It is well known that two-dimensional states, that is, qubits, can be represented using the Bloch sphere: all pure states lie on the surface of a three-dimensional sphere, while all the mixed states are in its interior.

This is consistent with a simple parameter counting argument: the space of all qubits is the space of all $2\times 2$ positive Hermitian operators with unit trace, which has dimension $2^2-1=3$, while for the pure states there are only $2\times2 - 2=2$ real degrees of freedom (two complex coefficients minus normalization and global phase). The same conclusion can be reached by considering that the pure states are a subset of all states satisfying the single additional constraint that $\operatorname{Tr}(\rho^2)=1$.

What happens for higher dimensional qudits? Let's consider the states living in a Hilbert space of dimension $D$. The space of all states is again the space of positive Hermitian $D\times D$ operators with unit trace, which has dimension $D^2-1$. The pure states can on the other hand be represented as length-$D$ complex vectors, so the number of real degrees of freedom, after having considered the global phase and the normalization condition, seems to be $2D-2$.

However, the set of pure states is also equal to the subset of all those states $\rho$ such that $\operatorname{Tr}(\rho^2)=1$. Now this seems like a single real condition, which therefore would make me think that the dimension of the space of pure $D$-dimensional states is $D^2-2$, in direct contrast with the argument above.

What is the correct way to think about this?


The Hilbert space counting which gets $2D-2$ is correct. When we think about parameter counting in the way you have in this question, we are implicitly assuming that the equations are sufficiently "generic" so that intersections work the way they do in linear algebra. This is not always the case, particularly when we consider equations with singularities or which are defined on spaces with boundaries. For an extreme example, in a normed $N$-dimensional real vector space, the equation $|\vec v|^2 = 0$ is a single equation, but it reduces the $N$-dimensional space down to a single (apparently $0$-dimensional) point.

When we write the equation $\text{Tr} \rho^2 = 1$, a slightly more complicated version of the same thing is happening. If you diagonalize the density operator, you will get a set of $D$ real eigenvalues $\lambda_i$. These must each be non-negative for a density operator. Additionally, we know that $\rho$ has unit trace, meaning that $\sum_i \lambda_i = 1$. This means that each $\lambda_i \in [0,1]$. Under these conditions, $\lambda_i \ge \lambda_i^2$ with equality only for $\lambda_i \in \{0,1\}$. Thus $\text{Tr} \rho^2 = \sum_i \lambda_i^2 \le \text{Tr} \rho = 1$, and the two are equal only if all the $\lambda_i$ are either $0$ or $1$, which means that exactly one is $1$ and the others are $0$. Note that this is just saying that $\rho$ is a projection onto the single eigenvector with eigenvalue $1$, meaning that $\rho = | \psi \rangle \langle \psi |$ for some $|\psi\rangle$.

Let us also note that, in general, the boundary of the set of mixed states is not the set of pure states. Being on the boundary means that (just) one inequality becomes an equality, which means we only need one $\lambda_i = 0$. Being a pure state is a much stronger condition. I think this may be part of your confusion as the boundary of the set of mixed states does have dimension $D^2 - 2$. The Bloch sphere is an unhelpful example in this case, because since the Hilbert space is only $2$ dimensional, one eigenvalue going to $0$ is equivalent to being a pure state, but for larger $D$ that is not true.

Note that this still looks like you only impose $D$ real equations, namely one per eigenvalue, meaning the naive dimension counting still appears to be wrong. Why is that? The answer is tied to the fact that our end result has a degeneracy; specifically we have $D-1$ eigenvectors of $\rho$ with eigenvalue $0$. Thus the system, described in this way, has a fictitious $U(D-1)$ symmetry rotating those vectors. If you apply such a rotation, the density operator does not change, but our naive counting would not realize that. We would think that we should subtract the dimensionality of $U(D-1)$, namely $(D-1)^2$. But this $U(D-1)$ does not act freely; a transformation which only changes the phase of a given $|\psi_i \rangle$ leaves $|\psi_i \rangle \langle \psi_i |$ invariant, meaning that any basis for the Hilbert space is stabilized by $U(1)^{D}$, so we actually have only $(D-1)^2 - D$ real equation redundancies. Now we can finally get the right counting:

$$(D^2 - 1) - D - ((D-1)^2 - D) = 2D-2.$$

The "trick" that makes this counting work, whereas the more naive counting fails, is that these equations can all be imposed on the affine space of trace-class unit trace operators without requiring positive semi-definiteness and still get the right set of pure states. Positive semidefiniteness is the set of inequalities that bounds the space of mixed states, and if we don't impose it we don't have the issues arising before where the solutions to the equations are on the boundary of the space.

  • $\begingroup$ great answer, thanks! Do you have any reference to suggest that treats these kinds of geometrical arguments for quantum states? $\endgroup$ – glS Apr 6 '18 at 16:34
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    $\begingroup$ I have skimmed the textbook "Geometry of Quantum States: An Introduction to Quantum Entanglement" by Bengtsson and Zyczkowski, and it seems like a reasonable introduction. arxiv.org/abs/quant-ph/0606228 is equivalent to some portion of the book and freely available. If you care primarily about pure states and complex projective geometry arxiv.org/abs/quant-ph/9906086 may have more information of interest. $\endgroup$ – Logan M Apr 6 '18 at 16:44

You can reach the correct conclusion starting from $$ \vert 1\rangle=\left(\begin{array}{c}1 \\ 0 \\ 0\\ \vdots \\ 0\end{array}\right) $$ and the simple density matrix $\rho_1=\vert 1\rangle\langle 1\vert$. Clearly, $\rho_1$ is left invariant by all transformations of the type $$ T= \left(\begin{array}{cc} e^{i\phi}&0\\ 0&U(n-1)\end{array}\right) $$ i.e. $T\rho_1 T^{-1}=\rho_1$. A general transformation of $\rho_1$ by unitary matrices depends on $n^2-1$ parameters: $U(n)$ has $n^2$ parameters but an overall phase will leave $\rho_1$ invariant. The transformation $T$ depends on $(n-1)^2$ parameters. Thus, the total of parameters to specify an $n$-dit is $$ n^2-1 - (n-1)^2 = 2n-2. $$


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