ENTROPIC CHARACTERISTICS OF QUANTUM CHANNELS AND THE ADDITIVITY PROBLEM A. S. Holevo Steklov Mathematical Institute, Moscow

Introduction: quantum information theory The classical capacity of quantum channel Hierarchy of additivity conjectures Global equivalence Partial results

INTRODUCTION A brief history of quantum information theory

Information Theory Born: middle of XX century, s (Shannon,…) Concepts: random source, entropy, typicality, code, channel, capacity: Tools: probability theory, discrete math, group theory,… Impact: digital data processing, data compression, error correction,…

Quantum Information Theory Born: second half of XX century Physics of quantum communication, s (Gabor, Gordon, Helstrom,…): FUNDAMENTAL QUANTUM LIMITATIONS ON INFORMATIOM TRANSMISSION ? Mathematical framework: s

Quantum Information Theory The early age ( s) Understanding quantum limits Concepts: random source, entropy, channel, capacity, coding theorem, …, entanglement Tools: noncommutative probability, operator algebra, random matrices (large deviations)… Implications: …, the upper bound for classical capacity of quantum channel: χ -capacity C ≤ C χ An overview in the book “Statistical structure of quantum theory” (Springer, 2001)‏

Quantum Information Theory (“Quantum Shannon theory”)‏ The new age ( s)‏ From quantum limitations to quantum advantages Q. data compression (Schumacher-Josza,…)‏ The quantum coding theorem for c-q channels: C = C χ (Holevo; Schumacher-Westmoreland)‏ Variety of quantum channel capacities/coding theorems (Shor, Devetak, Winter, ‏Hayden,…)‏ Summarized in recent book by Hayashi (Springer, 2006)‏

Additivity of channel capacity CLASSICAL INFORMATION CLASSICAL INFORMATION ? ? MEMORYLESS encodingdecoding

The χ -CAPACITY and the CLASSICAL CAPACITY of QUANTUM CHANNEL

Finite quantum system

Composite quantum systems – entanglement

Quantum channel Completely positive (CP) map, Σ( H )→ Σ( H ’): ρρ’

Product of channels

The minimal output entropy ?

The χ -capacity ensemble average conditional output entropy output entropy

The Additivity Conjecture ?

Separate encodings/ separate decodings CLASSICAL INFORMATION CLASSICAL INFORMATION.. n. separate q. encodings decodings ACCESSIBLE (SHANNON) INFORMATION

Separate encodings/ entangled decodings CLASSICAL INFORMATION CLASSICAL INFORMATION.. n. separate q. entangled encodings decodings HSW-theorem: χ - CAPACITY ! !

Entangled encodings/ entangled decodings CLASSICAL INFORMATION CLASSICAL INFORMATION.. n. entangled encodings decodings The full CLASSICAL CAPACITY ? ?

HIERARCHY of ADDITIVITY CONJECTURES - minimal output entropy - χ- capacity – convex closure/ constrained χ- capacity/EoF

Additivity of the minimal output entropy ?

Rényi entropies and p-norms

Rényi entropies for p<1

The χ -capacity ensemble average conditional output entropy output entropy convex closure

Convex closure EoF

Constrained capacity

Additivity with constraints ?

Equivalent forms of (CA )‏ THM

Partial results Qubit unital channel (King)‏ Entanglement-breaking channel (Shor) Depolarizing channel (King)‏ Lieb-Thirring inequality:

Recent work on special channels (2003-…) Alicki-Fannes; Datta-Fukuda-Holevo-Suhov; Giovannetti-Lloyd-Maccone-Shapiro-Yen; Hayashi-Imai-Matsumoto-Ruskai-Shimono; King-Nathanson-Ruskai; King-Koldan; Matsumoto-Yura; Macchiavello-Palma; Wolf-Eisert,… ALL ADDITIVE!

Transpose-depolarizing channel Numerical search for counterexamples

Breakthrough 2007 Multiplicativity breaks: p>2, large d (Winter); 1<p<2, large d (Hayden); p=0, large d (Winter); p close to 0. Method: random unitary (non-constructive)‏ It remains 0<p<1 and p=1 (the additivity!)‏... And many other questions

Random unitary channels

The basic Additivity Conjecture remains open

GLOBAL EQUIVALENCE of additivity conjectures (Shor, Audenaert-Braunstein, Matsumoto- Shimono-Winter, Pomeranski, Holevo- Shirokov)

“Global” proofs involving Shor’s channel extensions Discontinuity of In infinite dimensions

THM Proof: Uses Shor’s trick: extension of the original channel which has capacity obtained by the Lagrange method with a linear constraint

Channel extension 1 0

Lagrange Function

Set of states is separable metric space, not locally compact Entropy is “almost always” infinite and everywhere discontinuous BUT Entropy is lower semicontinuous Entropy is finite and continuous on “useful” compact subset of states (of bounded “mean energy”)‏

The χ -capacity ensemble average conditional output entropy output entropy Generalized ensemble (GE)=Borel probability measure on state space

THM In particular, for all Gaussian channels with energy constraints

Gaussian channels Canonical variables (CCR) Gaussian environment Gaussian states Gaussian states Energy constraint PROP For arbitrary Gaussian channel with energy constraint an optimal generalized ensemble (GE) exists. CONJ Optimal GE is a Gaussian probability measure supported by pure Gaussian states with fixed correlation matrix. (GAUSSIAN CHANNELS HAVE GAUSSIAN OPTIMIZERS?)‏ Holds for c-c, c-q, q-c Gaussian channels

CLASSES of CHANNELS

Complementary channels (AH, Matsumoto et al.,2005)‏ Observation: additivity holds for very classical channels; for very quantum channels Example:

Complementary channels

Entanglement-breaking channels

Entanglement-breaking channels -- additivity

Symmetric channels ?