7 research outputs found
Verification of state and entanglement with incomplete tomography
There exists, in general, a convex set of quantum state estimators that
maximize the likelihood for informationally incomplete data. We propose an
estimation scheme, catered to measurement data of this kind, to search for the
exact maximum-likelihood-maximum-entropy estimator using semidefinite
programming and a standard multi-dimensional function optimization routine.
This scheme can be used to infer the expectation values of a set of
entanglement witnesses that can be used to verify the entanglement of the
unknown quantum state for composite systems. Next, we establish an alternative
numerical scheme that is more computationally robust for the sole purpose of
maximizing the likelihood and entropy.Comment: 15 pages, 5 figure
Optimal, reliable estimation of quantum states
Accurately inferring the state of a quantum device from the results of
measurements is a crucial task in building quantum information processing
hardware. The predominant state estimation procedure, maximum likelihood
estimation (MLE), generally reports an estimate with zero eigenvalues. These
cannot be justified. Furthermore, the MLE estimate is incompatible with error
bars, so conclusions drawn from it are suspect. I propose an alternative
procedure, Bayesian mean estimation (BME). BME never yields zero eigenvalues,
its eigenvalues provide a bound on their own uncertainties, and it is the most
accurate procedure possible. I show how to implement BME numerically, and how
to obtain natural error bars that are compatible with the estimate. Finally, I
briefly discuss the differences between Bayesian and frequentist estimation
techniques.Comment: RevTeX; 14 pages, 2 embedded figures. Comments enthusiastically
welcomed
Entanglement quantification from incomplete measurements: Applications using photon-number-resolving weak homodyne detectors
The certificate of success for a number of important quantum information
processing protocols, such as entanglement distillation, is based on the
difference in the entanglement content of the quantum states before and after
the protocol. In such cases, effective bounds need to be placed on the
entanglement of non-local states consistent with statistics obtained from local
measurements. In this work, we study numerically the ability of a novel type of
homodyne detector which combines phase sensitivity and photon-number resolution
to set accurate bounds on the entanglement content of two-mode quadrature
squeezed states without the need for full state tomography. We show that it is
possible to set tight lower bounds on the entanglement of a family of two-mode
degaussified states using only a few measurements. This presents a significant
improvement over the resource requirements for the experimental demonstration
of continuous-variable entanglement distillation, which traditionally relies on
full quantum state tomography.Comment: 18 pages, 6 figure
Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators
Intuitively, if a density operator has small rank, then it should be easier
to estimate from experimental data, since in this case only a few eigenvectors
need to be learned. We prove two complementary results that confirm this
intuition. First, we show that a low-rank density matrix can be estimated using
fewer copies of the state, i.e., the sample complexity of tomography decreases
with the rank. Second, we show that unknown low-rank states can be
reconstructed from an incomplete set of measurements, using techniques from
compressed sensing and matrix completion. These techniques use simple Pauli
measurements, and their output can be certified without making any assumptions
about the unknown state.
We give a new theoretical analysis of compressed tomography, based on the
restricted isometry property (RIP) for low-rank matrices. Using these tools, we
obtain near-optimal error bounds, for the realistic situation where the data
contains noise due to finite statistics, and the density matrix is full-rank
with decaying eigenvalues. We also obtain upper-bounds on the sample complexity
of compressed tomography, and almost-matching lower bounds on the sample
complexity of any procedure using adaptive sequences of Pauli measurements.
Using numerical simulations, we compare the performance of two compressed
sensing estimators with standard maximum-likelihood estimation (MLE). We find
that, given comparable experimental resources, the compressed sensing
estimators consistently produce higher-fidelity state reconstructions than MLE.
In addition, the use of an incomplete set of measurements leads to faster
classical processing with no loss of accuracy.
Finally, we show how to certify the accuracy of a low rank estimate using
direct fidelity estimation and we describe a method for compressed quantum
process tomography that works for processes with small Kraus rank.Comment: 16 pages, 3 figures. Matlab code included with the source file
Rank-based model selection for multiple ions quantum tomography
The statistical analysis of measurement data has become a key component of
many quantum engineering experiments. As standard full state tomography becomes
unfeasible for large dimensional quantum systems, one needs to exploit prior
information and the "sparsity" properties of the experimental state in order to
reduce the dimensionality of the estimation problem. In this paper we propose
model selection as a general principle for finding the simplest, or most
parsimonious explanation of the data, by fitting different models and choosing
the estimator with the best trade-off between likelihood fit and model
complexity. We apply two well established model selection methods -- the Akaike
information criterion (AIC) and the Bayesian information criterion (BIC) -- to
models consising of states of fixed rank and datasets such as are currently
produced in multiple ions experiments. We test the performance of AIC and BIC
on randomly chosen low rank states of 4 ions, and study the dependence of the
selected rank with the number of measurement repetitions for one ion states. We
then apply the methods to real data from a 4 ions experiment aimed at creating
a Smolin state of rank 4. The two methods indicate that the optimal model for
describing the data lies between ranks 6 and 9, and the Pearson test
is applied to validate this conclusion. Additionally we find that the mean
square error of the maximum likelihood estimator for pure states is close to
that of the optimal over all possible measurements.Comment: 24 pages, 6 figures, 3 table
Permutationally invariant state reconstruction
Feasible tomography schemes for large particle numbers must possess, besides
an appropriate data acquisition protocol, also an efficient way to reconstruct
the density operator from the observed finite data set. Since state
reconstruction typically requires the solution of a non-linear large-scale
optimization problem, this is a major challenge in the design of scalable
tomography schemes. Here we present an efficient state reconstruction scheme
for permutationally invariant quantum state tomography. It works for all common
state-of-the-art reconstruction principles, including, in particular, maximum
likelihood and least squares methods, which are the preferred choices in
today's experiments. This high efficiency is achieved by greatly reducing the
dimensionality of the problem employing a particular representation of
permutationally invariant states known from spin coupling combined with convex
optimization, which has clear advantages regarding speed, control and accuracy
in comparison to commonly employed numerical routines. First prototype
implementations easily allow reconstruction of a state of 20 qubits in a few
minutes on a standard computer.Comment: 25 pages, 4 figues, 2 table