Geometric Quantum Mechanics

What is the “state” of a quantum system? More concretely, what do we mean when we use the word state? 

The answer can be approached from different angles but, at its core, I believe the notion of “state of a system” can be synthesized as follows. Given a certain system, its state is a mathematical representation of the knowledge that we can gain about the system itself. In more practical terms, a system has a certain number of physical observables–variables that the system naturally possess and whose operational meaning is given by the fact that a system interacts with its surroundings through channels that are mediated by such variables. 

The collection of all possible states of a certain system is usually called the “state space”. In the case of quantum mechanics, the space of the states of a quantum system is a Hilbert space which, here, will be considered to have finite dimension. However, it is well-known that such system contains a redundancy, given by the fact that states that differ for a phase or by normalization describe the same physical state. Thus, it is more appropriate to consider the space of quantum states, which is the collection of physically different quantum states. This manifold has been studied by mathematicians and physicists: It is a  Complex Projective space CPn. Most of the visual examples I will go through refer to CP1, which happens to be essentially equivalent to the so-called Rimemann Sphere or, in the context of quantum mechanics, Bloch Sphere. These manifolds have interesting mathematical properties and the formulation of quantum mechanics which leverages them goes under the name of Geometric Quantum Mechanics. References on GQM which I enjoyed very much are 

Among all the interesting aspects of GQM, the one I find most fascinating is that by using these tools one comes to understand how to encode the quantumness of a certain system in the geometry of its manifold of states. This is done in a very interesting way by Complex Projective spaces, which have two different notions of geometry, deeply intertwined with each other. The first one is Riemannian geometry. This means that the manifold has a preferred notion of distance between points, which allows us to define geodesics, compute lengths,  areas and volumes. This is the geometry we learn when we want to do calculations in General Relativity and the relevant metric here is called Fubini-Study metric, from the names of the first two people who studied it. The second one is Symplectic geometry, which is the fundamental notion of geometry necessary to describe phase-spaces in classical Hamiltonian mechanics. Without entering too much in the details, this has two important consequences. First, it means there exist a notion of Poisson Brackets, which implicitly define the geometry of Hamiltonian flows. Remarkably, one thus find that Schroedinger’s equation is nothing by Hamilton’s equations of motion in disguise. Second, exhibiting the properties of a phase-space, this allows to define a fundamental notion of measure to perform integrals and compute probabilities.

Statistical States

Once we identified the underlying geometric properties of our space of quantum states, the modern developments of probability theory suggest that an appropriate definition for the notion of state is as a probability distribution on the underlying space. A very interesting book that goes through various aspects of a very similar idea is Ensembles of Configuration Spaces, by Michael Hall and Marcel Reginatto. The statistical interpretation of the state is given using ensemble theory: The probability distribution (call it Q) essentially describes an ensemble of Independent and Identically Distributed systems, with probability distribution given by Q. This is the interpretation of density matrices, for example. A pure state would be a Dirac-delta distribution, while a mixed state is represented by a density matrix. However, it becomes clear that these are very specific assumptions about the structure of the underlying probability distribution on the space of quantum states which, in principle, is more general. We call this probability distribution Geometric Quantum State and, in Beyond Density Matrices: Geometric Quantum States, I go through the details of this idea and show how and why the geometric quantum state underlies the notion of density matrix.

Here you can see an example of two geometric quantum states. I have chosen them because if you compute the density matrix from them, you will get the same result. Since it is fairly clear that they are describing situations with physically appreciable differences, the fact that the density matrix is the same suggests that a density matrix is an incomplete notion of state. In particular, one realizes that the missing information can be understood in terms of interaction with an unknown environment. Indeed, while the density matrix does contain all the relevant information about the result of all possible quantum measurements one can perform on the system, it does not contain (1) the detailed information about how these results are realized due to the exchange of information of the system with its surroundings and, as a consequence (2) it does not allow us to understand the underlying physical and information-theoretic resources associated with it, thus creating fundamental misattributions. Here is a concrete example of what this mean.

Imagine two similar but slightly different situations, A and B. In both cases we are pulling states from two classically distinguishable boxes, red and blue, according to what the outcome of a classical coin throw is.

In Situation A, our boxes contain orthogonal states |0> and |1> , but we use a biased coin, with p0 – p1 = λ. The bias λ is essentially a measure of the randomness in the process, and therefore connected to memory.

In Situation B the resources are inverted. We use a fair coin to decide which box to use to pull quantum states, but the states inside each box are not orthogonal. They are |ψ0> and |ψ1>, with their definition given in the figure.

Now, if we pull a large number of states from both situations and we look at the ensemble that is formed, it is fairly easy to see that the density matrices we obtain are exactly the same one.

What we have done here is fairly easy to understand, we have traded a feature of classical randomness, the bias, for one of quantum nature. And yet, this kind of resource-oriented statement can not be captured by the density matrix formalism. It can, however, easily be captured by the geometric formalism. Indeed, while these two physical situations have the same density matrix, they have different underlying geometric quantum state.

One of the advantages of the geometric formalism is that one can use arbitrary coordinates to describe the manifold of the states. In this example we are looking a geometric quantum state on CP1, which is the manifold of quit states. The coordinates (p,φ) we are using are defined as follows. An arbitrary quit state is described by c0|0> + c1 |1>, where |c0|2 + |c1|2 = 1. However, thanks to the invariance under a global phase we can always choose c0 to be real. Moreover, by calling c1=peiφ and using normalization we find c0=(1-p)1/2. Thus, a pure state is completely specified by two real numbers, p ∈ [0,1] and φ ∈ [0,2π] and a geometric quantum state is a probability distribution on such manifold, as the one you can see in the figure.  

This formulation of quantum mechanics, and the associated notion of geometric quantum state,  opens the door to a plethora of interesting novel tools and research directions, which I am currently exploring. In the first one Geometric Quantum Thermodynamics, I explore a different idea of Quantum Thermalization and the consequence this has on Quantum Thermodynamics. Here we propose a definition of Quantum Heat and Quantum Work which differ from what is currently used in the literature. These quantities, however, can obviously measured and satisfy interesting fluctuation theorems as Crooks’ relation and Jarzynski’s equality.

In the second one, which currently is a reformulation of a previous result by Brody, we explore the idea of a maximum entropy inference principle, based on a notion of entropy which we dub Geometric Entropy: Geometric Quantum State Estimation.