We demonstrate that incomplete quantum tomography can give conclusive information in experimental realizations. We divide the state space into a union of multiple disjoint subsets and determine conclusively to which of the subsets a system, prepared in completely unknown state, belongs. In other words, we construct and solve membership problems. Our membership problems are partitions of the state space into a union of four disjoint sets formed by fixing two maximally entangled reference states and boundary values of a fidelity function ``radius'' between the reference states and the unknown preparation. We study the necessary and sufficient conditions of the measurements that solve these membership problems conclusively. We construct and experimentally implement such informationally incomplete measurement on two-photon polarization states with combined one-qubit measurements, and we solve the membership problem in example cases.
We demonstrate that incomplete quantum tomography can give conclusive information in experimental realizations. We divide the state space into a union of multiple disjoint subsets and determine conclusively to which of the subsets a system, prepared in completely unknown state, belongs. In other words, we construct and solve membership problems. Our membership problems are partitions of the state space into a union of four disjoint sets formed by fixing two maximally entangled reference states and boundary values of a fidelity function ``radius'' between the reference states and the unknown preparation. We study the necessary and sufficient conditions of the measurements that solve these membership problems conclusively. We construct and experimentally implement such informationally incomplete measurement on two-photon polarization states with combined one-qubit measurements, and we solve the membership problem in example cases.
State of a d -dimensional quantum system can only be inferred by performing an informationally complete measurement with m⩾d2 outcomes. However, an experimentally accessible measurement can be informationally incomplete. Here we show that a single informationally incomplete measuring apparatus is still able to provide all the information about the quantum system if applied several times in a row. We derive a necessary and sufficient condition for such a measuring apparatus and give illustrative examples for qubits, qutrits, general d -level systems, and composite systems of n qubits, where such a measuring apparatus exists. We show that projective measurements and Lüders measurements with 2 outcomes are useless in the considered scenario.
Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography. However, quantum tomography suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high-fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. For this situation we design a set of experiments and an optimization algorithm that alternates between maximizing the likelihood with respect to the states and measurements to produce estimates of each. In some cases, the procedure does not enable unique estimation of the states. For these cases, we show how one may identify a set of density matrices compatible with the measurements and use a semi-definite program to place bounds on the state's expectation values. We demonstrate the procedure on data from a simulated experiment with two trapped ions.
We introduce informationally complete measurements whose outcomes are entanglement witnesses and so answer the question of how many witnesses need to be measured to decide whether an arbitrary state is entangled or not: as many as the dimension of the state space. The witnesses can be measured successively; if all of them give an inconclusive result, one exploits their tomographic completeness for a reconstruction of the quantum state and can then determine its entanglement properties by data processing. There are witnesses that are optimal for this purpose. The optimized witness-based measurement can provide exponential improvement with respect to witness efficiency in high-dimensional Hilbert spaces, at the price of a reduction in the tomographic efficiency. We describe a systematic construction and illustrate the matter with the example of two qubits. For the case of two polarization qubits of photons, we show how existing technology can be used to implement the optimized witnesses in a very efficient way. Owing to the details of the implementation, which actually measures the eigenstate basis of the witness rather than solely determining the expectation value of the witness, one does not need to measure more than six witnesses in this example of a 16-dimensional state space.
Determining the state of a quantum system is a consuming procedure. For this reason, whenever one is interested only in some particular property of a state, it would be desirable to design a measurement set-up that reveals this property with as little effort as possible. Here, we investigate whether, in order to successfully complete a given task of this kind, one needs an informationally complete measurement, or if something less demanding would suffice. The first alternative means that in order to complete the task, one needs a measurement which fully determines the state. We formulate the task as a membership problem related to a partitioning of the quantum state space and, in doing so, connect it to the geometry of the state space. For a general membership problem, we prove various sufficient criteria that force informational completeness, and we explicitly treat several physically relevant examples. For the specific cases that do not require informational completeness, we also determine bounds on the minimal number of measurement outcomes needed to ensure success in the task.
Many experiments in quantum information aim at creating multi-partite entangled states. Quantifying the amount of entanglement that was actually generated can, in principle, be accomplished using full-state tomography. This method requires the determination of a parameter set that is growing exponentially with the number of qubits and becomes infeasible even for moderate numbers of particles. Non-trivial bounds on experimentally prepared entanglement can however be obtained from partial information on the density matrix. As introduced in the paper of K.M.R. Audenaert and M.B. Plenio [New J. Phys. 2006, 8, 266], the fundamental question is then formulated as: What is the entanglement content of the least entangled quantum state that is compatible with the available measurement data? We formulate the problem mathematically [Audenaert, K.M.R.; Plenio, M.B. New J. Phys. 2006, 8, 266] employing methods from the theory of semi-definite programming and then address this problem for the case, where the goal of the experiment is the creation of graph states. The observables that we consider are the generators of the stabilizer group, thus the number of measurement settings grows only linearly in the number of qubits. We provide analytical solutions as well as numerical methods that may be applied directly to experiments, and compare the obtained bounds with results from full-state tomography for simulated data.
Determining the state of a quantum system is a consuming procedure. For this reason, whenever one is interested only in some particular property of a state, it would be desirable to design a measurement set-up that reveals this property with as little effort as possible. Here, we investigate whether, in order to successfully complete a given task of this kind, one needs an informationally complete measurement, or if something less demanding would suffice. The first alternative means that in order to complete the task, one needs a measurement which fully determines the state. We formulate the task as a membership problem related to a partitioning of the quantum state space and, in doing so, connect it to the geometry of the state space. For a general membership problem, we prove various sufficient criteria that force informational completeness, and we explicitly treat several physically relevant examples. For the specific cases that do not require informational completeness, we also determine bounds on the minimal number of measurement outcomes needed to ensure success in the task.