Extracting tomographic information about quantum states is a crucial task in the quest towards devising high-precision quantum devices. Current schemes typically require measurement devices for tomography that are a priori calibrated to high precision. Ironically, the accuracy of the measurement calibration is fundamentally limited by the accuracy of state preparation, establishing a vicious cycle. Here, we prove that this cycle can be broken and the dependence on the measurement device's calibration significantly relaxed. We show that exploiting the natural low-rank structure of quantum states of interest suffices to arrive at a highly scalable `blind' tomography scheme with a classically efficient post-processing algorithm. We further improve the efficiency of our scheme by making use of the sparse structure of the calibrations. This is achieved by relaxing the blind quantum tomography problem to the de-mixing of a sparse sum of low-rank matrices. We prove that the proposed algorithm recovers a low-rank quantum state and the calibration provided that the measurement model exhibits a restricted isometry property. For generic measurements, we show that it requires a close-to-optimal number of measurement settings. Complementing these conceptual and mathematical insights, we numerically demonstrate that robust blind quantum tomography is possible in a practical setting inspired by an implementation of trapped ions.
We propose a numerical method for compensating dispersion effects in optical coherence tomography that does not require a priori knowledge of dispersive properties of the sample. The method is based on the generalized autoconvolution function, and its principle of operation can be intuitively visualized using the Wigner distribution function formalism.
Quantum entangled states have shown distinguished features beyond any classical state. Many methods like quantum state tomography have been presented to verify entanglement. In this work, we aim to identify unknown entanglements with partial information of the state space by developing a nonlinear entanglement witness. The witness consists of a generalized Greenberger-Horne-Zeilinger-like paradox expressed by Pauli observables, and a nonlinear inequality expressed by density matrix elements. First, we verify unknown bipartite entanglements and study the robustness of entanglement witnesses aga…
Using the the convex semidefinite programming method and superoperator formalism we obtain the finite quantum tomography of some mixed quantum states such as: qudit tomography, N-qubit tomography, phase tomography and coherent spin state tomography, where that obtained results are in agreement with those of References \cite{schack,Pegg,Barnett,Buzek,Weigert}.
Matrix permanents are hard to compute or even estimate in general. It had been previously suggested that the permanents of Positive Semidefinite (PSD) matrices may have efficient approximations. By relating PSD permanents to a task in quantum state tomography, we show that PSD permanents are NP-hard to approximate within a constant factor, and so admit no FPTAS (unless P=NP). We also establish that several natural tasks in quantum state tomography, even approximately, are NP-hard in the dimension of the Hilbert space. These state tomography tasks therefore remain hard even with only logarithmi…
We analyze quantum state tomography in scenarios where measurements and states are both constrained. States are assumed to live in a semi-algebraic subset of state space and measurements are supposed to be rank-one POVMs, possibly with additional constraints. Specifically, we consider sets of von Neumann measurements and sets of local observables. We provide upper bounds on the minimal number of measurement settings or outcomes that are required for discriminating all states within the given set. The bounds exploit tools from real algebraic geometry and lead to generic results that do not only…
We propose an approach to reconstruct two-electron spin qubit states in semiconductor quantum dots by employing tomographic techniques. This procedure exploits the combination of fast gate operations on electron spins trapped in dots and dynamical nuclear polarization of the underlying Ga and As nuclei. The presented method can be an important tool in solid state quantum computation for complete characterization of qubit states and theirs correlations.
In this paper, we discuss the minimal number of observables, where expectation values at some time instant determine the trajectory of a d-level quantum system ("qudit") governed by the Gaussian semigroup. We assume that the macroscopic information about the system in question is given by the mean values of n selfadjoint operators $Q_1,...,Q_n$ at some time instants $t_1<t_2<...<t_r$, where $n<d^2-1$ and $r\leq {\rm deg} μ(λ,\bBBL)$. Here $μ(λ,\bBBL)$ stands for the minimal polynomial of the generator of the Gaussian flow.
Characterizing complex quantum systems is a vital task in quantum information science. Quantum tomography, the standard tool used for this purpose, uses a well-designed measurement record to reconstruct quantum states and processes. It is, however, notoriously inefficient. Recently, the classical signal reconstruction technique known as "compressed sensing" has been ported to quantum information science to overcome this challenge: accurate tomography can be achieved with substantially fewer measurement settings, thereby greatly enhancing the efficiency of quantum tomography. Here we show that …
We introduce a new method to reconstruct unknown quantum states out of incomplete and noisy information. The method is a linear convex optimization problem, therefore with a unique minimum, which can be efficiently solved with Semidefinite Programs. Numerical simulations indicate that the estimated state does not overestimate purity, and neither the expectation value of optimal entanglement witnesses. The convergence properties of the method are similar to compressed sensing approaches, in the sense that, in order to reconstruct low rank states, it needs just a fraction of the effort correspon…
Recent advances in theoretical and experimental quantum computing bring us closer to scalable quantum computing devices. This makes the need for protocols that verify the correct functionality of quantum operations timely and has led to the field of quantum verification. In this paper we address key challenges to make quantum verification protocols applicable to experimental implementations. We prove the robustness of the single server verifiable universal blind quantum computing protocol of Fitzsimons and Kashefi (2012) in the most general scenario. This includes the case where the purificati…
We propose an iterative algorithm that computes the maximum-likelihood estimate in quantum state tomography. The optimization error of the algorithm converges to zero at an $O ( ( 1 / k ) \log D )$ rate, where $k$ denotes the number of iterations and $D$ denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is $O ( D ^ 3 + N D ^2 )$, where $N$ denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the $R ρR$ method [A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homod…
Genuine multipartite entanglement underlies correlation experiments corroborating quantum mechanics and it is an expedient empowering many quantum technologies. One of many counterintuitive facets of genuine multipartite entanglement is its ability to exhibit an emergent character, that is, one can infer its presence in some multipartite states merely from a set of its separable marginals. Here, we show that the effect can be found also in the context of Gaussian states of bosonic systems. Specifically, we construct examples of multimode Gaussian states carrying genuine multipartite entangleme…
We discuss methods of quantum state tomography for solid-state systems with a large nuclear spin $I=3/2$ in nanometer-scale semiconductors devices based on a quantum well. Due to quadrupolar interactions, the Zeeman levels of these nuclear-spin devices become nonequidistant, forming a controllable four-level quantum system (known as quartit or ququart). The occupation of these levels can be selectively and coherently manipulated by multiphoton transitions using the techniques of nuclear magnetic resonance (NMR) [Yusa et al., Nature (London) 434, 101 (2005)]. These methods are based on an uncon…
The term "machine learning" especially refers to algorithms that derive mappings, i.e. intput/output transforms, by using numerical data that provide information about considered transforms. These transforms appear in many problems, related to classification/clustering, regression, system identification, system inversion and input signal restoration/separation. We here first analyze the connections between all these problems, in the classical and quantum frameworks. We then focus on their most challenging versions, involving quantum data and/or quantum processing means, and unsupervised, i.e. …
We propose and experimentally demonstrate a quantum state tomography protocol that generalizes the Wallentowitz-Vogel-Banaszek-Wódkiewicz point-by-point Wigner function reconstruction. The full density operator of an arbitrary quantum state is efficiently reconstructed in the Fock basis, using semidefinite programming, after interference with a small set of calibrated coherent states. This new protocol is resource- and computationally efficient, is robust against noise, does not rely on approximate state displacements, and ensures the physicality of results.
We introduce an efficient method to reconstruct the Wigner function of many-mode continuous variable systems. It is based on convex optimization with semidefinite programs, and also includes a version of the maximum entropy principle, in order to yield unbiased states. A key ingredient of the proposed approach is the representation of the state in a truncated Fock basis. As a bonus, the discrete finite representation allows to easily quantify the entanglement.
Whenever we do not have an informationally complete set of measurements, the estimate of a quantum state can not be uniquely determined. In this case, among the density matrices compatible with the available data, it is commonly preferred that one which is the most uncommitted with the missing information. This is the purpose of the Maximum Entropy estimation (MaxEnt) and the Variational Quantum Tomography (VQT). Here, we propose a variant of Variational Quantum Tomography and show its relationship with Maximum Entropy methods in quantum tomographies with incomplete set of measurements. We pro…
Spin-entaglement has been proposed and extensively used in the case of correlated triplet pairs which are intermediate states in singlet fission process in select organic semiconductors. Here, we employ quantum process tomography of polarization entangled photon-pairs resonant with the excited state absorption of these states to investigate the nature of the inherent quantum correlations and to explore for an unambiguous proof for the existence of exciton entanglement.
When basic tools of quantum information are applied to the quantum tomography data presented in Nature 439, 179 (2006), none of their devices appears to be a source of entangled photons.
We present a novel method for quantum tomography of multi-qubit states. We apply the method to spin-multi-photon states, which we produce by periodic excitation of a semiconductor quantum-dot- confined spin every 1/4 of its coherent precession period. These timed excitations lead to the deterministic generation of strings of entangled photons in a cluster state. We show that our method can be used for characterizing the periodic process map, which produces the photonic cluster. From the measured process map, we quantify the robustness of the entanglement in the cluster. The 3-fold enhanced gen…
In this work we have explored few tools in Quantum State Tomography for Continuous Variable Systems. The concept of quantum states in phase space representation is introduced in a simple manner by using a few statistical concepts. Unlike most texts of Quantum information in which the Wigner function for a single mode is often more used, in this text the multi-modes state Wigner function is also approached. Our numerical investigations indicate that the reconstructed method using back-projection add some error due the choice of cutoff frequency, therefore it is necessary to use data post-proces…
Quantum Process Tomography (QPT) methods aim at identifying, i.e. estimating, a given quantum process. QPT is a major quantum information processing tool, since it especially allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. However, usual QPT procedures are complicated, since they set several constraints on the quantum states used as inputs of the process to be characterized. In this paper, we extend QPT so as to avoid two such constraints. On the one hand, usual QPT methods requires one to know, hence to precisely control (i.…
There exists, in general, a convex set of quantum state estimators that maximize the likelihood for informationally incomplete data. We propose an estimation scheme, catered to measurement data of this kind, to search for the exact maximum-likelihood-maximum-entropy estimator using semidefinite programming and a standard multi-dimensional function optimization routine. This scheme can be used to infer the expectation values of a set of entanglement witnesses that can be used to verify the entanglement of the unknown quantum state for composite systems. Next, we establish an alternative numeric…
We consider the problem of quantum-state tomography under the assumption that the state is pure, and more generally that its rank is bounded by a given value. In this scenario, new notions of informationally complete POVMs emerge, which allow for high-fidelity state estimation with fewer measurement outcomes than are required for an arbitrary rank state. We study this in the context of matrix completion, where the POVM outcomes determine only a few of the density matrix elements. We give an analytic solution that fully characterizes informational completeness and elucidates the important role …
Semiconductor quantum dots are probably the preferred choice for interfacing anchored, matter spin qubits and flying photonic qubits. While full tomography of a flying qubit or light polarization is in general straightforward, matter spin tomography is a challenging and resource-consuming task. Here we present a novel all-optical method for conducting full tomography of quantum-dot-confined spins. Our method is applicable for electronic spin configurations such as the conduction-band electron, the valence-band hole, and for electron-hole pairs such as the bright and the dark exciton. We excite…
State and measurement tomography make assumptions about the experimental states or measurements. These assumptions are often not justified because state preparation and measurement errors are unavoidable in practice. Here we describe how the Gram matrix associated with the states and measurement operators can be estimated via semidefinite programming if the states and the measurements are so called globally completable. This is for instance the case if the unknown measurements are known to be projective and non-degenerate. The computed Gram matrix determines the states, and the measurement ope…
Continuous variable quantum state tomography is one of the most powerful techniques to study the properties of light fields in quantum optics. However, the need for a fixed phase reference has so far prevented widespread usage in other fields such as semiconductor spectroscopy. Here, we introduce non-stationary quantum state tomography, which adapts the technique to the special requirements of ultrafast spectroscopy. In detail, we gain access to the amplitude and phase of light fields with a temporal resolution of about 100\,fs without the need for a fixed phase reference. Further, we show how…
