As increasingly impressive quantum information processors are realized in laboratories around the world, robust and reliable characterization of these devices is now more urgent than ever. These diagnostics can take many forms, but one of the most popular categories is tomography, where an underlying parameterized model is proposed for a device and inferred by experiments. Here, we introduce and implement efficient operational tomography, which uses experimental observables as these model parameters. This addresses a problem of ambiguity in representation that arises in current tomographic approaches (the gauge problem). Solving the gauge problem enables us to efficiently implement operational tomography in a Bayesian framework computationally, and hence gives us a natural way to include prior information and discuss uncertainty in fit parameters. We demonstrate this new tomography in a variety of different experimentally-relevant scenarios, including standard process tomography, Ramsey interferometry, randomized benchmarking, and gate set tomography.
A novel operational method for estimating the efficiency of quantum state tomography protocols is suggested. It is based on a-priori estimation of the quality of an arbitrary protocol by means of universal asymptotic fidelity distribution and condition number, which takes minimal value for better protocol. We prove the adequacy of the method both with numerical modeling and through the experimental realization of several practically important protocols of quantum state tomography.
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 POVM 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 $d^2(d^2-1)$ "gauge" parameters that can never be determined by any such experiment, irrespecti…
A central aim of physics is to describe the dynamics of physical systems. Schrodinger's equation does this for isolated quantum systems. Describing the time evolution of a quantum system that interacts with its environment, in its most general form, has proved to be difficult because the dynamics is dependent on the state of the environment and the correlations with it. For discrete processes, such as quantum gates or chemical reactions, quantum process tomography provides the complete description of the dynamics, provided that the initial states of the system and the environment are independe…
The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. In this paper, we motivate and review two results that generalize de Finetti's theorem to the quantum mechanical setting: Namely a de Finetti theorem for quantum states and a de Finetti theorem for quantum operations. The quantum-state theorem, in a closely analogous fashion to the original de Finetti theorem, deals with exchangeable density-operator assignments a…
We demonstrate quantum detector tomography of a commercial 2x2 array of superconducting nanowire single photon detectors. We show that detector-specific figures of merit including efficiency, dark-count and cross-talk probabilities can be directly extracted, without recourse to the underlying detector physics. These figures of merit are directly identified from just four elements of the reconstructed positive operator valued measure (POVM) of the device. We show that the values for efficiency and dark-count probability extracted by detector tomography show excellent agreement with independent …
Entanglement without Quantum Mechanics: Operational Constraints on the Quantum Signature
University of Vienna（维也纳大学）；Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；University of Geneva（日内瓦大学）；Constructor University（Constructor 大学）
Temporal Kirkwood-Dirac Quasiprobability Distribution and Unification of Temporal State Formalisms through Temporal Bloch Tomography
Institute of Quantum Physics, School of Physics, Central South University（中南大学物理学院量子物理研究所）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Physics, National University of Singapore（新加坡国立大学物理系）；Science, Mathematics and Technology Cluster, Singapore University of Technology and Design（新加坡科技设计大学科学、数学与技术集群）
We consider the implementation of a symmetric informationally complete probability-operator measurement (SIC POM) in the Hilbert space of a d-level system by a two-step measurement process: a diagonal-operator measurement with high-rank outcomes, followed by a rank-1 measurement in a basis chosen in accordance with the result of the first measurement. We find that any Heisenberg-Weyl group-covariant SIC POM can be realized by such a sequence where the second measurement is simply a measurement in the Fourier basis, independent of the result of the first measurement. Furthermore, at least for t…
Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.
Quantum memory -- the capacity to store and faithfully recover unknown quantum states -- is essential for quantum-enhanced technology. There is thus a pressing need for operationally meaningful means to benchmark candidate memories across diverse physical platforms. Here we introduce a universal benchmark distinguished by its relevance across multiple key operational settings, exactly quantifying (1) the memory's robustness to noise, (2) the number of noiseless qubits needed for its synthesis, (3) its potential to speed up statistical sampling tasks, and (4) performance advantage in non-local …
Standard Bayesian credible-region theory for constructing an error region on the unique estimator of an unknown state in general quantum-state tomography to calculate its size and credibility relies on heavy Monte~Carlo sampling of the state space followed by sample rejection. This conventional method typically gives negligible yield for very small error regions originating from large datasets. We propose an operational reformulated theory to compute both size and credibility from region-average quantities that in principle convey information about behavior of these two properties as the credi…
Current techniques in quantum process tomography typically return a single point estimate of an unknown process based on a finite albeit large amount of measurement data. Due to statistical fluctuations, however, other processes close to the point estimate can also produce the observed data with near certainty. Unless appropriate error bars can be constructed, the point estimate does not carry any sound operational interpretation. Here, we provide a solution to this problem by constructing a confidence region estimator for quantum processes. Our method enables reliable estimation of essentiall…
Does chaos in the dynamics enable information gain in quantum tomography or impede it? We address this question by considering continuous measurement tomography in which the measurement record is obtained as a sequence of expectation values of a Hermitian observable evolving under the repeated application of the Floquet map of the quantum kicked top. For a given dynamics and Hermitian observables, we observe completely opposite behavior in the tomography of well-localized spin coherent states compared to random states. As the chaos in the dynamics increases, the reconstruction fidelity of spin…
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-measurment (SPAM) 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 non-holonomic co…
Quantum features of correlated optical modes define a major aspect of the nonclassicality in quantized radiation fields. However, the phase-sensitive detection of a two-mode light field is restricted to interferometric setups and local intensity measurements. Even the full reconstruction of the quantum state of a single radiation mode relies on such detection layouts and the preparation of a well-defined reference light field. In this work, we establish the notion of the essential quantum correlations of two-mode light fields. It refers to those quantum correlations which are measurable by a g…
We present a comprehensive theory and an easy to follow method for the design and construction of a wideband homodyne detector for time-domain quantum measurements. We show how one can evaluate the performance of a detector in a specific time-domain experiment based on electronic spectral characteristic of that detector. We then present and characterize a high-performance detector constructed using inexpensive, commercially available components such as low-noise high-speed operational amplifiers and high-bandwidth photodiodes. Our detector shows linear behavior up to a level of over 13 dB clea…
Quantum state tomography is the task of inferring the state of a quantum system by appropriate measurements. Since the frequency distributions of the outcomes of any finite number of measurements will generally deviate from their asymptotic limits, the estimates computed by standard methods do not in general coincide with the true state, and therefore have no operational significance unless their accuracy is defined in terms of error bounds. Here we show that quantum state tomography, together with an appropriate data analysis procedure, yields reliable and tight error bounds, specified in ter…
Long-term quantum coherence constitutes one of the main challenges when engineering quantum devices. However, easily accessible means to quantify complex decoherence mechanisms are not readily available, nor are sufficiently stable systems. We harness novel phase-space methods - expressed through non-Gaussian convolutions of highly singular Glauber-Sudarshan quasiprobabilities - to dynamically monitor quantum coherence in polariton condensates with significantly enhanced coherence times. Via intensity- and time-resolved reconstructions of such phase-space functions from homodyne detection data…
We investigate quantum state reconstruction of a superconducting qubit threaded by an Aharonov-Bohm flux, with particular attention to the local geometric phase. A state reconstruction scheme is introduced with a proper account of the local geometric phase generated by Faraday's law of induction. Our scheme is based on measurement of three complementary quantities, that is, the extra charge and two local currents. Incorporating time-reversal symmetry and the Faraday's law, we show that the full density matrix can be reconstructed without ambiguity in the choice of gauge. This procedure clearly…
We introduce a hierarchical classification of theories that describe systems with fundamentally limited information content. This property is introduced in an operational way and gives rise to the existence of mutually complementary measurements, i.e. a complete knowledge of future outcome in one measurement is at the expense of complete uncertainty in the others. This is characteristic feature of the theories and they can be ordered according to the number of mutually complementary measurements which is also shown to define their computational abilities. In the theories multipartite states ma…
This paper charts a very direct path between the categorical approach to quantum mechanics, due to Abramsky and Coecke, and the older convex-operational approach based on ordered vector spaces (recently reincarnated as "generalized probabilistic theories"). In the former, the objects of a symmetric monoidal category C are understood to represent physical systems and morphisms, physical processes. Elements of the monoid C(I,I) are interpreted somewhat metaphorically as probabilities. Any monoid homomorphism from the scalars of a symmetric monoidal category C gives rise to a covariant functor V_…
Gauge freedom in quantum electrodynamics (QED) outside of textbook regimes is reviewed. It is emphasized that QED subsystems are defined relative to a choice of gauge. Each definition uses different gauge-invariant observables. This relativity is eliminated only if a sufficient number of Markovian and weak-coupling approximations are employed. All physical predictions are gauge invariant, including subsystem properties such as photon number and entanglement. However, subsystem properties naturally differ for different physical subsystems. Gauge ambiguities arise not because it is unclear how t…
Quantum process tomography, the standard procedure to characterize any quantum channel in nature, is affected by a circular argument: in order to characterize the channel, the tomographic preparation and measurement need in turn to be already characterized. We break this loop by designing an operational framework able to optimally characterize any given unknown quantum channel in a device-independent fashion, namely, by only looking at its input-output statistics, under the sole assumption that quantum theory is valid. We provide explicit solutions, in closed form, for practically relevant cas…
Spin-dependent electrical injection has found useful applications in storage devices, but fully operational spin-dependent semiconductor electronics remain a challenging task because of weak spin-orbit couplings and/or strong spin relaxations. These limitations are lifted considering atoms instead of electrons or holes as spin carriers. In this emerging field of atomtronics, we demonstrate the equivalent of a Datta-Das transistor using a degenerate Fermi gas of strontium atoms as spin carriers in interaction with a tripod laser-beams scheme. We explore the dependence of spin rotation, and we i…
The time-symmetric formalism endows the weak measurement and its outcome, the weak value,many unique features. In particular, it allows a direct tomography of quantum states without resort to complicated reconstruction algorithms and provides an operational meaning to wave functions and density matrices. Here, we propose and experimentally demonstrate the direct tomography of a measurement apparatus by taking the backward direction of weak measurement formalism. Our protocol works rigorously with the arbitrary measurement strength, which offers an improved accuracy and precision. The precision…
We discuss the effects of a gauge freedom in representing quantum information processing devices, and its implications for characterizing these devices. We demonstrate with experimentally relevant examples that there exists equally valid descriptions of the same experiment which distribute errors differently among objects in a gate-set, leading to different error rates. Consequently, it can be misleading to attach a concrete operational meaning to figures of merit for individual gate-set elements. We propose an alternative operational figure of merit for a gate-set, the mean variation error, a…
Entanglement is assuming a central role in modern quantum many-body physics. Yet, for lattice gauge theories its certification remains extremely challenging. A key difficulty stems from the local gauge constraints underlying the gauge theory, which separate the full Hilbert space into a direct sum of subspaces characterized by different superselection rules. In this work, we develop the theoretical framework of entanglement witnessing for lattice gauge theories that takes this subtlety into account. We illustrate the concept at the example of a $\mathrm{U}(1)$ lattice gauge theory in 2+1 dimen…
In the Aharonov-Bohm (AB) effect, a superposed charge acquires a detectable phase by enclosing an infinite solenoid, in a region where the solenoid's electric and magnetic fields are zero. Its generation seems therefore explainable only by the local action of gauge-dependent potentials, not of gauge-independent fields. This was recently challenged by Vaidman, who explained the phase by the solenoid's current interacting with the electron's field (at the solenoid). Still, his model has a residual non-locality: it does not explain how the phase, generated at the solenoid, is detectable on the ch…
Local Operations and Field Mediated Entanglement without a Local Tensor Product Structure
University of Vienna（维也纳大学）；Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences（奥地利科学院量子光学和量子信息研究所）；ETH Zurich（苏黎世联邦理工学院）；Università di Roma Tor Vergata（罗马第二大学）；Sezione INFN Roma2（INFN罗马第二分部）
In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of "physical experiment" and assuming "experimental accessibility and simplicity" as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper quant-ph/0506034. Pivotal roles are played by the "local observability principle", which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of "informationally complete observables" and of a …
Classical shadows for sample-efficient measurements of gauge-invariant observables
United States Naval Academy（美国海军学院）；Harvard University（哈佛大学）；University of Washington（华盛顿大学）；Paul Scherrer Institute（保罗谢尔研究所）；ETH Zürich（苏黎世联邦理工学院）；University of New Mexico（新墨西哥大学）
