If you have a Hilbert space $\mathcal{H}$, you can make a projective Hilbert space by modding out by the $U(1)$ group action of multiplying by a phase, $\mathcal{H}/U(1)$.

While we usually talk about states as vectors in a Hilbert space, really they are rays in a projective Hilbert space because phases are unobservable. This has some physical consequences like the existence of half integer spin.

However, there seems to me to be something you cannot do in a projective Hilbert space: add states. While there is no difference between $|\psi\rangle$ and $-|\psi\rangle$, there is a big difference between $|\phi\rangle + |\psi\rangle$ and $|\phi\rangle - |\psi\rangle$. So how can states be added together in a self consistent way in a projective Hilbert space?

  • $\begingroup$ I was sort of hoping for an answer along the lines of "here is the extra structure we endow a projective Hilbert space with that allows us to add states" without ever having to actually leave the projective Hilbert space in order to do it $\endgroup$ Apr 12, 2020 at 18:58

1 Answer 1


Normally, and in a finite number of dimensions, projective Hilbert space is not a vector space, but a (complex) manifold. (I say normally because I am not sure if a contrived topology and structure could be "cooked up" on it as a bare set to make it a vector space, say, by removing or adding a single point, etc.)

Consider a pure state, say $\alpha |0\rangle + \beta |1\rangle$ with $|\alpha|^2 + |\beta|^2 = 1$. Once you switch to the projective space perspective, then there is no need to think as if in vector space. In fact, one may argue that working in the projective space is even more intuitive than in the conventional representation of the state in the vector space because it provides a direct link to "constructible" states (the Hilbert space dimension essentially explodes as the number of degrees of freedom increases, so the Hilbert space of states is really a fiction that a real physical system can never hope of visiting or be prepared at but a tiny fraction). What does it mean to have the superposition $\alpha |0\rangle + \beta |1\rangle$? What is the relation between the mathematical abstraction of $\alpha, \, \beta \in \mathbb{C}$, normalized to satisfy $|\alpha|^2 + |\beta|^2 = 1$, with settings and knobs in an apparatus that attempts to create this kind of state? Depending on the kind of quantum system we are discussing, this may be accomplished using interaction ("manipulation") between the system and the apparatus (to name just a few: lasers, RF, Rabi "flopping", adiabatic methods, etc -- or, "hard interaction" that gives "measurement") that we would mathematically represent with some operators that may be unitary or "measurement" operators. Therefore, to answer your question "So how can states be added together in a self-consistent way in a projective Hilbert space", it would be done with group action (unitaries) and measurement operators. This includes "passive transformations" (e.g., rotating the apparatus). In fact, there are well-developed theories of both the group actions on homogeneous spaces such as here as well as quantum operations. (Also, experimentalists are of course getting ever better at manipulating quantum systems.) Note that even an observable (that we ordinarily associate with a self-adjoint operator barring superselection rules) can be represented as a (one-parameter) group action (and, if we are lucky, we may not even need to get entangled with tricky issues of a domain when it comes to operators defined on infinite-dimensional spaces). To wrap it up, with non-pure states the story goes along similar lines except that at some point it is easier to just completely switch to algebras of operators and group actions (and other quantum operations) on them.


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