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Reconstruction of quantum states based on measurements

Quantum tomography or quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. The source of these states may be any device or system which prepares quantum states either consistently into quantum pure states or otherwise into general mixed states. To be able to uniquely identify the state, the measurements must be tomographically complete. That is, the measured operators must form an operator basis on the Hilbert space of the system, providing all the information about the state. Such a set of observations is sometimes called a quorum. The term tomography was first used in the quantum physics literature in a 1993 paper introducing experimental optical homodyne tomography. In quantum process tomography on the other hand, known quantum states are used to probe a quantum process to find out how the process can be described. Similarly, quantum measurement tomography works to find out what measurement is being performed. Whereas, randomized benchmarking scalably obtains a figure of merit of the overlap between the error prone physical quantum process and its ideal counterpart. The general principle behind quantum state tomography is that by repeatedly performing many different measurements on quantum systems described by identical density matrices, frequency counts can be used to infer probabilities, and these probabilities are combined with Born's rule to determine a density matrix which fits the best with the observations. This can be easily understood by making a classical analogy. Consider a harmonic oscillator (e. g. a pendulum). The position and momentum of the oscillator at any given point can be measured and therefore the motion can be completely described by the phase space. This is shown in figure 1. By performing this measurement for a large number of identical oscillators we get a probability distribution in the phase space (figure 2). This distribution can be normalized (the oscillator at a given time has to be somewhere) and the distribution must be non negative. So we have retrieved a function which gives a description of the chance of finding the particle at a given point with a given momentum. For quantum mechanical particles the same can be done. The only difference is that the Heisenberg's uncertainty principle mustn't be violated, meaning that we cannot measure the particle's momentum and position at the same time. The particle's momentum and its position are called quadratures (see Optical phase space for more information) in quantum related states. By measuring one of the quadratures of a large number of identical quantum states will give us a probability density corresponding to that particular quadrature. This is called the marginal distribution, or (see figure 3). In the following text we will see that this probability density is needed to characterize the particle's quantum state, which is the whole point of quantum tomography.

Quantum tomography is applied on a source of systems, to determine the quantum state of the output of that source. Unlike a measurement on a single system, which determines the system's current state after the measurement (in general, the act of making a measurement alters the quantum state), quantum tomography works to determine the state(s) prior to the measurements. Quantum tomography can be used for characterizing optical signals, including measuring the signal gain and loss of optical devices, as well as in quantum computing and quantum information theory to reliably determine the actual states of the qubits. One can imagine a situation in which a person Bob prepares many identical objects (particles or fields) in the same quantum states and then gives them to Alice to measure. Not confident with Bob's description of the state, Alice may wish to do quantum tomography to classify the state herself.

One of the primary problems with using linear inversion to solve for the density matrix is that in general the computed solution will not be a valid density matrix. For example, it could give negative probabilities or probabilities greater than 1 to certain measurement outcomes. This is particularly an issue when fewer measurements are made. Another issue is that in infinite dimensional Hilbert spaces, an infinite number of measurement outcomes would be required. Making assumptions about the structure and using a finite measurement basis leads to artifacts in the phase space density. The methods are an active topic of research.

One can imagine a situation in which an apparatus performs some measurement on quantum systems, and determining what particular measurement is desired. The strategy is to send in systems of various known states, and use these states to estimate the outcomes of the unknown measurement. Also known as "quantum estimation", tomography techniques are increasingly important including those for quantum measurement tomography and the very similar quantum state tomography. Since a measurement can always be characterized by a set of POVM's, the goal is to reconstruct the characterizing POVM's The simplest approach is linear inversion. As in quantum state observation, use Exploiting linearity as above, this can be inverted to solve for the
Not surprisingly, this suffers from the same pitfalls as in quantum state tomography: namely, non physical results, in particular negative probabilities. Here the will not be valid POVM's, as they will not be positive. Bayesian methods as well as Maximum likelihood estimation of the density matrix can be used to restrict the operators to valid physical results.

Quantum process tomography (QPT) deals with identifying an unknown quantum dynamical process. The first approach, introduced in 1996 and sometimes known as standard quantum process tomography (SQPT) involves preparing an ensemble of quantum states and sending them through the process, then using quantum state tomography to identify the resultant states. Other techniques include ancilla assisted process tomography (AAPT) and entanglement assisted process tomography (EAPT) which require an extra copy of the system. Each of the techniques listed above are known as indirect methods for characterization of quantum dynamics, since they require the use of quantum state tomography to reconstruct the process. In contrast, there are direct methods such as direct characterization of quantum dynamics (DCQD) which provide a full characterization of quantum systems without any state tomography. The number of experimental configurations (state preparations and measurements) required for full quantum process tomography grows exponentially with the number of constituent particles of a system. Consequently, in general, QPT is an impossible task for large scale quantum systems. However, under weak decoherence assumption, a quantum dynamical map can find a sparse representation. The method of compressed quantum process tomography (CQPT) uses the compressed sensing technique and applies the sparsity assumption to reconstruct a quantum dynamical map from an incomplete set of measurements or test state preparations.