Given an ensemble of qubits, which we are told consists of a mixture of two pure states, one with probability $η_{0}$ and one with probability $η_{1}$, we want to find a POVM that will discriminate between the two states by measuring the qubits. We do not know the states, and for any given qubit, we do not know which of the two states it is in. This can be viewed as learning a POVM from quantum data. Once found, the POVM can be used to separate the remaining qubits in the ensemble into two groups, corresponding to the two states present in the ensemble. In order to find the POVM, we need more information about the possible states. We examine several cases. First, we suppose that we know that the Bloch vectors of the states lie in the x-z plane and their \emph{a priori} probabilities are equal. We next keep the restriction to the x-z plane, but allow the \emph{a priori} probabilities to be different. Finally, we consider the case in which the Bloch vectors of the states have the same z component.
There are many different definitions of what a Bell-Kochen-Specker proof with POVMs might be. Here we present and discuss the minimal proof on qubits for three of these definitions and show that they are indeed minimal.
We address the class of positive operator-valued measures (POVMs) for qubit systems that are obtained by coupling the signal qubit with a probe qubit and then performing a projective measurement on the sole probe system. These POVMs, which represent the simplest class of qubit POVMs, depends on 3+3+2=8 free parameters describing the initial preparation of the probe qubit, the Cartan representative of the unitary coupling, and the projective measurement at the output, respectively. We analyze in some details the properties of the POVM matrix elements, and investigate their values for given rang…
A bipartite Bell inequality is derived which is maximally violated on the two-qubit state space if measurements describable by positive operator valued measure (POVM) elements are allowed rather than restricting the possible measurements to projective ones. In particular, the presented Bell inequality requires POVMs in order to be maximally violated by a maximally entangled two-qubit state. This answers a question raised by N. Gisin.
In a recent paper, Buscemi and al. defined a notion of clean positive operator valued measures (POVMs). We here characterize which POVMs are clean in some class that we call quasi-qubit POVMs, namely POVMs whose elements are all rank-one or full-rank. We give an algorithm to check whether a given quasi-qubit POVM satisfies to this condition. We describe explicitely all the POVMs that are clean for the qubit. On the way we give a sufficient condition for a general POVM to be clean.
Symmetric informationally complete positive operator-valued measurement (SIC-POVM) is one important class of quantum measurement which is crucial for various quantum information processing tasks. SIC-POVMs have the advantage of providing an unbiased estimator for quantum states with the minimal number of outcomes needed for full tomography. We present an experimental approach on a photonic quantum walk which can be used to implement SIC-POVMs on a single-qbubit. The projection measurements of single-photons' positions correspond to elements of SIC-POVM on the polarization of single-photons.
Many quantum measurements, such as photodetection, can be destructive. In photodetection, when the detector clicks a photon has been absorbed and destroyed. Yet the lack of a click also gives information about the presence or absence of a photon. In monitoring the emission of photons from a source, one decomposes the strong measurement into a series of weak measurements, which describe the evolution of the state during the measurement process. Motivated by this example of destructive photon detection, a simple model of destructive weak measurements using qubits was studied in [1]. It has shown…
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…
In the four-dimensional Hilbert space, there exist 16 Heisenberg--Weyl (HW) covariant symmetric informationally complete positive operator valued measures (SIC~POVMs) consisting of 256 fiducial states on a single orbit of the Clifford group. We explore the structure of these SIC~POVMs by studying the symmetry transformations within a given SIC~POVM and among different SIC~POVMs. Furthermore, we find 16 additional SIC~POVMs by a regrouping of the 256 fiducial states, and show that they are unitarily equivalent to the original 16 SIC~POVMs by establishing an explicit unitary transformation. We t…
We derive a deterministic protocol to implement a general single-qubit POVM on near-term circuit-based quantum computers. The protocol has a modular structure, such that an $n$-element POVM is implemented as a sequence of $(n-1)$ circuit modules. Each module performs a $2$-element POVM. Two variations of the protocol are suggested, one optimal in terms of number of ancilla qubits, the other optimal in terms of number of qubit gate operations and quantum circuit depth. We use the protocol to implement $2$- and $3$-element POVMs on two publicly available quantum computing devices. The results we…
We report an experimental implementation of a single-qubit generalised measurement scenario(POVM) based on a quantum walk model. The qubit is encoded in a single-photon polarisation. The photon performs a quantum walk on an array of optical elements, where the polarisation-dependent translation is performed via birefringent beam displacers and a change of the polarisation is implemented with the help of wave-plates. We implement: (i) Trine-POVM, i.e., the POVM elements uniformly distributed on an equatorial plane of the Bloch sphere; (ii) Symmetric-Informationally- Complete (SIC) POVM; and (ii…
Eigenstates of permutation gates are either stabilizer states (for gates in the Pauli group) or magic states, thus allowing universal quantum computation [M. Planat and Rukhsan-Ul-Haq, Preprint 1701.06443]. We show in this paper that a subset of such magic states, when acting on the generalized Pauli group, define (asymmetric) informationally complete POVMs. Such IC-POVMs, investigated in dimensions $2$ to $12$, exhibit simple finite geometries in their projector products and, for dimensions $4$ and $8$ and $9$, relate to two-qubit, three-qubit and two-qutrit contextuality.
We show that a one-dimensional discrete time quantum walk can be used to implement a generalized measurement in terms of positive operator value measure (POVM) on a single qubit. More precisely, we show that for a single qubit any set of rank 1 and rank 2 POVM elements can be generated by a properly engineered quantum walk. In such a scenario the measurement of particle at position x=i corresponds to a measurement of a POVM element E_i on a qubit. We explicitly construct quantum walks implementing unambiguous state discrimination and SIC-POVM.
We show that non-maximal entangled states can be used for implementing, with unit probability, remote generalized measurements (POVMs). We show how any n-qubit POVM can be applied remotely and derive its entanglement cost. The later turns out to be equal to the entanglement capability for a class of POVMs. This suggests a one-to-one relation between sub-sets of POVM operations and entanglement.
We present adaptive measurement techniques tailored for variational quantum algorithms on near-term small and noisy devices. In particular, we generalise earlier "learning to measure" strategies in two ways. First, by considering a class of adaptive positive operator valued measures (POVMs) that can be simulated with simple projective measurements without ancillary qubits, we decrease the amount of required qubits and two-qubit gates. Second, by introducing a method based on Quantum Detector Tomography to mitigate the effect of noise, we are able to optimise the POVMs as well as to infer expec…
Each platonic solid defines a single-qubit positive operator valued measure (POVM) by interpreting its vertices as points on the Bloch sphere. We construct simple circuits for implementing this kind of measurements and other simple types of symmetric POVMs on one qubit. Each implementation consists of a discrete Fourier transform and some elementary quantum operations followed by an orthogonal measurement in the computational basis.
We study extreme points of the set of finite-outcome positive-operator-valued measures (POVMs) on finite-dimensional Hilbert spaces and particularly the possible ranks of the effects of an extreme POVM. We give results discussing ways of deducing new rank combinations of extreme POVMs from rank combinations of known extreme POVMs and, using these results, show ways to characterize rank combinations of extreme POVMs in low dimensions. We show that, when a rank combination together with a given dimension of the Hilbert space solve a particular packing problem, there is an extreme POVM on the Hil…
In this paper, we propose concurrence classes for an arbitrary multi-qubit state based on orthogonal complement of a positive operator valued measure, or POVM in short, on quantum phase. In particular, we construct concurrence for an arbitrary two-qubit state and concurrence classes for the three- and four-qubit states. And finally, we construct $W^{m}$ and $GHZ^{m}$ class concurrences for multi-qubit states. The unique structure of our POVM enables us to distinguish different concurrence classes for multi-qubit states.
Measurements in quantum theory exhibit incompatibility, i.e., they can fail to be jointly measurable. An intuitive way to represent the (in)compatibility relations among a set of measurements is via a hypergraph representing their joint measurability structure: its vertices represent measurements and its hyperedges represent (all and only) subsets of compatible measurements. Projective measurements in quantum theory realize (all and only) joint measurability structures that are graphs. On the other hand, general measurements represented by positive operator-valued measures (POVMs) can realize …
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…
We present an efficient method to solve the quantum discord of two-qubit X states exactly. A geometric picture is used to clarify whether and when the general POVM measurement is superior to von Neumann measurement. We show that either the von Neumann measurement or the three-element POVM measurement is optimal, and more interestingly, in the latter case the components of the postmeasurement ensemble are invariant for a class of states.
Self-testing represents the strongest form of certification of a quantum system. Here we investigate theoretically and experimentally the question of self-testing non-projective quantum measurements. That is, how can one certify, from observed data only, that an uncharacterised measurement device implements a desired non-projective positive-operator-valued-measure (POVM). We consider a prepare-and-measure scenario with a bound on the Hilbert space dimension, which we argue is natural for this problem since any measurement can be made projective by artificially increasing the Hilbert space dime…
The properties of deterministic LOCC transformations of three qubit pure states are studied. We show that the set of states in the GHZ class breaks into an infinite number of disjoint classes under this type of transformation. These classes are characterized by the value of a quantity that is invariant under these transformations, and is defined in terms of the coefficients of a particular canonical form in which only states in the GHZ class can be expressed. This invariant also imposes a strong constraint on any POVM that is part of a deterministic protocol. We also consider a transformation …
Recently, Vértesi and Bene [Phys. Rev. A. {\bf 82}, 062115 (2010)] derived a two-qubit Bell inequality, $I_{CH3}$, which they show to be maximally violated only when more general positive operator valued measures (POVMs) are used instead of the usual von Neumann measurements. Here we consider a general parametrization for the three-element-POVM involved in the Bell test and obtain a higher quantum bound for the $I_{CH3}$-inequality. With a higher quantum bound for $I_{CH3}$, we investigate if there is an experimental setup that can be used for observing that POVMs give higher violations in Bel…
Quantifying nonclassicality in qubit systems via positive operator-valued measures
Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院斐兹克量子中心）；Department of Physics, Isfahan University of Technology（伊斯法罕理工大学物理系）
Analytic Qubit Separation between POVMs and Projective Measurements
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Quantum Science Center of Guangdong–Hong Kong–Macao Greater Bay Area（粤港澳大湾区量子科学中心）
The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continu…
There has been much interest in so-called SIC-POVMs: rank 1 symmetric informationally complete positive operator valued measures. In this paper we discuss the larger class of POVMs which are symmetric and informationally complete but not necessarily rank 1. This class of POVMs is of some independent interest. In particular it includes a POVM which is closely related to the discrete Wigner function. However, it is interesting mainly because of the light it casts on the problem of constructing rank 1 symmetric informationally complete POVMs. In this connection we derive an extremal condition alt…
We introduce and implement a technique to extend the quantum computational power of cluster states by replacing some projective measurements with generalized quantum measurements (POVMs). As an experimental demonstration we fully realize an arbitrary three-qubit cluster computation by implementing a tunable linear-optical POVM, as well as fast active feedforward, on a two-qubit photonic cluster state. Over 206 diferent computations, the average output fidelity is 0.9832 +/- 0.0002; furthermore the error contribution from our POVM device and feedforward is only of order 10e-3, less than some re…
We consider general prepare-and-measure scenarios in which Alice can transmit qubit states to Bob, who can perform general measurements in the form of positive operator-valued measures (POVMs). We show that the statistics obtained in any such quantum protocol can be simulated by the purely classical means of shared randomness and two bits of communication. Furthermore, we prove that two bits of communication is the minimal cost of a perfect classical simulation. In addition, we apply our methods to Bell scenarios, which extends the well-known Toner and Bacon protocol. In particular, two bits o…
Quantum discord is a measure based on local projective measurements which captures quantum correlations that may not be fully captured by entanglement. A change in the measurement process, achieved by replacing rank-one projectors with a weak positive operator-valued measure (POVM), allows one to define weak variants of quantum discord. In this work, we experimentally simulate the effect of a weak POVM on a nuclear magnetic resonance quantum information processor. The two-qubit system under investigation is part of a three-qubit system, where one of the qubits is used as an ancillary to implem…
We describe the structure of the extended Clifford Group (defined to be the group consisting of all operators, unitary and anti-unitary, which normalize the generalized Pauli group (or Weyl-Heisenberg group as it is often called)). We also obtain a number of results concerning the structure of the Clifford Group proper (i.e. the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete POVMs (or SIC-POVMs) covariant relative to the action of the gener…
We use quantum detector tomography to characterize the qubit readout in terms of measurement POVMs on IBM Quantum Computers IBM Q 5 Tenerife and IBM Q 5 Yorktown. Our results suggest that the characterized detector model deviates from the ideal projectors by a few percent. Further improvement on this characterization can be made by adopting two- or more-qubit detector models instead of independent single-qubit detectors for all the qubits in one device. An unexpected behavior was seen in the physical qubit labelled as qubit 3 of IBM Q 5 Tenerife, which can be a consequence of detector crosstal…
We propose a scheme to implement general quantum measurements, also known as Positive Operator Valued Measures (POVMs) in dimension $d$ using only classical resources and a single ancillary qubit. Our method is based on the probabilistic implementation of $d$-outcome measurements which is followed by postselection of some of the received outcomes. We conjecture that the success probability of our scheme is larger than a constant independent of $d$ for all POVMs in dimension $d$. Crucially, this conjecture implies the possibility of realizing arbitrary nonadaptive quantum measurement protocol o…
We mainly study the block-coherence measures based on resource theory of block-coherence and the coherence measures based on positive-operator-valued measures (POVM). Several block-coherence measures including a block-coherence measure based on maximum relative entropy, the one-shot block coherence cost under the maximally block-incoherent operations, and a coherence measure based on coherent rank have been introduced and the relationships between these block-coherence measures have been obtained. We also give the definition of the maximally block-coherent state and describe the deterministic …
It was presented by Cabello and Nakamura [A. Cabello, Phys. Rev. Lett. 90, 190401 (2003)], that the Kochen-Specker theorem applies to two dimensions if one uses Positive Operator-Valued Measures. We show that contextuality in their models is not of the Kochen-Specker type. It is rather the result of not keeping track of the whole system on which the measurement is performed. This is connected to the fact that there is no one-to-one correspondence between POVM elements and projectors on the extended Hilbert space and the same POVM element has to originate from two different projectors when used…
Tomography of the two qubit density matrix shared by Alice and Bob is an essential ingredient for guaranteeing an acceptable margin of confidentiality during the establishment of a secure fresh key through the Quantum Key Distribution (QKD) scheme. We show how the Singapore protocol for key distribution is optimal from this point of view, due to the fact that it is based on so called SIC POVM qubit tomography which allows the most accurate full tomographic reconstruction of an unknown density matrix on the basis of a restricted set of experimental data. We illustrate with the help of experimen…
We describe a technique for self consistently characterizing both the quantum state of a single qubit system, and the positive-operator-valued measure (POVM) that describes measurements on the system. The method works with only ten measurements. We assume that a series of unitary transformations performed on the quantum state are fully known, while making minimal assumptions about both the density operator of the state and the POVM. The technique returns maximum-likely estimates of both the density operator and the POVM. To experimentally demonstrate the method, we perform reconstructions of o…
We consider joint measurement of two and three unsharp qubit observables through an Arthur-Kelly type joint measurement model for qubits. We investigate the effect of initial state of the detectors on the unsharpness of the measurement as well as the post-measurement state of the system. Particular emphasis is given on a physical understanding of the POVM to PVM transition in the model and entanglement between system and detectors.Two approaches for characterizing the unsharpness of the measurement and the resulting measurement uncertainty relations are considered.The corresponding measures of…
We discuss a state discriminator that unambiguously distinguishes between two quantum registers prepared with multiple copies of two unknown qubits. This device achieves the optimal performance by von Neumann measurement and general POVM in different ranges of the input preparation probabilities, respectively, and in the limit of very large program registers the optimal measurement reduces to only von Neumann measurements.
In this paper we present a protocol for the implementation of a positive-operator-valued measure (POVM) on massive fermionic qubits. We present methods for implementing non-dispersive qubit transport, spin rotations and spin polarizing beam-splitter operations. Our scheme attains linear optics-like control of the spatial extent of the qubits by considering groundstate electrons trapped in the minima of surface acoustic waves in semiconductor heterostructures. Furthermore, we numerically simulate a high-fidelity POVM that carries out Procrustean entanglement distillation in the framework of our…
Quantum coherence is a fundamental feature of quantum mechanics and an underlying requirement for most quantum information tasks. In the resource theory of coherence, incoherent states are diagonal with respect to a fixed orthonormal basis, i.e., they can be seen as arising from a von Neumann measurement. Here, we introduce and study a generalization to a resource theory of coherence defined with respect to the most general quantum measurement, i.e., to an arbitrary positive-operator-valued measure (POVM). We establish POVM-based coherence measures and POVM-incoherent operations which coincide…
We present a method for measuring quantum states encoded in the temporal modes of photons. The basis for the multilevel quantum states is defined by the use of modes propagating in a dispersive medium, which is a fiber in this case. The propagation and time-resolved single photon detection allow us to define a positive-operator valued measure (POVM). The POVM depends on the amount of dispersion and the characteristics of a detector. This framework is numerically tested by performing quantum state tomography on a large number of states for a set of realistic experimental settings. Finally, the …
Randomness is a central feature of quantum mechanics and an invaluable resource for both classical and quantum technologies. Commonly, in Device-Independent and Semi-Device-Independent scenarios, randomness is certified using projective measurements and the amount of certified randomness is bounded by the dimension of the measured quantum system. In this work, we propose a new Source-Device-Independent protocol, based on Positive Operator Valued Measurement (POVM), which can arbitrarily increase the number of certified bits for any fixed dimension. A tight lower-bound on the quantum conditiona…
