It is shown that mean value of any observable with bounded spectrum can be uniquely determined from binary statistics of the measurement performed on {\it single} qubit ancilla coupled to a given system. The observable structure is fully encoded in the corresponding POVM. The method is generalised to the case of distant labs paradigm and discussed in the context of entanglement detection with few local measurements. The results are also discussed in the context of quantum programming.
A trade-off relation on our knowledge about two noncommuting observables of a qubit system in simultaneous measurement is formulated. The obtained inequality offers a quantitative information-theoretic representation of Bohr's principle of complementarity, and can be interpreted as a trade-off relation on the asymptotic accuracy of the maximum-likelihood estimation of the probability distributions of observables.
Joint measurements of qubit observables have recently been studied in conjunction with quantum information processing tasks such as cloning. Considerations of such joint measurements have until now been restricted to a certain class of observables that can be characterized by a form of covariance. Here we investigate conditions for the joint measurability of arbitrary pairs of qubit observables. For pairs of noncommuting sharp qubit observables, a notion of approximate joint measurement is introduced. Optimal approximate joint measurements are shown to lie in the class of covariant joint measu…
We present a single inequality as the necessary and sufficient condition for two unsharp observables of a two-level system to be jointly measurable in a single apparatus and construct explicitly the joint observables. A complementarity inequality arising from the condition of joint measurement, which generalizes Englert's duality inequality, is derived as the trade-off between the unsharpnesses of two jointly measurable observables.
A quantum observable and a channel are considered compatible if they form parts of the same measurement device, otherwise they are incompatible. Constrains on compatibility between observables and channels can be quantified via relations highlighting the necessary trade-offs between noise and disturbance within quantum measurements. In this paper we shall discuss the general properties of these compatibility relations, and then fully characterize the compatibility conditions for an unbiased qubit observable and a Pauli channel. The implications of the characterization are demonstrated on some …
Whereas complementarity manifests itself via two incompatible observables, quantum contextuality can only be revealed via the joint measurements among at least three observables. By incorporating unsharp measurements and joint measurements into a realistic model, we reestablish an inequality due to Liang, Spekkens, and Wiseman rigorously based on the assumption of noncontextuality alone. Its violation therefore unambiguously pinpoints the quantum contextuality of a two-level system. The maximal violation is attained by three triplewise jointly measurable observables that are pairwise jointly m…
A tight information-theoretic measurement uncertainty relation is experimentally tested with neutron spin-1/2 qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum …
The Pauli operators (tensor products of Pauli matrices) provide a complete basis of operators on the Hilbert space of N qubits. We prove that the set of 4^N-1 Pauli operators may be partitioned into 2^N+1 distinct subsets, each consisting of 2^N-1 internally commuting observables. Furthermore, each such partitioning defines a unique choice of 2^N+1 mutually unbiased basis sets in the N-qubit Hilbert space. Examples for 2 and 3 qubit systems are discussed with emphasis on the nature and amount of entanglement that occurs within these basis sets.
Heisenberg's uncertainty relations for measurement quantify how well we can jointly measure two complementary observables and have attracted much experimental and theoretical attention recently. Here we provide an exact tradeoff between the worst-case errors in measuring jointly two observables of a qubit, i.e., all the allowed and forbidden pairs of errors, especially asymmetric ones, are exactly pinpointed. For each pair of optimal errors we provide an optimal joint measurement that is realizable without introducing any ancilla and entanglement. Possible experimental implementations are disc…
Maximally entangled states (MES) represent a valuable resource in quantum information processing. In $N$-qubit systems the MES are $N$-GHZ states, i.e. the collection of $\ket{GHZ_N}=\frac{1}{\sqrt{2}}(\ket{00...0}+\ket{11...1})$ and its local unitary (LU) equivalences. While it is well-known that such states are uniquely stabilized by $N$ commuting observables, in this Letter we consider the minimum number of non-commuting observables needed to characterize an $N$-qubit MES as the unique common eigenstate. Here, we prove that in this general case, any $N$-GHZ state can be uniquely stabilized …
Cloning of observables, unlike standard cloning of states, aims at copying the information encoded in the statistics of a class of observables rather then on quantum states themselves. In such a process the emphasis is on the quantum operation (evolution plus measurement) necessary to retrieve the original information. We analyze, for qubit systems, the cloning of a class generated by two noncommuting observables, elucidating the relationship between such a process and joint measurements. This helps in establishing an optimality criterion for cloning of observables. We see that, even if the cl…
Experimental determination of entanglement is important not only to characterize the state and use it in quantum information, but also in understanding complicated phenomena such as phase transitions. In this paper we show that in many cases, it is possible to determine entanglement of a two qubit state, as represented by concurrence, with a few observables, most of which are local. In particular, rank 1 and rank 2 states need exclusively measurement of local observables while rank 3 states need measurement of just one correlation observable in addition to local observables. Only the rank 4 st…
Recently, Meyer and Wallach [D.A. Meyer and N.R. Wallach (2002), J. of Math. Phys., 43, pp. 4273] proposed a measure of multi-qubit entanglement that is a function on pure states. We find that this function can be interpreted as a physical quantity related to the average purity of the constituent qubits and show how it can be observed in an efficient manner without the need for full quantum state tomography. A possible realization is described for measuring the entanglement of a chain of atomic qubits trapped in a 3D optical lattice.
We address the problem of unambiguous discrimination and identification among quantum observables. We set a general framework and investigate in details the case of qubit observables. In particular, we show that perfect discrimination with two shots is possible only for sharp qubit observables (e.g. Stern-Gerlach apparatuses) associated with mutually orthogonal directions. We also show that for sharp qubit observables associated to nonorthogonal directions unambiguous discrimination with an inconclusive result is always possible.
We construct a multi-observable uncertainty equality as well as an inequality based on the sum of standard deviations in the qubit system. The obtained equality indicates that the uncertainty relation can be expressed more accurately, and also can be used to detect the mixedness of the system. Meanwhile, the new uncertainty inequality can provide a tighter lower bound, and the tightness can be maintained at a high level even in an open system. Furthermore, the deficiency of the uncertainty relation, that the lower bound of the product form uncertainty relations can be null even for two incompa…
We construct a single observable measurement of which mean value on four copies of an {\it unknown} two-qubit state is sufficient for unambiguous decision whether the state is separable or entangled. In other words, there exists a universal collective entanglement witness detecting all two-qubit entanglement. The test is directly linked to a function which characterizes to some extent the entanglement quantitatively. This function is an entanglement monotone under so--called local pure operations and classical communication (pLOCC) which preserve local dimensions. Moreover it provides tight up…
We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the N-qubit Pauli group for N >= 4, thus adding to the proofs that have been presented earlier for the two- and three-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large numbe…
We derive schemes to measure the so-called weak values of quantum system observables by coupling of the system to a qubit meter system. We highlight, in particular, the meaning of the imaginary part of the weak values, and show how it can be measured directly on equal footing with the real part of the weak value. We present compact expressions for the weak value of single qubit observables and of product observables on qubit pairs. Experimental studies of the results are suggested with cold trapped ions.
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 consider the kinematics of bi-partite quantum states as determined by observable quantities, in particular the Bloch vectors of the subsystems. In examining the simplest case of a pair of two-level systems, there is a remarkable connection between the presence of non-classical correlations and the chirality of the two bases generated by the singular value decomposition of the correlation matrix of the Bloch vectors. We investigate the limits imposed by quantum mechanics of this effect and it relationship with other methods on quantifying the system's non-classical behaviour.
Monitoring a quantum observable continuously in time produces a stochastic measurement record that noisily tracks the observable. For a classical process such noise may be reduced to recover an average signal by minimizing the mean squared error between the noisy record and a smooth dynamical estimate. We show that for a monitored qubit this usual procedure returns unusual results. While the record seems centered on the expectation value of the observable during causal generation, examining the collected past record reveals that it better approximates a moving-mean Gaussian stochastic process …
We consider a hypothetical apparatus that implements measurements for arbitrary 4-local quantum observables A on n qubits. The apparatus implements the ``measurement algorithm'' after receiving a classical description of A. We show that a few precise measurements, applied to a basis state would provide a probabilistic solution of PSPACE problems. The error probability decreases exponentially with the number of runs if the measurement accuracy is of the order of the spectral gaps of A. Moreover, every decision problem which can be solved on a quantum computer in T time steps can be encoded into…
The information obtained from the operation of a quantum gate on only two complementary sets of input states is sufficient to estimate the quantum process fidelity of the gate. In the case of entangling gates, these conditions can be used to predict the multi qubit entanglement capability from the fidelities of two non-entangling local operations. It is then possible to predict highly non-classical features of the gate such as violations of local realism from the fidelities of two completely classical input-output relations, without generating any actual entanglement.
Greenberger-Horne-Zeilinger (GHZ) theorem asserts that there is a set of mutually commuting nonlocal observables with a common eigenstate on which those observables assume values that refute the attempt to assign values only required to have them by the local realism of Einstein, Podolsky, and Rosen (EPR). It is known that for a three-qubit system there is only one form of the GHZ-Mermin-like argument with equivalence up to a local unitary transformation, which is exactly Mermin's version of the GHZ theorem. In this paper, however, for a four-qubit system which was originally studied by GHZ, w…
We propose and analyze quantum state estimation (tomography) using continuous quantum measurements with resource limitations, allowing the global state of many qubits to be constructed from only measuring a few. We give a proof-of-principle investigation demonstrating successful tomographic reconstruction of an arbitrary initial quantum state for three different situations: single qubit, remote qubit, and two interacting qubits. The tomographic reconstruction utilizes only a continuous weak probe of a single qubit observable, a fixed coupling Hamiltonian, together with single-qubit controls. I…
Entanglement with a mode observable via a tunable interaction with a qubit
FZU - Institute of Physics of the Czech Academy of Sciences（捷克科学院物理研究所）；Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院）；Department of Optics, Palacký University（帕尔卡光学系）
We analyze the continuous measurement of two non-commuting observables for a qubit, and investigate whether the simultaneously observed noisy signals are consistent with the evolution of an equivalent classical system. Following the approach outlined by Leggett and Garg, we show that the readouts violate macrorealistic inequalities for arbitrarily short temporal correlations. Moreover, the derived inequalities are manifestly violated even in the absence of Hamiltonian evolution, unlike for Leggett-Garg inequalities that use a single continuous measurement. Such a violation should indicate the …
The number of qubits used by a quantum algorithm will be a crucial computational resource for the foreseeable future. We show how to obtain the classical query complexity for continuous problems. We then establish a simple formula for a lower bound on the qubit complexity in terms of the classical query complexity
Being one of the centroidal concepts in quantum theory, the fundamental constraint imposed by Heisenberg uncertainty relations has always been a subject of immense attention and challenging in the context of joint measurements of general quantum mechanical observables. In particular, the recent extension of the original uncertainty relations has grabbed a distinct research focus and set a new ascendent target in quantum mechanics and quantum information processing. In the present work we explore the joint measurements of three incompatible observables, following the basic idea of a newly propo…
We introduce an approximate description of an $N$-qubit state, which contains sufficient information to estimate the expectation value of any observable with precision independent of $N$. We show, in fact, that the error in the estimation of the observables' expectation values decreases as the inverse of the square root of the number of the system's identical preparations and increases, at most, linearly in a suitably defined, $N$-independent, seminorm of the observables. Building the approximate description of the $N$-qubit state only requires repetitions of single-qubit rotations followed by…
We experimentally generate arbitrary time-bin qubits using continuous-wave light. The advantage unique to our qubit is its compatibility with deterministic continuous-variable quantum information processing. This compatibility comes from its optical coherence with continuous waves, well-defined spatio-temporal mode, and frequency spectrum within the operational bandwidth of the current continuous-variable technology. We also demonstrate an efficient scheme to characterize time-bin qubits via eight-port homodyne measurement. This enables the complete characterization of the qubits as two-mode s…
We investigate how entanglement can be transferred between continuous variable and qubit systems. We find that a two-mode squeezed vacuum state and a continuous variable Werner state can be converted to the product states of infinitive number of two-qubit states while keeping the entanglement. The reverse process is also possible.
We consider multi-time correlators for output signals from linear detectors, continuously measuring several qubit observables at the same time. Using the quantum Bayesian formalism, we show that for unital (symmetric) evolution in the absence of phase backaction, an $N$-time correlator can be expressed as a product of two-time correlators when $N$ is even. For odd $N$, there is a similar factorization, which also includes a single-time average. Theoretical predictions agree well with experimental results for two detectors, which simultaneously measure non-commuting qubit observables.
Unsharp measurements are increasingly important for foundational insights in quantum theory and quantum information applications. Here, we report an experimental implementation of unsharp qubit measurements in a sequential communication protocol, based on a quantum random access code. The protocol involves three parties; the first party prepares a qubit system, the second party performs operations which return both a classical and quantum outcome, and the latter is measured by the third party. We demonstrate a nearly-optimal sequential quantum random access code that outperforms both the best …
Uncertainty relations for more than two observables have found use in quantum information, though commonly known relations pertain to a pair of observables. We present novel uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Tsallis $α$-entropies. For all real $α\in(0;1]$ and integer $α\geq2$, lower bounds on the sum of three $α$-entropies are obtained. These bounds are tight in the sense that they are always reached with certain pure states. The necessary and sufficient condition for equality is that the qubit state is an eigenstate o…
We investigate how quantum state can be converted between continuous variable and qubits systems. Non-linear Jaynes-Cumings interaction Hamiltonian is introduced to accomplish the conversion. Detail analysis on the conversion of thermal state exhibits that pretty good fidelity can be achieved.
In quantum mechanics, continuously measuring an observable steers the system into one eigenstate of that observable. This property has interesting and useful consequences when the observable is a joint property of two remotely separated qubits. In particular, if the measurement of the two-qubit joint observable is performed in a way that is blind to single-qubit information, quantum back-action generates correlation of the discord type even if the measurement is weak and inefficient. We demonstrate the ability to generate these quantum correlations in a circuit-QED setup by performing a weak j…
It was shown by Ahn, Wiseman, and Milburn [PRA {\bf 67}, 052310 (2003)] that feedback control could be used as a quantum error correction process for errors induced by weak continuous measurement, given one perfectly measured error channel per qubit. Here we point out that this method can be easily extended to an arbitrary number of error channels per qubit. We show that the feedback protocols generated by our method encode $n-2$ logical qubits in $n$ physical qubits, thus requiring just one more physical qubit than in the previous case.
We consider the simultaneous and continuous measurement of qubit observables $σ_z$ and $σ_z\cosφ+ σ_x\sinφ$, focusing on the temporal correlations of the two output signals. Using quantum Bayesian theory, we derive analytical expressions for the correlators, which we find to be in very good agreement with experimentally measured output signals. We further discuss how the correlators can be applied to parameter estimation, and use them to infer a small residual qubit Hamiltonian arising from calibration inaccuracy in the experimental data.
The measurement of a quantum two-level system, or a qubit in modern terminology, often involves an electromagnetic field that interacts with the qubit, before the field is measured continuously and the qubit state is inferred from the noisy field measurement. During the measurement, the qubit may undergo spontaneous transitions, further obscuring the initial qubit state from the observer. Taking advantage of some well known techniques in stochastic detection theory, here we propose a novel signal processing protocol that can infer the initial qubit state optimally from the measurement in the p…
The necessary and sufficient criteria for violating the Mermin and Svetlichny inequalities by arbitrary three-qubit states are presented. Several attempts have been made, earlier, to find such criteria, however, those extant criteria are neither tight for most of the instances, nor fully general. We generalize the existing criteria for Mermin and Svetlichny inequalities which are valid for the local projective measurement observables as well as for the arbitrary ones. We obtain the maximal achievable bounds of the Mermin and Svetlichny operators with unbiased measurement observables for arbitr…
It is possible to prepare a composite qubit-environment system so that its time evolution will guarantee the conservation of a preselected qubit's observable. In general, this observable is not associated with a symmetry. The latter may not even be present in the subsystem. The initial states which lead to such a quantity conserved dynamics form a subspace of the qubit-environment space of states. General construction of this subspace is presented and illustrated by two examples. The first one is the exactly soluble Jaynes-Cummings model and the second is the multi-photon Rabi model.
We present a protocol for transferring arbitrary continuous-variable quantum states into a few discrete-variable qubits and back. The protocol is deterministic and utilizes only two-mode Rabi-type interactions which are readily available in trapped-ion and superconducting circuit platforms. The inevitable errors caused by transferring an infinite-dimensional state into a finite-dimensional register are suppressed exponentially with the number of qubits. Furthermore, the encoded states exhibit robustness against noise, such as dephasing and amplitude damping, acting on the qubits. Our protocol …
We consider the evolution of a spin 1/2 (qubit) under the simultaneous continuous measurement of three non-commuting qubit operators sigma_x, sigma_y, sigma_z. For identical ideal detectors the qubit state evolves by approaching a pure state with a random direction in the Bloch vector space and by undergoing locally isotropic diffusion in the perpendicular directions. The quantum state conditioned on the complete detector record is used to assess the fidelity of classically inspired estimates based on running time averages and discrete time bin detector outputs.
We consider the temporal correlations of the quantum state of a qubit subject to simultaneous continuous measurement of two non-commuting qubit observables. Such qubit state correlators are defined for an ensemble of qubit trajectories, which has the same fixed initial state and can also be optionally constrained by a fixed final state. We develop a stochastic path integral description for the continuous quantum measurement and use it to calculate the considered correlators. Exact analytic results are possible in the case of ideal measurements of equal strength and are also shown to agree with…
Continuous-variable (CV) qubits can be created on an optical longitudinal mode in which quantum information is encoded by the superposition of even and odd Schroedinger's cat states with quadrature amplitude. Based on the analogous features of paraxial optics and quantum mechanics, we propose a system to generate and detect CV qubits on an optical transverse mode. As a proof-of-principle experiment, we generate six CV qubit states and observe their probability distributions in position and momentum space. This enabled us to prepare a non-Gaussian initial state for CV quantum computing. Other p…
The canonical commutation relations of quantum field theory require all pairs of observables located in spacelike-separated regions to commute. In the theory as it is currently constituted, this implies that the information-carrying capacity of a finite volume of space is infinite. Yet Bekenstein's bound gives us strong reason to believe that it is finite. A class of quantum field theories is presented in which observables localised in spacelike-separated regions do not necessarily commute, but which nevertheless has no physical pathologies.
A field configuration utilizing local static and oscillating fields is constructed to achieve universal (but low-order) protection of two-qubit states. That is, two-qubit states can be protected against arbitrary system-environment coupling if the driving field frequency, as compared with the cutoff frequency of the environment, is sufficiently large. Equally important, we show that it is possible to construct driving fields to protect two-qubit entangling gates against decoherence, without assuming any particular form of system-environment coupling. Using a non-Markovian master equation, we f…
The concept of a modular value of an observable of a pre- and post-selected quantum system is introduced. It is similar in form and in some cases has a close connection to the weak value of an observable, but instead of describing an effective interaction when the coupling is weak, it describes a coupling of any strength but only to qubit meters. The generalization of the concept for a coupling of a composite system to a multi-qubit meter provides an explanation of some current experiments.
We consider the problem of obtaining maximally entangled photon states at distance in the presence of loss. We compare the efficiency of two different schemes in establishing $N$ shared ebits: i) $N$ single ebit states with the qubit encoded on polarization; ii) a single continuous variable entangled state (emode) assisted by optimal local operation and classical communication (LOCC) protocol in order to obtain a $2^N$-dimensional maximally entangled state, with qubits encoded on the photon number.
We present an experimental scheme to measure the full distribution of many-body observables in spin systems, both in and out of equilibrium, using an auxiliary qubit as a probe. We focus on the determination of the magnetization and the kink-number statistics at thermal equilibrium. The corresponding characteristic functions are related to the analytically-continued partition function. Thus, both distributions can be directly extracted from experimental measurements of the coherence of a probe qubit that is coupled to an Ising-type bath, as reported in [X. Peng et al., Phys. Rev. Lett. 114, 01…
Protecting quantum states from the decohering effects of the environment is of great importance for the development of quantum computation devices and quantum simulators. Here, we introduce a continuous dynamical decoupling protocol that enables us to protect the entangling gate operation between two qubits from the environmental noise. We present a simple model that involves two qubits which interact with each other with a strength that depends on their mutual distance and generates the entanglement among them, as well as in contact with an environment. The nature of the environment, that is,…
It is known that any two-outcome quantum measurement can be decomposed into a continuous stochastic process using a feedback loop. In this article, we characterize which of these decompositions are possible when each iteration of the feedback loop consists of a weak measurement caused by an interaction with a probe system. We restrict ourselves to the case when the probe is a qubit and the interaction Hamiltonian between the probe and system is constant. We find that even given the ability to perform arbitrary unitary pulses throughout the continuous decomposition, only generalized measurement…
