使用改进相等查询的无公共币量子同时协议

Optimal one-shot quantum algorithm for EQUALITY and AND

We study the computation complexity of Boolean functions in the quantum black box model. In this model our task is to compute a function $f:\{0,1\}\to\{0,1\}$ on an input $x\in\{0,1\}^n$ that can be accessed by querying the black box. Quantum algorithms are inherently probabilistic; we are interested in the lowest possible probability that the algorithm outputs incorrect answer (the error probability) for a fixed number of queries. We show that the lowest possible error probability for $AND_n$ and $EQUALITY_{n+1}$ is $1/2-n/(n^2+1)$.

Conditions for equality between entanglement-assisted and unassisted classical capacities of a quantum channel

Several relations between the Holevo capacity and the entanglement-assisted classical capacity of a quantum channel are proved, necessary and sufficient conditions for their coincidence are obtained. In particular, it is shown that these capacities coincide if (correspondingly, only if) the channel (correspondingly, the $χ$--essential part of the channel) belongs to the class of classical-quantum channels (the $χ$-essential part is a restriction of a channel obtained by discarding all states useless for transmission of classical information). The obtained conditions and their corollaries are g…

小误差下相等性问题的随机与量子通信复杂度的紧界

Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

研究相等性函数在小误差下的随机与量子通信复杂度，给出私有硬币协议改进及纯态、混合态单向协议的紧界，并涉及EPR对数量与近似秩上界。

Monotonicity of the Holevo quantity: a necessary condition for equality in terms of a channel and its applications

A condition for reversibility (sufficiency) of a channel with respect to a given countable family of states with bounded rank is obtained. This condition shows that a quantum channel preserving the Holevo quantity of at least one (discrete or continuous) ensemble of states with rank $\leq r$ has the r-partially entanglement-breaking complementary channel. Several applications of this result are considered. In particular, it is shown that coincidence of the constrained Holevo capacity and the quantum mutual information of a quantum channel at least at one full rank state implies that this chann…

On equalities in two entropic inequalities

A simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel is obtained. It implies that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown that for a Gaussian channel having no completely depolarizing components the above characteristics may coincide only at non-Gaussian mixed states and a criterion of existence of such states is given. All the obtained results may be reformulated as condi…

Equality conditions for internal entropies of certain classical and quantum models

Mathematical models use information from past observations to generate predictions about the future. If two models make identical predictions the one that needs less information from the past to do this is preferred. It is already known that certain classical models (certain Hidden Markov Models called ε-machines which are often optimal classical models) are not in general the preferred ones. We extend this result and show that even optimal classical models (models with minimal internal entropy) in general are not the best possible models (called ideal models). Instead of optimal classical mod…