In attribute-based access control, users with certain verified attributes will gain access to some particular data. Concerning with privacy of the users' attributes, we study the problem of distributed attribute-based private access control (DAPAC) with multiple authorities, where each authority will learn and verify only one of the attributes. To investigate its fundamental limits, we introduce an information theoretic DAPAC framework, with $N \in \mathbb{N}$, $N\geq 2$, replicated non-colluding servers (authorities) and some users. Each user has an attribute vector $\mathbf{v^*}=(v_1^*, ..., v_N^*)$ of dimension $N$ and is eligible to retrieve a message $W^{\mathbf{v}^*}$, available in all servers. Each server $n\in [N]$ is able to only observe and verify the $n$'th attribute of a user. In response, it sends a function of its data to the user. The system must satisfy the following conditions: (1) Correctness: the user with attribute vector $\mathbf{v^*}$ is able to retrieve his intended message $W^{\mathbf{v}^*}$ from the servers' response, (2) Data Secrecy: the user will not learn anything about the other messages, (3) Attribute Privacy: each Server~$n$ learns nothing beyond attribute $n$ of the user. The capacity of the DAPAC is defined as the ratio of the file size and the aggregated size of the responses, maximized over all feasible schemes. We obtain a lower bound on the capacity of this problem by proposing an achievable algorithm with rate $\frac{1}{2K}$, where $K$ is the size of the alphabet of each attribute.