Equivalence of Additivity Questions in Quantum Information Theory

By Peter W. Shor. 2003

234 citations recorded by the source corpus.

We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. We show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of formation, and strong superadditivity of the entanglement of formation, are either all true or all false.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/82aba3b6acb0f8148f5898470d0e44c4cca4734e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. John Watrous
2. Andreas J. Winter
3. Karol Zyczkowski
4. Mary Beth Ruskai
5. Ion Nechita
6. Benoît Collins
7. Ingemar Bengtsson
8. Renato Renner

In Atomic and Molecular Physics, and Optics

1. Andreas J. Winter
2. John Watrous
3. Karol Zyczkowski
4. Ingemar Bengtsson
5. Mary Beth Ruskai
6. Renato Renner
7. Patrick M. Hayden
8. Ion Nechita

In Computational Theory and Mathematics

1. Andreas J. Winter
2. Renato Renner
3. Ingemar Bengtsson
4. Ion Nechita
5. Alexander S. Holevo
6. John Watrous
7. Dennis Kretschmann
8. Karol Zyczkowski

Citations in the corpus

• The Theory of Quantum Information (2018, background): “Shor [192] proved that the non-additivity of Holevo capacity follows from the non-additivity of minimum output entropy.”
• Geometry of Quantum States: Index (2007, methodology): “In a seminal paper [837] Shor showed that additivity of the Holevo information is equivalent to the additivity of other relevant quantities, including the minimal output entropy Smin( 1⊗ 2) = Smin( 1)+Smin( 2), where Smin( ) = minρ S( (ρ)).”
• Quantum channels with memory (2005, background): “(Whether such examples exist in the memoryless setting is still an open question, and presently considered one of the most eminent open problems of quantum information theory, with wide implications for other problems in the field [16, 17].)”
• Symmetry of large physical systems implies independence of subsystems (2007, result): “Essentially, the problem is to prove the following conjecture, called additivity of the minimum output entropy of a quantum channel, which has been shown to be equivalent to various other additivity questions [5].”
• Complementary Channels and the Additivity Problem (2005, background): “if one holds for all channels, another holds for all channels as well [17].”
• Properties of Conjugate Channels with Applications to Additivity and Multiplicativity (2005, background): “Shor [29] has shown that several of these conjectures are globally equivalent to (17).”
• Random matrix techniques in quantum information theory (2015, background): “ with non-zero probability. At this moment, we are not aware of any explicit, non-random choices for ; in the case 1 p2. The additivity property for the minimum output entropy Hmin() was related in [Sho04] to the additivity of another important entropic quantity, the Holevo quantity ˜() = max fp i;X”
• Aspects of Generic Entanglement (2004, background): “In the first case, ρAB is a (rather drastic) counterexample to the additivity conjecture for the entanglement of formation: ∀σ Ef (σ AB) = nEf (σAB) (see, for example, [45]).”
• Quantum entanglement (2007, unknown): “Quite interestingly, these problems have even further connection with entanglement: they are equivalent to additivity of one of the most important measures of entanglement — so called entanglement of formation, see (Audenaert and Braunstein, 2004; Koashi and Winter, 2004; Matsumoto, 2005; Shor, 2003”