# How to make sense of phases of individual qubits in the context of multiple entangled qubits?

I hear that phase information of qubits is important and you can clearly see the phase of a qubit when represented on the Bloch sphere.

But, I am not sure what to think of the phase of individual qubits in the context of an entangled state of multiple qubits.

For example, when talking about the GHZ state,one could have a relative phase relation between $$|000\rangle$$ and $$|111\rangle$$ in the form of $$\frac{1}{\sqrt{2}}|000\rangle+e^{i\phi}|111\rangle$$

But, that's just one phase as opposed to multiple phases which come from the phases of individual qubits.

Is there a simple way to make sense of the fact that there is only one relative phase despite the fact that there are 3 individual qubits?

• "But, that's just one phase as opposed to multiple phases which come from the phases of individual qubits." could you clarify what you mean with "phases coming from the phases of the individual qubits"? The relative phases are attached to pairs of basis states, not individual qubits.
– glS
Sep 16 at 8:07

To set a context for my understanding of your question let's start with a single qubit. Then, the state can be described by two complex numbers $$|\psi\rangle=a|0\rangle+b|1\rangle$$ subject to a constraint $$|a|^2+|b|^2=1$$ and up to an equivalence $$|\psi\rangle \sim e^{i\phi} |\psi\rangle$$. Absolute values of $$a$$ and $$b$$ determine probabilities to obtain either state $$|0\rangle$$ or $$|1\rangle$$ when measured in the computational basis ($$|a|^2$$ and $$|b|^2$$, respectively). These probabilities are insensitive to phases of $$a$$ and $$b$$. However, for other measurements the relative phase between $$a$$ and $$b$$ does matter. For example, states $$|\pm\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle\pm |1\rangle\right)$$ are orthogonal and have different probabilities when measured in the basis $$|\pm\rangle$$.
For two qubits there will be four coefficientn $$|\psi\rangle = a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$$ subject to $$|a|^2+|b|^2+|c|^2+|d|^2=1$$ and up to a global phase $$|\psi\rangle\sim e^{i\phi}|\psi\rangle$$. Again, for measurments in the computational basis only absolute values of $$a,b,c,d$$ are relevant. However, for any generic measurement phases of these complex numbers are also essential. For multi-qubit states you can assign phases to each vector in the computational basis (the phase of the corresponding coefficient) but that is not the same as phases of individual qubits. In fact, I do not think there any sense to talk about phases of individual qubits when they are part of the multi-qubit system.
If I understand correctly, you assume that in a multi-qubit state individual qubits still must have some "phases". This is incorrect, as individual qubits do not even have (pure) states! In sharp contrast to classical mechanics a state of a composite system in quantum mechanics can not generally be described by describing each part of the system individually. This is only true for unentangled states. If a qubit is entangled with others it generally can only be described by a density matrix. In your example, the GHZ state is maximally entangled. The density matrix of any individual qubit is $$\rho=\begin{pmatrix}\frac12&0\\0&\frac12\end{pmatrix}$$ and it is as far from any pure state as you can get. Operationally, measuring such qubit in any basis gives completely random results (for pure states there is always a basis with definite results of the measurements).
• Thank you. I think the clarification definitely helped me understand it better. So, if you project the GHZ state and it results in $|111\rangle$, only then it makes sense to start thinking about the phases of the individual qubits. Is that a correct statement? Sep 16 at 19:25