# How to measure the arbitrariness of a quantum state?

An arbitrary qubit is represented as $\alpha|0\rangle+\beta|1\rangle$ with $|\alpha|^2+|\beta|^2=1$. If we know either $\alpha$ or $\beta$, the state can be completely identified. The 'arbitrariness' of an arbitrary qubit thus can be said to be '1'. For a known state this is '0'. Now if we consider an arbitrary two qubit state, the total number of variables are 3 (actually 4 but 1 less due to normalization). If any two of these variables are equal or one of them is known, the state is still arbitrary but from a given restricted class. We can say that the 'arbitrariness' of this quantum state is something between 0 and 1. Can anyone suggest a measure for the arbitrariness of this quantum state? I want to extend this measure to higher order states. Can anyone suggest any reference for this?

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It's not true that "If we know either α or β, the state can be completely identified." There is an important relative phase which has to be identified yet. – Ali Jul 29 '13 at 22:32
Also, I think using density matrices will give you the result you are looking for. – Ali Jul 29 '13 at 22:34
Remember $\alpha,\beta$ are complex numbers, so each is two real numbers. There are $4-2=2$ real number parameters for a one cubit state: one magnitude and one relative phase ($\alpha$ alone is not enough). For a two cubit state there are four complex numbers but a normalization constraint and an unobservable overall phase so a total of $8-2=6$ real variables. In general it's $2\times 2^n - 2$ real variables for an $n$ cubit state. – Michael Brown Jul 30 '13 at 6:59