# How many measurements does it take to determine a quantum state?

Suppose I have a wave-function over a Hilbert-space of (complex) dimension $$N$$. It has $$2 N-2$$ real degrees of freedom, after normalization and removing the phase. It seems to me that I can measure these degrees of freedom with $$2N-2$$ measurements, first by projecting on each of the basis states, and then by making a measurement for each of the relative phases.

Lucien Hardy in his 2001 paper about the "reasonable axioms" (https://arxiv.org/abs/quant-ph/0101012) says it takes instead $$N^2 -2$$ measurements to completely determine the state, by which he means the density matrix (I am referring to normalized pure states, otherwise it's $$N^2$$). He calls these "fiducial" measurments.

I understand where the $$N^2$$ comes from -- it's the number of real entries that you need to specify a generic hermitian matrix over a complex vector space of dimension $$N$$.

What I don't understand is why do I need $$N^2-2$$ measurements if I know that the density matrix of a pure state can be written as a tensor product of the wave-function and therefore has only $$2N-2$$ degrees of freedom? Why do the measurements that I mentioned above (projection on basis plus relative angles) not entirely determine the density matrix? Or do I misunderstand what the "fiducial" measurements are?

• Could you point to where in the paper this statement is? – user2723984 May 26 '20 at 10:28
• It says on page 2 "The number of degrees of freedom, $K$, is defined as the minimum number of probability measurements needed to determine the state, or, more roughly, as the number of real parameters required to specify the state." He then goes on to show that in quantum mechanics $K=N^2$ (states are not normalized and this includes mixed states, hence the difference of 2 to my question). The "fiducial" measurements are defined at the beginning of section 6.3: "We will call the probability measurements labeled by $k = 1$ to $K$ used in determining the state the fiducial measurements" – WIMP May 26 '20 at 10:33
• As far as I read the paper, the statement $K=N^2$ is never claimed to be a lower bound for quantum states, but an upper bound, since they include mixed, pure, normalized, and unnormalized states. So I don't think the paper implies that you need $N^2 - 2$ measurements if you know that the state you are looking at is a pure state. – Marius Ladegård Meyer May 26 '20 at 11:23
• See the top of page 9 for a discussion. – Marius Ladegård Meyer May 26 '20 at 11:42
• about the parameter counting using the "one constraint" of $\operatorname{tr}(\rho^2)$, @NorbertSchuch might be thinking of this or maybe this threads – glS May 27 '20 at 11:39