Experimental demonstration of loop state-preparation-and-measurement tomography

Abstract

We have performed an experiment demonstrating that loop state-preparation-and-measurement (SPAM) tomography [C. Jackson and S. J. van Enk, Phys. Rev. A 92, 042312 (2015)] is capable of detecting correlated errors between the preparation and the measurement of a quantum system. Specifically, we have prepared pure and mixed states of single qubits encoded in the polarization of heralded individual photons. By performing measurements using multiple state preparations and multiple measurement device settings we are able to detect if there are any correlated errors between them, and are also able to determine which state preparations are correlated with which measurements. This is accomplished by going around a "loop" in parameter space, which allows us to check for self-consistency. No assumptions are made concerning either the state preparations or the measurements, other than that the dimensions of the states and the positiveoperator-valued measures (POVMs) describing the detector are known. In cases where no correlations are found we are able to perform quantum state tomography of the polarization qubits by using knowledge of the detector POVMs, or quantum detector tomography by using knowledge of the state preparations. Note, however, that the detection of correlated errors does not require estimating any state or measurement parameters.Another form of tomography is quantum-detector tomography (QDT), which estimates the positive-operator-valued measure (POVM) that describes a detector [8][9][10][11][12]. Here the detector is illuminated with many different probe states, and the detector is characterized by measuring its response to these states. In QDT it is assumed that the states are known and well-characterized, but the detector POVM is initially unknown.In quantum-process tomography (QPT) the process that transforms an open quantum system from one state to another is fully characterized [13][14][15][16][17]. This is done by having known input states and performing QST on the output states.

Loop state-preparation-and-measurement tomography of a two-qubit system

“…We have shown that loop SPAM tomography is capable of detecting correlated errors between measurements performed on two spatially separated qubits. This, combined with the fact that previous experiments have demonstrated its ability to detected correlated errors between state preparations and measurements [4], further demonstrates the utility of loop SPAM tomography. Because it works in a device-independent manner, with a minimum of assumptions, loop SPAM tomography promises to be a useful and effective tool for detecting correlated errors in a wide variety of quantum information processing systems.…”

“…It is called the partial determinant because it is mathematically similar to the determinant, but it is not a scalar quantity-it is a matrix of smaller size than E . It can be shown that the measured data are internally consistent as described above, and free of correlated SPAM errors, if and only if ( ) 1 E D = , where 1 is the 3x3 identity matrix [1][2][3][4]. No knowledge of the state ρ or the detector operators is necessary to make this determination.…”

Self-consistent tomography and measurement-device independent cryptography

“…One can go one step further than self-consistent tomography and perform overcomplete sets of measurements on overcomplete sets of states that allow one to check the assumption that a given state-preparation procedure indeed produces a single specific state, and that a given measurement procedure indeed produces a single specific measurement. In particular, "holonomic" SPAM tomography or "loop" SPAM tomography denote a procedure to check for correlations between measurement and statepreparation [10,11]. For example, suppose one uses a laser to perform a measurement on a system whose state was prepared using that same laser just a microsecond ago.…”

Photodetector figures of merit in terms of POVMs

“…Furthermore, various different forms of quantum tomography have been developed recently for experimentally determining individual POVM elements P k . Detector tomography [3][4][5][6][7][8][9], self-consistent tomography [10], and SPAM tomography [11][12][13] use different sorts of assumptions to estimate from experimental data what POVM element corresponds to a given outcome of a quantum measurement.…”

“…we find the normalized a posteriori probability distribution over ω that outcome k implies as å Analogously, we can define an a posteriori probability distribution over detection times of the photon as i (t) defined in(13). These probability distributions are over continuous quantities.…”