The Operational Meaning of Min- and Max-Entropy Christian Schaffner – CWI Amsterdam, NL joint work with Robert König – Caltech Renato Renner – ETH Zürich, Switzerland TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AAA A A A A A A AA A A A A A

2 /34 Agenda Shannon / von Neumann Entropy Min- and Max-Entropies Operational Meaning Conclusion

3 /34 Information Theory quantify the acquisition, transmission, storage of data often used: Shannon entropy Example: data compression minimal encoding length: [Shannon]: for iid

4 /34 quantum setting: finite-dimensional Hilbert spaces classical-quantum setting: classical setting: Notation

5 /34 Operational Interpretation of Shannon Entropy data compression of a source: transmission rate of a channel: secret-key rate of a correlation: …

6 /34 Operational Interpr of van Neumann Entropy data compression of a source: randomness-extraction rate of a cq-state: secret-key rate of a cqq-state: …

7 /34 van Neumann Entropy simple definition: for state “handy” calculus: chain rule: strong subadditivity: …

8 /34 van Neumann Entropy data compression: randomness extraction: Shannon entropy: … simple definition “handy” calculus operational: many iid settings one-shot setting? 

9 /34 Conditional Min- and Max-Entropy conditional van Neumann entropy: conditional min-entropy: conditional max-entropy: for pure [Renner 05] for pure Goal of this talk: Understanding these quantities!

10 /34 for product state: measure for the rank of ½ A Warm-Up Calculations for a product state classically:

11 /34 Warm-Up Calculations for a pure state

12 /34 Classical Min-Entropy without Conditioning suggests “smoothing”: … … 

13 /34 Smooth Min- and Max-Entropy where ± (, ) is the trace distance or (squared) fidelity for a purification [Renner 05]

14 /34 Smooth-Min-Entropy Calculus von Neumann entropy as special case: strong subadditivity: additivity: chain rules: 

15 /34 Single-Shot Data Compression [Renner,Wolf 04] with i.e. up to additive constant of order minimal encoding length: [Shannon]: for iid *

16 /34 Privacy Amplification [Renner, König 07] with maximum number of extractable bits such that

17 /34 Decoupling [Renner, Winter, Berta 07] with maximum size of A’ such that completely mixed state on A’

18 /34 State Merging with [Renner, Winter, Berta 07] minimal number of ebits required to transmit ½ A to B with LOCC LOCC maximal number of ebits generated by transmitting ½ A to B with LOCC [Horodecki, Oppenheim, Winter 05]

19 /34 Agenda von Neumann Entropy Min- and Max-Entropies Operational Meaning Conclusion

20 /34 Conditional Min- and Max-Entropy conditional van Neumann entropy: conditional min-entropy: conditional max-entropy: for pure [Renner 05] for pure Goal of this talk: Understanding these quantities!

21 /34 Warm-Up: Classical Case for classical states: corresponds to average probability of guessing X given Y notion used by [Dodis, Smith]

22 /34 The Operational Meaning of Min-Entropy for classical states: guessing probability for cq-states: guessing probability for a POVM {M x }

23 /34 The Operational Meaning of Min-Entropy for cq-states: guessing probability for qq-states: achievable singlet fraction F(, ) 2

24 /34 F(, ) 2 The Operational Meaning of Max-Entropy for cq-states: security of a key for

25 /34 F(, ) 2 The Operational Meaning of Max-Entropy for cq-states: security of a key for qq-states: decoupling accuracy for

26 /34 Proof: using Duality of SDPs for cq-states: guessing probability primal semi-definite program (SDP)

27 /34 Proof II: Choi-Jamiolkowski isomorphism bijective quantum operations

28 /34 Proof III: Putting It Together CPTP maps bijective

29 /34 Proof: Operational Interpr of Max-Entropy for follows using monotonicity of fidelity unitary relation of purifications

30 /34 connections between operational quantities, e.g. randomness extraction additivity of min-/max-entropies: · follows from definition Implications of our Results

31 /34 subadditivity of min-entropy: Implications of our Results implies subadditivity of von Neumann entropy

32 /34 van Neumann entropy: easy definition calculus shortcomings in single-shot setting replaced by min- and max-entropies scenarios for smooth versions operational interpretation of non-smooth versions Conclusions 

33 /34 Summary

34 /34 Summary

35 /34 operational meaning of smooth-min entropy calculus for fidelity-based smooth min-entropy Open questions

36 /34 Example: Channel Capacity maximum number of transmittable bits: [Shannon] (noisy-channel coding):

37 /34 Single-Shot Channel Capacity [Renner,Wolf,Wullschleger 06]: with maximum number of transmittable bits: [Shannon] (noisy-channel coding):