# Can you encode two values in a single qubit state, say its phase and orientation?

I had an interesting talk with someone who said that we can use a single qubit to encode two different values. We encode the first value in its orientation (up/down) of particle/wave (electron, photon, etc.) and we encode the second value in the phase of that particle/wave. What do you think about that?

• – glS
Jul 13 '20 at 15:55

Each qubit can be described as $$|\psi\rangle = \alpha|0\rangle+\mathrm{e}^{i\varphi}\beta|1\rangle,$$ where $$\alpha^2+\beta^2=1$$. Both $$\alpha$$ and $$\beta$$ are real numbers. These values are so-called amplitudes and it holds that $$P(|0\rangle) = \alpha^2$$ and $$P(|1\rangle) = \beta^2$$. So, amplitudes describe probability that you find your qubit in either state after measurement (up or down in your question or $$|0\rangle$$ or $$|1\rangle$$ in example above). An angle $$\varphi$$ is the phase, the person you mentioned talked about. It is another purely quantum parameter of qubit (i.e. there is no analog in classical world). The existence of phase is one of reasons why quantum computer can be superior to classical one in some tasks.

Appendix:

You can also write the qubit as $$|\psi\rangle = \alpha|0\rangle+\beta|1\rangle,$$ where $$|\alpha|^2+|\beta|^2=1$$. Both $$\alpha$$ and $$\beta$$ are complex numbers. These numbers are again called amplitudes and it holds that $$P(|0\rangle) = |\alpha|^2$$ and $$P(|1\rangle) = |\beta|^2$$. The phase is hidden in amplitudes in this case.

Both above mentioned forms can be transformed to each other (write complex numbers $$\alpha$$ and $$\beta$$ in expontential form).

No. Even though you can encode information in both the amplitude and phase degrees of freedom independently, it is important that the two values'' be encoded in a way that they can be successfully distinguished. This nuance is captured by the so-called Holevo's theorem which essentially states that a qubit can contain at most one bit of information.

This is a fairly counterintuitive result since a single qubit state can take infinitely many values, corresponding to various choices of the amplitudes and phases. But since there are only two orthogonal states (for the pure state case), one can retrieve only a single bit of information from a qubit.

What is "a single qubit state" in your question?

If it is really a single qubit in a unique unknown state and there are no other qubits in the same state (for example, prepared in the same way), then in the answer it is already quite explained for this case.

If it is a single qubit in a quantum program (circuit) designed to be executed on a NISQ computer (or on its simulator), then usually such a circuit is run several times (or shots) up to obtain stable results. Thus, although one speaks of measuring a single qubit state, each shot a "new" state of the qubit is measured. Accordingly, using multiple runs, you can get the orientation (e.g. up/down) in different proportions, which can "encode" various values. Also two "encoded" values can be "decoded" from a single qubit, measuring the orientation (up/down) in half the cases, the phase in the other half. And more than two values, if measuring this qubit in more then two various basises, for details see e.g. here.