## 5.8 Why qubits, subsystems, and entanglement?

One question that is rather natural to ask at this point is the following:

If entanglement is so fragile and difficult to control, then why bother? Why not perform the whole computation in one physical system that has as many quantum states as we normally have labels for the states of qubits, so that we can label these quantum states in the same way as we normally label the qubits, and give them computational meaning?

This suggestion, although possible, gives a very inefficient way of representing data (it is describing the unary encoding). For serious computations we need subsystems. Here is why.

Suppose you have n physical objects, and each object has k distinguishable states. If you can access each object separately and put it into any of the k states, then, with only n operations, you can prepare any of the k^{n} different configurations of the combined systems. Without any loss of generality, let us take k=2 and refer to each object of this type as a physical bit. We label the two states of a physical bit as 0 and 1. Any collection of n physical bits can be prepared in 2^{n} different configurations, which can be used to store up to 2^{n} messages/binary strings/different numbers. In order to represent numbers from 0 to N-1 we just have to choose n such that N\leqslant 2^n.

Suppose the two states in the physical bit are separated by the energy difference \Delta E. Then a preparation of any particular configuration will cost no more than E=n \Delta E=(\log_2 N)\Delta E units of energy.

In contrast, if we choose to encode N configurations into one chunk of matter, say, into the first N energy states of a single harmonic oscillator with the interstate energy separation \Delta E then, in the worst case, one has to use E=N\Delta E units of energy, e.g. to go from the ground state (labelled as 0) to the most excited state (labelled as N). For large N this gives an exponential gap in the energy expenditure between the binary encoding using physical bits and unary encoding, using energy levels of harmonic oscillators.

One can, of course, try to switch from harmonic oscillators to quantum systems which have a finite spread in the energy spectrum. For example, by operating on the energy states of the hydrogen atom one can encode any number from 0 to N-1, and one is guaranteed not to spend more than E_{\mathrm{max}}= 13.6\,\mathrm{eV} (otherwise the atom is ionised). The snag is that, in this case, some of the electronic states will be separated by the energy difference to the order of E_{\mathrm{max}}/N, and to drive the system selectively from one state to another one has to tune into the frequency E_{\mathrm{max}}/\hbar N, which requires a sufficiently long wave-packet (so that the frequency is well defined), and consequently the interaction time is of order N(\hbar/E_{\mathrm{max}}).

That is, we have to trade energy for time.

It turns out that whichever way we try to represent the number N using the unary encoding (i.e. using N different states of a single chunk of matter), we end up depleting our physical resources (such as energy, time, space) at a much greater rate than in the case when we use subsystems. This plausibility argument indicates that, for efficient processing of information, the system must be divided into subsystems — for example, into physical bits.