There are at least two choices to initialize your record type as you want. One is using an initialization function, and the other is using the value of N in an aggregate.
Functions are a nice way to initialize custom data types. In your case, you could create a function default_entry_from_width(n), returning an entry_type value:
type entry_type is record
state: std_logic_vector;
end record;
function default_entry_from_width(width: natural) return entry_type is
variable return_vector: std_logic_vector(width-1 downto 0);
begin
for i in return_vector'range loop
return_vector(i) := '1' when i <= width/2 else '0';
end loop;
return (state => return_vector);
end;
constant ENTRY_1: entry_type := default_entry_from_width(3); -- return 011
constant ENTRY_2: entry_type := default_entry_from_width(4); -- return 0011
The other alternative is to initialize the constant with an aggregate, using the previsouly defined value of N:
constant N: natural := 4;
constant ENTRY_3: entry_type := (
state => (
N-1 downto N/2 => '1',
N/2-1 downto 0 => '0'
)
);