Simple minimal but informationally complete positive operator-valued measures are constructed out of the expectation-value representation for qudits. Upon suitable modification, the procedure transforms any set of d^2 linearly independent hermitean operators into such an observable. Minor changes in the construction lead to closed-form expressions for informationally complete positive measures in d-dimensional Hilbert spaces.
Quantum measurements play a fundamental role in quantum mechanics and quantum information processing, but it is not easy to implement generalized measurements, the most powerful measurements allowed by quantum mechanics. Here we propose a simple recipe for implementing generalized measurements on a qudit via quantum walks. With this recipe, any discrete quantum measurement can be implemented via a one-dimensional discrete quantum walk; the number of steps is only two times the number of measurement outcomes. As an illustration, we present a unified solution for implementing arbitrary symmetric…
We study the Einstein-Podolsky-Rosen (EPR) steering and present steerability criteria for arbitrary qubit-qudit (qudit-qubit) systems based on mutually unbiased measurements (MUMs) and general symmetric informationally complete measurements (general SIC-POVMs). Avoiding the usual complicated steering inequalities, these criteria can be more operational than some existing criteria and implemented experimentally. Detailed examples are given to illustrate the efficiency of the criteria in both computation and experimental implementation.
We consider measurements, described by a positive-operator-valued measure (POVM), whose outcome probabilities determine an arbitrary pure state of a D-dimensional quantum system. We call such a measurement a pure-state informationally complete (PSI-complete) POVM. We show that a measurement with 2D-1 outcomes cannot be PSI-complete, and then we construct a POVM with 2D outcomes that suffices, thus showing that a minimal PSI-complete POVM has 2D outcomes. We also consider PSI-complete POVMs that have only rank-one POVM elements and construct an example with 3D-2 outcomes, which is a generalizat…
Informationally complete (IC) positive operator-valued measures (POVMs) are generalized quantum measurements that offer advantages over the standard computational basis readout of qubits. For instance, IC-POVMs enable efficient extraction of operator expectation values, a crucial step in many quantum algorithms. POVM measurements are typically implemented by coupling one additional ancilla qubit to each logical qubit, thus imposing high demands on the device size and connectivity. Here, we show how to implement a general class of IC-POVMs without ancilla qubits. We exploit the higher-dimension…
Generalized symmetric informationally complete (SIC) measurements are SIC measurements that are not necessarily rank one. They are interesting originally because of their connection with rank-one SICs. Here we reveal several merits of generalized SICs in connection with quantum state tomography and Lie algebra that are interesting in their own right. These properties uniquely characterize generalized SICs among minimal IC measurements although, on the face of it, they bear little resemblance to the original definition. In particular, we show that in quantum state tomography generalized SICs ar…
In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions, which are obtained via minimal informationally complete quantum measurements. The suggested method for probability representation of quantum dynamics preserves the tensor product structure, which makes it favourable for the analysis of multi-qubit systems. A key advantage of the suggested approach is that minimal informationally complete positive operator-valued measures (MIC-POVMs) are easier to construct in comparison…
We study informationally overcomplete measurements for quantum state estimation so as to clarify their tomographic significance as compared with minimal informationally complete measurements. We show that informationally overcomplete measurements can improve the tomographic efficiency significantly over minimal measurements when the states of interest have high purities. Nevertheless, the efficiency is still too limited to be satisfactory with respect to figures of merit based on monotone Riemannian metrics, such as the Bures metric and quantum Chernoff metric. In this way, we also pinpoint th…
We show that there are informationally complete joint measurements of two conjugated observables on a finite quantum system, meaning that they enable to identify all quantum states from their measurement outcome statistics. We further demonstrate that it is possible to implement a joint observable as a sequential measurement. If we require minimal noise in the joint measurement, then the joint observable is unique. If the dimension d is odd, then this observable is informationally complete. But if d is even, then the joint observable is not informationally complete and one has to allow more no…
Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, "probability" distributions that can go negative. Informationally complete POVMs, a much younger subject than phase space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born Rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonaliza…
Using the stabilizer formalism we construct the minimal code into a D-dimensional Hilbert space (qudit) to protect a qubit against phase damping. The effectiveness of this code is then studied by means of input-output fidelity.
We obtain a simple formula for the average gate fidelity of a linear map acting on qudits. It is given in terms of minimal sets of pure state preparations alone, which may be interesting from the experimental point of view. These preparations can be seen as the outcomes of certain minimal positive operator valued measures. The connection of our results with these generalized measurements is briefly discussed.
We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, "as close as possible" to orthonormal bases for the space of quantum states. These measures are distinguished by an exceptionally simple state-reconstruction formula which allows "painless" quantum state tomography. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter being the unique minimal rank-one members. Recast as ensembles of pure quantum states, the rank-one me…
A quantum measurement is Fisher symmetric if it provides uniform and maximal information on all parameters that characterize the quantum state of interest. Using (complex projective) 2-designs, we construct measurements on a pair of identically prepared quantum states that are Fisher symmetric for all pure states. Such measurements are optimal in achieving the minimal statistical error without adaptive measurements. We then determine all collective measurements on a pair that are Fisher symmetric for the completely mixed state and for all pure states simultaneously. For a qubit, these measurem…
We prove necessary and sufficient conditions for the informational completeness of an arbitrary set of Gaussian observables on continuous variable systems with finite number of degrees of freedom. In particular, we show that an informationally complete set either contains a single informationally complete observable, or includes infinitely many observables. We show that for a single informationally complete observable, the minimal outcome space is the phase space, and the observable can always be obtained from the quantum optical $Q$-function by linear postprocessing and Gaussian convolution, …
C. A. Fuchs and M. Sasaki defined the quantumness of a set of quantum states in \cite{Quantumness}, which is closely related to the fidelity loss in transmission of the quantum states through a classical channel. In \cite{Fuchs}, Fuchs showed that in $d$-dimensional Hilbert space, minimum quantumness is $\frac{2}{d+1}$, and this can be achieved by all rays in the space. He left an open problem, asking whether fewer than $d^2$ states can achieve this bound. Recently, in a different context, A. J. Scott introduced a concept of generalized $t$-design in \cite{GenSphet}, which is a natural general…
It has been shown that classes of (minimal asymmetric) informationally complete POVMs in dimension d can be built using the multiparticle Pauli group acting on appropriate fiducial states [M. Planat and Z. Gedik, R. Soc. open sci. 4, 170387 (2017)]. The latter states may also be derived starting from the Poincaré upper half-plane model H. For doing this, one translates the congruence (or non-congruence) subgroups of index d of the modular group into groups of permutation gates whose some of the eigenstates are the seeked fiducials. The structure of some IC-POVMs is found to be intimately relat…
Minimal informationally complete positive operator-valued measures (MIC-POVMs) are special kinds of measurement in quantum theory in which the statistics of their $d^2$-outcomes are enough to reconstruct any $d$-dimensional quantum state. For this reason, MIC-POVMs are referred to as standard measurements for quantum information. Here, we report an experiment with entangled photon pairs that certifies, for what we believe is the first time, a MIC-POVM for qubits following a device-independent protocol (i.e., modeling the state preparation and the measurement devices as black boxes, and using o…
A general symmetric-informationally-complete (GSIC)-positive-operator-valued measure (POVM) is known to provide an optimal quantum state tomography among minimal IC POVMs with a fixed average purity. In this paper we provide a general construction of a GSIC POVM by means of a complete orthogonal basis (COB), also interpreted as a normal quasiprobability representation. A spectral property of a COB is shown to play a key role in the construction of SIC POVMs and also for the bound of the mean-square error of the state tomography. In particular, a necessary and sufficient condition to construct …
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.
We propose a normal form for single-qudit gates composed of Clifford and $T$-gates for qudits of odd prime dimension $p\geq 5$. We prove that any single-qudit Clifford+$T$ operator can be re-expressed in this normal form in polynomial time. We also provide strong numerical evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an algorithm for exact synthesis of any single-qudit Clifford+$T$ operator with minimal $T$-count.
It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose is symmetric informationally complete positive operator-valued measurements (SIC-POVMs). SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliar…
What is the minimal number of elements in a rank-1 positive-operator-valued measure (POVM) which can uniquely determine any pure state in $d$-dimensional Hilbert space $\mathcal{H}_d$? The known result is that the number is no less than $3d-2$. We show that this lower bound is not tight except for $d=2$ or 4. Then we give an upper bound of $4d-3$. For $d=2$, many rank-1 POVMs with four elements can determine any pure states in $\mathcal{H}_2$. For $d=3$, we show eight is the minimal number by construction. For $d=4$, the minimal number is in the set of $\{10,11,12,13\}$. We show that if this n…
Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, numbe…
Generating multiple entangled qubit pairs between distributed nodes is a prerequisite for a future quantum internet. To achieve a practicable generation rate, standard protocols based on photonic qubits require multiple long-term quantum memories, which remains a significant experimental challenge. In this paper, we propose a novel protocol based on $2^m$-dimensional time-bin photonic qudits that allow for the simultaneous generation of multiple ($m$) entangled pairs between two distributed qubit registers and outline a specific implementation of the protocol based on cavity-mediated spin-phot…
Efficient Experimental Qudit State Estimation via Point Tomography
Universidad de Concepción（康塞普西翁大学）；Millennium Institute for Research in Optics（光学研究千年研究所）；University of Vienna（维也纳大学）；Vienna Center for Quantum Science and Technology (VCQ)（维也纳量子科学与技术中心）；Christian Doppler Laboratory for Photonic Quantum Computer（光子量子计算克里斯蒂安·多普勒实验室）；ICFO - Institut de Ciencies Fotoniques（ICFO-光子科学研究所）；The Barcelona Institute of Science and Technology（巴塞罗那科学技术研究院）；Universidad Católica de la Santísima Concepción（神圣康塞普西翁天主教大学）；Fundação Getulio Vargas（热图利奥·瓦加斯基金会）
We present the experimental quantum tomography of 7- and 8-dimensional quantum systems based on projective measurements in the mutually unbiased basis (MUB-QT). One of the advantages of MUB-QT is that it requires projections from a minimal number of bases to be performed. In our scheme, the higher dimensional quantum systems are encoded using the propagation modes of single photons, and we take advantage of the capabilities of amplitude- and phase-modulation of programmable spatial light modulators to implement the MUB-QT.
A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters $κ$ of a finite group $G$. Here we introduce a $d$-dimensional Hilbert space with $d=cc(G)$, the number of conjugacy classes of $G$. Groups under consideration should follow two rules, (a) the character table contains both two- and three-dimensional representations with at least one of them faithful and (b) there are minimal informationally complete measurements under the action of a $d$-dimensional Pauli group over the characters of these representations. Groups with small $d$ th…
We introduce informationally complete measurements whose outcomes are entanglement witnesses and so answer the question of how many witnesses need to be measured to decide whether an arbitrary state is entangled or not: as many as the dimension of the state space. The optimized witness-based measurement can provide exponential improvement with respect to witness efficiency in high-dimensional Hilbert spaces, at the price of a reduction in the tomographic efficiency. We describe a systematic construction, and illustrate the matter at the example of two qubits.
Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gröbner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where …
We address the trade-off between information gain and state disturbance in measurement performed on qudit systems and devise a class of optimal measurement schemes that saturate the ultimate bound imposed by quantum mechanics to estimation and transmission fidelities. The schemes are minimal, i.e. they involve a single additional probe qudit, and optimal, i.e. they provide the maximum amount of information compatible with a given level of disturbance. The performances of optimal single-user schemes in extracting information by sequential measurements in a N -user transmission line are also inv…
We introduce an operational and statistically meaningful measure, the quantum tomographic transfer function, that possesses important physical invariance properties for judging whether a given informationally complete quantum measurement performs better tomographically in quantum-state estimation relative to other informationally complete measurements. This function is independent of the unknown true state of the quantum source, and is directly related to the average optimal tomographic accuracy of an unbiased state estimator for the measurement in the limit of many sampling events. For the ex…
We show that quantum state tomography with perfect knowledge of the measurement apparatus proves to be, in some instances, inferior to strategies discarding all information about the measurement at hand, as in the case of data pattern tomography. In those scenarios, the larger uncertainty about the measurement is traded for the smaller uncertainty about the reconstructed signal. This effect is more pronounced for minimal or nearly minimal informationally complete measurement settings, which are of utmost practical importance.
We characterize minimal measurement setups for validating the quantum coherence of an unknown quantum state. We show that for a $d$-level system, the optimal strategy consists of measuring $d$ orthonormal bases such that each measured basis is mutually unbiased with respect to the reference basis, and together with the reference basis they form an informationally complete set of measurements. We show that, in general, any strategy capable of validating quantum coherence allows one to evaluate also the exact value of coherence. We then give an explicit construction of the optimal measurements f…
Bell nonlocality as a resource for device independent certification schemes has been studied extensively in recent years. The strongest form of device independent certification is referred to as self-testing, which given a device certifies the promised quantum state as well as quantum measurements performed on it without any knowledge of the internal workings of the device. In spite of various results on self-testing protocols, it remains a highly nontrivial problem to propose a certification scheme of qudit-qudit entangled states based on violation of a single $d$-outcome Bell inequality. Her…
Quasiprobability representations, such as the Wigner function, play an important role in various research areas. The inevitable appearance of negativity in such representations is often regarded as a signature of nonclassicality, which has profound implications for quantum computation. However, little is known about the minimal negativity that is necessary in general quasiprobability representations. Here we focus on a natural class of quasiprobability representations that is distinguished by simplicity and economy. We introduce three measures of negativity concerning the representations of qu…
We propose an experiment that realizes a symmetric informationally complete (SIC) probability-operator measurement (POM) in the four-dimensional Hilbert space of a qubit pair. The qubit pair is carried by a single photon as a polarization qubit and a path qubit. The implementation of the SIC POM is accomplished with the means of linear optics. The experimental scheme exploits a new approach to SIC POMs that uses a two-step process: a measurement with full-rank outcomes, followed by a projective measurement on a basis that is chosen in accordance with the result of the first measurement. The ba…
Symmetric informationally complete (SIC) POVMs are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: 1) every POVM element is rank one, 2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and 3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to d…
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…
We introduce a reliable compressive procedure to uniquely characterize any given low-rank quantum measurement using a minimal set of probe states that is based solely on data collected from the unknown measurement itself. The procedure is most compressive when the measurement constitutes pure detection outcomes, requiring only an informationally complete number of probe states that scales linearly with the system dimension. We argue and provide numerical evidence showing that the minimal number of probe states needed is even generally below the numbers known in the closely-related classical ph…
Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes
Digital Technologies Research Centre, National Research Council of Canada（加拿大国家研究委员会数字技术研究）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）
We show that for any positive integer $n$, the maps $x \in \mathbb{C}^n \mapsto \{\left|\langle x, z_i \rangle \right|^2\}_{i=1}^{4n} \in \mathbb{R}^{4n}$, where $z_i$ are the columns of four $n\times n$ unitary matrices, are generically injective modulo multiplication by a global phase factor, yielding a family of embeddings of $\mathbb{C}P^{n-1}$ into $\mathbb{R}^{4n-4}$. In particular, this implies that distribution measurements about a pure state with four generic full-rank observables are informationally complete, which is sharp for $n \geq 6$. To complement this information-theoretic stu…
Mutually unbiased bases (MUB) are interesting for various reasons. The most attractive example of (a complete set of) MUB is the one constructed by Ivanović as well as Wootters and Fields, which is referred to as the canonical MUB. Nevertheless, little is known about anything that is unique to this MUB. We show that the canonical MUB in any prime power dimension is uniquely determined by an extremal orbit of the (restricted) Clifford group except in dimension 3, in which case the orbit defines a special symmetric informationally complete measurement (SIC), known as the Hesse SIC. Here the extr…
The Clifford operators are an important and well-studied subset of quantum operations, in both the qubit and higher-dimensional qudit cases. While there are many ways to characterize this set, this paper aims to provide an ideal characterization, in the sense that it has the same characterization in every finite dimension, is characterized by a minimal set of gates, is constructive, and does not make any assumptions about non-Clifford operations or resources (such as the use of ancillas or the ability to make measurements). While most characterizations satisfy some of these properties, this ap…
Determining the state of a quantum system is a consuming procedure. For this reason, whenever one is interested only in some particular property of a state, it would be desirable to design a measurement setup that reveals this property with as little effort as possible. Here we investigate whether, in order to successfully complete a given task of this kind, one needs an informationally complete measurement, or if something less demanding would suffice. The first alternative means that in order to complete the task, one needs a measurement which fully determines the state. We formulate the tas…
