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arXiv:1702.00118 · Physical Review A

Nonholonomic tomography. I. The Born rule as a connection between experiments

Christopher S. Jackson, S. J. van Enk

Published May 12, 2017• 8 citations
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Abstract

In the context of quantum tomography, we recently introduced a quantity called a partial determinant [C. Jackson and S. J. van Enk, Phys. Rev. A 92, 042312 (2015)]. Partial determinants (PDs) are explicit functions of the collected data that are sensitive to the presence of state preparation and measurement correlated errors. As such, PDs bypass the need to estimate state preparation or measurement parameters individually. In the present work, we suggest a theoretical perspective for the PD. We show that the PD is a holonomy and that the notions of state, measurement, and tomography can be generalized to nonholonomic constraints. To illustrate and clarify these abstract concepts, direct analogies are made to parallel transport, thermodynamics, and gauge field theory. This paper is the first of a two-part series where the following paper is about scalable generalizations of the PD in multiqudit systems, with possible applications for debugging a quantum computer.

DOI
10.1103/physreva.95.052327

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Nonholonomic tomography. II. Detecting correlations in multiqudit systems

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In the context of quantum tomography, quantities called partial determinants [C. Jackson and S. J. van Enk, Phys. Rev. A 92, 042312 (2015)] were recently introduced. Partial determinants (PDs) are explicit functions of the collected data that are sensitive to the presence of state preparation and measurement (SPAM) correlations. In this paper we demonstrate further applications of the PD and its generalizations. In particular, we construct methods for detecting various types of SPAM correlation in multiqudit systems, e.g., measurement-measurement correlations. The relationship between the PDs of each method and the correlations they are sensitive to is topological. We give a complete classification scheme for all such methods but focus on the explicit details of only the most scalable methods, for which the number of settings scales as $O({d}^{4})$. This paper is the second of a two-part series where the preceding paper [C. Jackson and S. van Enk, preceding paper, Phys. Rev. A 95, 052327 (2017)] is about a theoretical perspective for the PD, particularly its interpretation as a holonomy.

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In the context of quantum tomography, we recently introduced a quantity called a partial determinant \cite{jackson2015detecting}. PDs (partial determinants) are explicit functions of the collected data which are sensitive to the presence of state-preparation-and-measurement (SPAM) correlations. Importantly, this is done without any need to estimate state-preparation or measurement parameters. In the present work, we wish to better explain our theoretical perspective behind the PD. Further, we would like to demonstrate that there is an overwhelming variety of applications and generalizations of the PD. In particular we will construct methods for detecting SPAM correlations in multiqudit systems. The relationship between the PDs of each method and the correlations they are sensitive to is topological. We give a classification of all such methods but focus on explicitly detailing only the most scalable methods, $\mathcal{O}(d^4)$.

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The following dissertation investigates a problem related to the practice of quantum tomography, where one usually estimates the parameters associated with quantum states or measurements. \n \nIn particular, the question answered is whether and how one could detect if states and measurements are correlated. \n \nA similar question answered is how one could detect state-preparation non-localities and measurement non-localities in multiqudit systems. \n \nThe solution involves an analysis of certain matrix quantities called \\emph{partial determinants}. \n \nPartial determinants are an application of the Born rule that can be interpreted as tomography over a loop in the space of state and measurement settings. \n \nFrom this perspective, the notion of state and observable become \\emph{non-holonomic} \n \n| that is, state and observable parameters can be defined ``locally'' over each setting but not globally over all settings. \n \nAs such, state and measurement parameters are not estimated because such estimated values don't exist in correlated systems, but rather the inability to estimate such values is quantified. \n \nTherefore, partial determinants are a measure of the amount of contradiction that would result from any claim of such a estimated values by propagating these estimates through a `tomography loop' of data collected by various experiments. \n \nSuch measures of contradiction are generally known as \\emph{holonomies}.

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Whereas in standard quantum-state tomography one estimates an unknown state by performing various measurements with known devices, and whereas in detector tomography one estimates the positive-operator-valued-measurement elements of a measurement device by subjecting to it various known states, we consider here the case of SPAM (state preparation and measurement) tomography where neither the states nor the measurement device are assumed known. For $d$-dimensional systems measured by $d$-outcome detectors, we find there are at most ${d}^{2}({d}^{2}\ensuremath{-}1)$ ``gauge'' parameters that can never be determined by any such experiment, irrespective of the number of unknown states and unknown devices. For the case $d=2$ we find gauge-invariant quantities that can be accessed directly experimentally and that can be used to detect and describe SPAM errors. In particular, we identify conditions whose violations detect the presence of correlations between SPAM errors. From the perspective of SPAM tomography, standard quantum-state tomography and detector tomography are protocols that fix the gauge parameters through the assumption that some set of fiducial measurements is known or that some set of fiducial states is known, respectively.

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

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We have performed an experiment demonstrating that loop state-preparation-and-measurement tomography [Jackson and 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 we 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 positive-operator-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.

arXiv:1701.06498•Physical Review A•Apr 20, 2017•10 citations
In the context of quantum tomography, we recently introduced a quantity called a partial determinant [C. Jackson and S. J. van Enk, Phys. Rev. A 92, 042312 (2015)]. Partial determinants (PDs) are explicit functions of the collected data that are sensitive to the presence of state preparation and measurement correlated errors. As such, PDs bypass the need to estimate state preparation or measurement parameters individually. In the present work, we suggest a theoretical perspective for the PD. We show that the PD is a holonomy and that the notions of state, measurement, and tomography can be generalized to nonholonomic constraints. To illustrate and clarify these abstract concepts, direct analogies are made to parallel transport, thermodynamics, and gauge field theory. This paper is the first of a two-part series where the following paper is about scalable generalizations of the PD in multiqudit systems, with possible applications for debugging a quantum computer.
In the context of quantum tomography, quantities called partial determinants [C. Jackson and S. J. van Enk, Phys. Rev. A 92, 042312 (2015)] were recently introduced. Partial determinants (PDs) are explicit functions of the collected data that are sensitive to the presence of state preparation and measurement (SPAM) correlations. In this paper we demonstrate further applications of the PD and its generalizations. In particular, we construct methods for detecting various types of SPAM correlation in multiqudit systems, e.g., measurement-measurement correlations. The relationship between the PDs of each method and the correlations they are sensitive to is topological. We give a complete classification scheme for all such methods but focus on the explicit details of only the most scalable methods, for which the number of settings scales as O(d4)O({d}^{4})O(d4). This paper is the second of a two-part series where the preceding paper [C. Jackson and S. van Enk, preceding paper, Phys. Rev. A 95, 052327 (2017)] is about a theoretical perspective for the PD, particularly its interpretation as a holonomy.
In the context of quantum tomography, we recently introduced a quantity called a partial determinant \cite{jackson2015detecting}. PDs (partial determinants) are explicit functions of the collected data which are sensitive to the presence of state-preparation-and-measurement (SPAM) correlations. Importantly, this is done without any need to estimate state-preparation or measurement parameters. In the present work, we wish to better explain our theoretical perspective behind the PD. Further, we would like to demonstrate that there is an overwhelming variety of applications and generalizations of the PD. In particular we will construct methods for detecting SPAM correlations in multiqudit systems. The relationship between the PDs of each method and the correlations they are sensitive to is topological. We give a classification of all such methods but focus on explicitly detailing only the most scalable methods, O(d4)\mathcal{O}(d^4)O(d4).
The following dissertation investigates a problem related to the practice of quantum tomography, where one usually estimates the parameters associated with quantum states or measurements. \n \nIn particular, the question answered is whether and how one could detect if states and measurements are correlated. \n \nA similar question answered is how one could detect state-preparation non-localities and measurement non-localities in multiqudit systems. \n \nThe solution involves an analysis of certain matrix quantities called \\emph{partial determinants}. \n \nPartial determinants are an application of the Born rule that can be interpreted as tomography over a loop in the space of state and measurement settings. \n \nFrom this perspective, the notion of state and observable become \\emph{non-holonomic} \n \n| that is, state and observable parameters can be defined ``locally'' over each setting but not globally over all settings. \n \nAs such, state and measurement parameters are not estimated because such estimated values don't exist in correlated systems, but rather the inability to estimate such values is quantified. \n \nTherefore, partial determinants are a measure of the amount of contradiction that would result from any claim of such a estimated values by propagating these estimates through a `tomography loop' of data collected by various experiments. \n \nSuch measures of contradiction are generally known as \\emph{holonomies}.
Quantum holonomies are investigated in different contexts. A geometric phase is proposed for decomposition dependent evolution, where each component of a given decomposition of a mixed state evolves independently. It is shown that this geometric phase only depends on the path traversed in the space of decompositions. A holonomy is associated to general paths of subspaces of a Hilbert space, both discrete and continuous. This opens up the possibility of constructing quantum holonomic gates in the open path setting. In the discrete case it is shown that it is possible to associate two distinct holonomies to a given path. Interferometric setups for measuring both holonomies are provided. It is further shown that there are cases when the holonomy is only partially defined. This has no counterpart in the Abelian setting. An operational interpretation of amplitudes of density operators is provided. This allows for a direct interferometric realization of Uhlmann's parallelity condition, and the possibility of measuring the Uhlmann holonomy for sequences of density operators. Off-diagonal geometric phases are generalized to the non-Abelian case. These off-diagonal holonomies are undefined for cyclic evolution, but must contain members of non-zero rank if all standard holonomies are undefined. Experimental setups for measuring the off-diagonal holonomies are proposed. The concept of nodal free geometric phases is introduced. These are constructed from gauge invariant quantities, but do not share the nodal point structure of geometric phases and off-diagonal geometric phases. An interferometric setup for measuring nodal free geometric phases is provided, and it is shown that these phases could be useful in geometric quantum computation. A holonomy associated to a sequence of quantum maps is introduced. It is shown that this holonomy is related to the Uhlmann holonomy. Explicit examples are provided to illustrate the general idea.
Whereas in standard quantum-state tomography one estimates an unknown state by performing various measurements with known devices, and whereas in detector tomography one estimates the positive-operator-valued-measurement elements of a measurement device by subjecting to it various known states, we consider here the case of SPAM (state preparation and measurement) tomography where neither the states nor the measurement device are assumed known. For ddd-dimensional systems measured by ddd-outcome detectors, we find there are at most ${d}^{2}({d}^{2}\ensuremath{-}1)$ ``gauge'' parameters that can never be determined by any such experiment, irrespective of the number of unknown states and unknown devices. For the case d=2d=2d=2 we find gauge-invariant quantities that can be accessed directly experimentally and that can be used to detect and describe SPAM errors. In particular, we identify conditions whose violations detect the presence of correlations between SPAM errors. From the perspective of SPAM tomography, standard quantum-state tomography and detector tomography are protocols that fix the gauge parameters through the assumption that some set of fiducial measurements is known or that some set of fiducial states is known, respectively.
We have performed an experiment demonstrating that loop state-preparation-and-measurement tomography [Jackson and 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 we 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 positive-operator-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.