Yet another additivity conjecture

It is known that the additivity conjecture of Holevo capacity, output minimum entoropy, and the entanglement of formation (EoF), are equivalent with each other. Among them, the output minimum entropy is simplest, and hence many researchers are focusing on this quantity. Here, we suggest yet another entanglement measure, whose strong superadditivity and additivity are equivalent to the additivity of the quantities mentioned above. This quantity is as simple as the output minimum entropy, and in existing proofs of additivity conjecture of the output minimum entropy for the specific examples, they are essentially proving the strong superadditivity of this quantity.Comment: corrections of typo, etc. minor revisio

Entanglement entropy and vacuum states in Schwarzschild geometry

Recently, it was proposed that there must be either large violation of the additivity conjecture or a set of disentangled states of the black hole in the AdS/CFT correspondence. In this paper, we study the additivity conjecture for quantum states of fields around the Schwarzschild black hole. In the eternal Schwarzschild spacetime, the entanglement entropy of the Hawking radiation is calculated assuming that the vacuum state is the Hartle-Hawking vacuum. In the additivity conjecture, we need to consider the state which gives minimal output entropy of a quantum channel. The Hartle-Hawking vacuum state does not give the minimal output entropy which is consistent with the additivity conjecture. We study the entanglement entropy in other static vacua and show that it is consistent with the additivity conjecture.Comment: 31 pages, 1 figure; v2: 33 pages, minor corrections, references adde

On Shor's channel extension and constrained channels

In this paper we give several equivalent formulations of the additivity conjecture for constrained channels, which formally is substantially stronger than the unconstrained additivity. To this end a characteristic property of the optimal ensemble for such a channel is derived, generalizing the maximal distance property. It is shown that the additivity conjecture for constrained channels holds true for certain nontrivial classes of channels. Recently P. Shor showed that conjectured additivity properties for several quantum information quantities are in fact equivalent. After giving an algebraic formulation for the Shor's channel extension, its main asymptotic property is proved. It is then used to show that additivity for two constrained channels can be reduced to the same problem for unconstrained channels, and hence, "global" additivity for channels with arbitrary constraints is equivalent to additivity without constraints.Comment: 19 pages; substantially revised and enhanced. To appear in Commun. Math. Phy

On Hastings' counterexamples to the minimum output entropy additivity conjecture

Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming majority of channels consisting of a Haar random isometry followed by partial trace over the environment, for an environment dimension much bigger than the output dimension. This makes Hastings' original reasoning clearer and extends the class of channels for which additivity can be shown to be violated.Comment: 17 pages + 1 lin

Convex Trace Functions on Quantum Channels and the Additivity Conjecture

We study a natural generalization of the additivity problem in quantum information theory: given a pair of quantum channels, then what is the set of convex trace functions that attain their maximum on unentangled inputs, if they are applied to the corresponding output state? We prove several results on the structure of the set of those convex functions that are "additive" in this more general sense. In particular, we show that all operator convex functions are additive for the Werner-Holevo channel in 3x3 dimensions, which contains the well-known additivity results for this channel as special cases.Comment: 9 pages, 1 figure. Published versio

Non-additivity of Renyi entropy and Dvoretzky's Theorem

The goal of this note is to show that the analysis of the minimum output p-Renyi entropy of a typical quantum channel essentially amounts to applying Milman's version of Dvoretzky's Theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden-Winter disproving the additivity conjecture for the minimal output p-Renyi entropy (for p>1).Comment: 8 pages, LaTeX; v2: added and updated references, minor editorial changes, no content change

The Power of Unentanglement

The class QMA(k). introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. * We give a protocol by which a verifier can be convinced that a 3SAT formula of size m is satisfiable, with constant soundness, given Õ (√m) unentangled quantum witnesses with O(log m) qubits each. Our protocol relies on the existence of very short PCPs. * We show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. * We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one