymptotics
The three cases: abbreviate Si ≡ S(ρi ), si ≡ s(ρi )
Case 1: S1 = S2, s1 < s2, ∈ (0, 1/2)
Case 2: S1 > S2, t >
Case 3: S1 > S2, t <
Main result
Let , t ∈ (0, 1) and ρ1, ρ2 ∈ D(H).
Case 1: mn, (ρ1, ρ2, t) = nS2 −
√
n s2Φ−1 ( ) + O(log n)
Case 2: mn, (ρ1, ρ2, t) = nS1 −
√
n s1Φ−1
t + O(log n)
Case 3: mn, (ρ1, ρ2, t) = nS2 −
√
n s2Φ−1 −t
1−t + O(log n)
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 13 / 1

14. References & Acknowledgements
References:
FL, Nilanjana Datta, “Source coding for a mixed source:
determination of second order asymptotics,” arXiv:1407.6616
[quant-ph], 2014.
Nilanjana Datta, FL, “Second-order asymptotics for source coding,
dense coding and pure-state entanglement conversions,”
arXiv:1403.2543 [quant-ph], 2014.
Marco Tomamichel, Masahito Hayashi, “A hierarchy of information
quantities for finite block length analysis of quantum tasks, IEEE
Trans. on Inf. Th. 59, no. 11, 76937710, 2013.
Special thanks to Francesco Buscemi, Nilanjana Datta, Will Matthews and
David Reeb for valuable feedback.
Thank you very much for your attention!
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 14 / 1

15. Second order asymptotics
How do we prove this?
Key observation: Growth of Hs(ρ⊗n
i ) is mainly governed by first
order term Si in asymptotic expansion
Hs(ρ⊗n
i ) = nSi −
√
n si Φ−1
( ) + O(log n).
Example: Case 2
By assumption, S1 > S2 and t > .
For sufficiently large n, this implies that
H
1
s ρ⊗n
1 > H
2
s ρ⊗n
2
for any 1, 2 ∈ (0, 1).
Hence, the first source ρ1 dominates in the second order asymptotics of
the minimum compression length.
Similar assertions hold in Cases 1 and 3.
Felix Leditzky (University of Cambridge) Source coding for a mixed source 27 August 2014 15 / 1