Abstract
In this chapter, based on the lectures by Andreas Winter, we survey four different areas in quantum information theory in which ideas from the theory of operator systems and operator algebras play a natural role. The first is the problem of zero-error communication over quantum channels, which makes use of concepts from operator systems and quantum error correction. Second, we study the strong subadditivity property of quantum entropy and the central role played by the structure theorem of matrix algebras in understanding its equality case. Next, we describe different norms on quantum states and the corresponding induced norms on quantum channels. Finally, we look at matrix-valued random variables, prove Hoeffding-type tail bounds and describe the applications of such matrix tail bounds in the quantum information setting.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A remark on notation: throughout this chapter we use I to denote the identity operator and \(\mathbb{I}\) to denote the identity map, for example, \(\mathbb{I}_{A}: \mathcal{B}(\mathcal{H}_{A}) \rightarrow \mathcal{B}(\mathcal{H}_{A})\).
- 2.
The Hilbert-Schmidt inner product between two operators A,B is simply the inner product defined by the trace, namely, \(\langle A,B\rangle = \text{Tr}[A^{\dag }B]\).
- 3.
- 4.
For any CPTP map \(\mathcal{T}\) in a Kraus representation {T i }, the adjoint map is another CPTP map with Kraus operators {T i †}.
- 5.
This is essentially a mean ergodic theorem for the dual of a quantum operation. See [11] for a proof.
- 6.
Any convex, centrally symmetric subset \(\mathbb{M} \subset [-\mathcal{I},\mathcal{I}] \subset \mathcal{B}(\mathcal{H})\) with the property \(-\mathbb{M} = \mathbb{M}\), gives rise to a norm, the dual of which is a measure of the distinguishability.
- 7.
We could equally well have considered unitaries of the type \(I_{A} \otimes U_{B}\), and the same results would hold.
References
R. Ahlswede, A. Winter, Strong converse for identification via quantum channels. IEEE Trans. Inf. Theory 48(3), 569–579 (2002)
S. Beigi, P. Shor, On the complexity of computing zero-error and Holevo capacity of quantum channels. ArXiv preprint arXiv:0709.2090 (2007)
S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)
S. Boyd, L. Vandenberghe, Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)
T.S. Cubitt, J. Chen, A.W. Harrow, Superactivation of the asymptotic zero-error classical capacity of a quantum channel. IEEE Trans. Inf. Theory 57(12), 8114–8126 (2011)
T.S. Cubitt, D. Leung, W. Matthews, A. Winter, Improving zero-error classical communication with entanglement. Phys. Rev. Lett. 104, 230503 (2010)
D.P. Divincenzo, D.W. Leung, B.M. Terhal, Quantum data hiding. IEEE Trans. Inf. Theory 48(3), 580–598 (2002)
R. Duan, Super-activation of zero-error capacity of noisy quantum channels. ArXiv preprint arXiv:0906.2527 (2009)
R. Duan, S. Severini, A. Winter, Zero-error communication via quantum channels, non-commutative graphs, and a quantum Lovász number. IEEE Trans. Inf. Theory 59, 1164–1174 (2013)
B. Groisman, S. Popescu, A. Winter, Quantum, classical, and total amount of correlations in a quantum state. Phys. Rev. A 72, 032317 (2005)
P. Hayden, R. Jozsa, D. Petz, A. Winter, Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys. 246(2), 359–374 (2004)
A.S. Holevo, Information-theoretical aspects of quantum measurement. Problemy Peredachi Informatsii 9(2), 31–42 (1973); C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976)
M. Horodecki, J. Openheim, A. Winter, Partial quantum information. Nature 436, 673–676 (2005)
M. Horodecki, J. Openheim, A. Winter, Quantum state merging and negative information. Commun. Math. Phys. 269, 107–136 (2007)
M. Koashi, N. Imoto, Operations that do not disturb partially known quantum states. Phys. Rev. A 66, 022318 (2002)
S. Kullback, R.A. Leibler, On information and sufficiency. Ann. Math. Stat. 22(1), 79–86 (1951)
E.H. Lieb, M.B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14, 1938–1941 (1973)
G. Lindblad, Completely positive maps and entropy inequalities. Commun. Math. Phys. 40(2), 147–151 (1975)
G. Lindblad, A general no-cloning theorem. Lett. Math. Phys. 47(2), 189–196 (1999)
L. Lovász, On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25(1), 1–7 (1979)
R.A.C. Medeiros, R. Alleaume, G. Cohen, F.M. de Assis, Quantum states characterization for the zero-error capacity. ArXiv pre-print:quant-ph/0611042 (2006)
D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Commun. Math. Phys. 105(1), 123–131 (1986)
D. Petz, Sufficiency of channels over von Neumann algebras. Q. J. Math. 39(1), 97 (1988)
M.B. Ruskai, Inequalities for quantum entropy: a review with conditions for equality. J. Math. Phys. 43, 4358 (2002)
B. Schumacher, Quantum coding. Phys. Rev. A 51(4), 2738 (1995)
M. Takesaki, Theory of Operator Algebras I (Springer, New York, 1979)
J.A. Tropp, User-friendly tail bounds for sums of random matrices. Found. Comput. Math. 12(4), 389–434 (2012)
A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Commun. Math. Phys. 54(1), 21–32 (1977)
H. Umegaki, Conditional expectation in an operator algebra. IV. Entropy and information. Kodai Math. J. 14(2), 59–85 (1962)
L. Vandenberghe, S. Boyd, Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Gupta, V.P., Mandayam, P., Sunder, V.S. (2015). Quantum Information Theory. In: The Functional Analysis of Quantum Information Theory. Lecture Notes in Physics, vol 902. Springer, Cham. https://doi.org/10.1007/978-3-319-16718-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-16718-3_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16717-6
Online ISBN: 978-3-319-16718-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)