How to write byte(s) to a file in C++?

The issue is operator<< is the text encoding method, even if you've specified std::ofstream::binary. You can use put to write a single binary character or write to output multiple characters. Note that you are responsible for the conversion of data to its char representation.

std::bitset<8> bits = foo();
std::ofstream outfile("compressed", std::ofstream::out | std::ofstream::binary);

// In reality, your conversion code is probably more complicated than this
char repr = bits.to_ulong();

// Use scoped sentries to output with put/write
{
    std::ofstream::sentry sentry(outfile);
    if (sentry)
    {
        outfile.put(repr);                  // <- Option 1
        outfile.write(&repr, sizeof repr);  // <- Option 2
    }
}