2. Laura has 2010 lamps and 2010 switches in front of her, with different switches controlling different lamps. She wants to find the correspondence between the switches and the lamps. For this, Charlie operates the switches. Each time Charlie presses some switches, and the number of lamps that light up is the same as the number of switches pressed (Note: The switches return to their original state after each operation). The questions are:
(1) What is the maximum number of different operations Charlie can perform so that Laura can accurately determine the correspondence between the lamps and the switches?
(2) If Laura operates the switches herself, what is the minimum number of operations she needs to find the correspondence between the lamps and the switches?
(1) Charlie selects a pair of switches $(A, B)$, each operation involves pressing them simultaneously or leaving them untouched, while other switches can be chosen arbitrarily, resulting in $2^{2009}$ different operations, but Laura cannot determine which lights are controlled by switches $A$ and $B$.

If Charlie performs $2^{2009}+1$ different operations, consider a particular switch $A$. Each operation either presses $A$ or leaves it untouched, dividing the operations into two categories, one of which must contain at least $2^{2008}+1$ operations (let's assume it's the category where $A$ is pressed). Consider the set of lights turned on by these operations, where one light is always on, indicating it is controlled by $A$.
(2) Since $2010<2^{11}$, we can use 11-bit binary numbers to number all the switches (from 00000000001 to 11111011010). Use 11 operations to number the lights (also using 11-bit binary numbers): the $k$-th operation ($k=1,2, \cdots, 11$) involves pressing the switches whose $k$-th bit is 1. Write 1 in the $k$-th bit of the number for all lights that are on, and 0 for those that are off. Thus, switches with the same number control the same light, establishing a one-to-one correspondence. Therefore, the switch-light correspondence can be determined in at most 11 operations.