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Theory

POVMs

Let \text{Her}(\mathcal{H}) \subset \mathcal{H} be the set of hermitian operators acting on a Hilbert space \mathcal{H}. Elements of \text{Her}(\mathcal{H}) may be written as double kets: | F \rangle \! \rangle. Such a space is also a real Hilbert space under the inner product

$$\langle \! \langle E | F \rangle \! \rangle= \text{Tr} E^\dagger F = \text{Tr} E F$$

The state of a quantum system is represented by an element \rho of the set of positive semi-definite operators \text{Pos}(\mathcal{H}) \subset \text{Her}(\mathcal{H}) such that \text{Tr} \rho = 1. Observables are represented by elements A \in \text{Her}(\mathcal{H}) so that its expectation value is given by \langle A \rangle = \langle \! \langle A | \rho \rangle \! \rangle. A Positive Operator Valued Measure (POVM) is a set of observables \{F_m\} \subset \text{Pos}(\mathcal{H}) with the property that \sum_m F_m = I, where I is the identity operator. These operators model the possible outcomes of an experiment: outcome m happens with probability p(m) = \langle \! \langle F_m | \rho \rangle \! \rangle. The POVM conditions ensure that p(m) \ge 0 and \sum_m p(m) = 1.

Linear Inversion

The simplest method which solves the problem of quantum state tomography is linear inversion, which we describe in what follows. A POVM with M elements induces a linear map T: \text{Her}(\mathcal{H}) \to \mathbb{R}^M defined by the expression

$$T| \Omega \rangle \! \rangle = \left(\langle \! \langle F_1 | \Omega \rangle \! \rangle,\ldots,\langle \! \langle F_M | \Omega \rangle \! \rangle\right).$$

In order to be suitable for tomography, we want that the measurement probabilities \mathbf{p} = T | \rho \rangle \! \rangle uniquely determine the state | \rho \rangle \! \rangle. This is assured if the transformation T is injective. A POVM with this property is said to be informationally complete.

By choosing a basis \{\Omega_n\} \subset \text{Her}(\mathcal{H}), we can specify an arbitrary state | \rho \rangle \! \rangle by a list of coefficients \mathbf{x} = (x_1,\ldots,x_N) such that | \rho \rangle \! \rangle = \sum_n x_n | \Omega_n \rangle \! \rangle. Then, denoting by \mathbb{T} the matrix of T with respect to the canonical basis of \mathbb{R}^M and \{\Omega_m\}, which has entries \mathbb{T}_{mn} = \langle \! \langle F_m | \Omega_n \rangle \! \rangle, we have \mathbf{p} = \mathbb{T}\mathbf{x}. When \mathbb{T} is injective, \mathbb{T}^\dagger\mathbb{T} is invertible, and then we can explicitly write the solution of the above equation as

$$\mathbf{x} = (\mathbb{T}^\dagger\mathbb{T})^{-1} \mathbb{T}^\dagger \mathbf{p},$$

which, in theory, solves the tomography problem.

Note that, to apply this method, one needs a reliable estimate of the probabilities \mathbf{p}. This can only be obtained by performing a large number of measurements.

Bayesian tomography

Bayesian tomography is the application of Bayesian inference to the problem of quantum state tomography 1. The goal is to estimate the posterior distribution of the quantum state given the observed data. The posterior distribution is given by Bayes' theorem

$$P(\rho | \mathcal{M}) = \frac{P(\mathcal{M} | \rho) P(\rho)}{P(\mathcal{M})},$$

where P(\mathcal{M} | \rho) is the likelihood of the observations \mathcal{M} given the state, P(\rho) is the prior distribution of the state, and P(\mathcal{M}) is the evidence. In the case of quantum state tomography, the likelihood is given by the Born rule

$$P(\mathcal{M} | \rho) = \prod_m p(m)^{n_m}, \ \ \ p(m) = \langle \! \langle F_m | \rho \rangle \! \rangle,$$

where n_m is the number of times outcome m was observed. The prior distribution is a probability distribution over the space of states, which encodes any prior knowledge about the state. The evidence is the normalization constant, given by P(\mathcal{M}) = \int P(\mathcal{M} | \rho) P(\rho) d\rho.

Bayesian inference does not provide a single estimate of the state, but a full distribution. The mean of the distribution is, in a sense, the best estimate of the state 1, and the variance gives an idea of the uncertainty of the estimate. Directly calculating the posterior distribution is infeasible, as it requires the computation of the evidence, which is a high-dimensional integral. The alternative is to sample the posterior distribution using Markov Chain Monte Carlo (MCMC) methods. The package provides an implementation of the Metropolis Adjusted Langevin Algorithm (MALA) 2 3 to sample the posterior distribution.

As shown in the video above, MALA generates a random walk in the space of valid density operators (we reject all proposals falling outside this set) whose statistics are given by the desired posterior distribution.

Footnotes

  1. Blume-Kohout, Robin. "Optimal, reliable estimation of quantum states." New Journal of Physics 12.4 (2010): 043034. ↩ ↩2

  2. Karagulyan, Avetik. Sampling with the Langevin Monte-Carlo. Diss. Institut polytechnique de Paris, 2021. ↩

  3. [Titsias, Michalis. "Optimal Preconditioning and Fisher Adaptive Langevin Sampling." Advances in Neural Information Processing Systems 36 (2024).] (https://arxiv.org/abs/2305.14442) ↩