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This work was funded by the ERC (grant agreement No 648913)
Quantum channels from subfactors
Pieter Naaijkens Universidad Complutense de Madrid 21 March 2019
Three cultures
Thanks to Reinhard Werner for this characterisation.
Quantum channels from subfactors Pieter Naaijkens Universidad - - PowerPoint PPT Presentation
Quantum channels from subfactors Pieter Naaijkens Universidad - - PowerPoint PPT Presentation
Quantum channels from subfactors Pieter Naaijkens Universidad Complutense de Madrid 21 March 2019 This work was funded by the ERC (grant agreement No 648913) Three cultures Type I n : everything finite dimensional (no infinite resources)
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use quantum systems to communicate main question: how much information can I transmit? will consider infinite systems here… … described by subfactors channel capacity is given by Jones index
Quantum information
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Outline
Classical information theory Subfactors and QI
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Classical information theory
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Image source: Alfred Eisenstaedt/The LIFE Picture Collection
Information theory Alice wants to communicate with Bob using a noisy channel. How much information can Alice send to Bob per use of the channel?
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Setup
Alice Bob input space
- utput space
How well can Bob recover the messages sent by Alice (small error allowed)?
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Shannon entropy
Def: Measure for the information content of X Coding: represent tuples in Xn by codewords (asymptotically, error goes to zero!)
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Relative entropy
Compare two probability distributions P, Q: Vanishes iff P=Q, otherwise positive
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Mutual information
`information’ due to noise here the conditional entropy is defined: some algebra gives:
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Channel capacities
What is the maximum amount of information we can send through the channel? Def: the Shannon capacity of the channel is defined as:
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Operational approach
encode n channels N messages decode Maximum error for all possible encodings:
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Coding theorem
Def: R is called an achievable rate if The supremum of all R is called the capacity C.
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Quantum information
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Quantum information
work mainly in the Heisenberg picture
- bservables modelled by von Neumann algebra
consider normal states on channels are normal unital CP maps Araki relative entropy
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Araki relative entropy
Let be faithful normal states: Def: Def:
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Distinguishing states Alice prepares a mixed state : …and sends it to Bob Can Bob recover ?
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Holevo 𝜓 quantity In general not exactly: Generalisation of Shannon information
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Infinite systems
Suppose is an infinite factor, say Type III, and a faithful normal state
Better to compare algebras!
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Comparing algebras
Want to compare and , with subfactor Shirokov & Holevo, arXiv:1608.02203
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A quantum channel
For finite index inclusion , say irreducible, quantum channel, describes the restriction of operations
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Jones index and entropy
gives an information-theoretic interpretation to the Jones index
Hiai, J. Operator Theory, ’90; J. Math. Soc. Japan, ‘91
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Quantum wiretapping
Alice Bob Eve
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Theorem (Devetak, Cai/Winter/Young) The rate of a wiretapping channel is given by
lim
n→∞
1 n max
{px,ρx}
- χ({px}, Φ⊗n
B (ρx)}) − χ({px}, Φ⊗n E (ρx)})
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A conjecture
The Jones index of a subfactor gives the classical capacity of the wiretapping channel that restricts from to .
[M : N] M N
- L. Fiedler, PN, T.J. Osborne, New J. Phys 19:023039 (2017)
PN, Contemp. Math. 717, pp. 257-279 (2018), arXiv:1704.05562
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