We discuss the problem of estimating a general (mixed) qubit state. We give the optimal guess that can be inferred from any given set of measurements. For collective measurements and for a large number $N$ of copies, we show that the error in the estimation goes as 1/N. For local measurements we focus on the simpler case of states lying on the equatorial plane of the Bloch sphere. We show that standard tomographic techniques lead to an error proportional to $1/N^{1/4}$, while with our optimal data processing it is proportional to $1/N^{3/4}$.
We propose a distribution scheme of polarization states of a single photon over collective-noise channel. By adding one extra photon with a fixed polarization, we can protect the state against collective noise via a parity-check measurement and post-selection. While the scheme succeeds only probabilistically, it is simpler and more flexible than the schemes utilizing decoherence-free subspace. An application to BB84 protocol through collective noise channel, which is robust to the Trojan horse attack, is also given.
We present a faithful qubit transmission scheme with linear optics against collective noise, not resorting to ancillary qubits. Its set-up is composed of three unbalanced polarization interferometers, based on a polarizing beam splitter, a beam splitter and a half-wave plate, which makes this scheme more feasible than others with present technology. The fidelity of successful transmission is 1, independent of the parameters of the collective noise, and the success probability for obtaining an uncorrupted state can be improved to 100% with some time delayers. Moreover, this scheme has some good…
We present a method of generating collective multi-qubit entanglement via global addressing of an ion chain following the guidelines of the Tavis-Cummings model, where several qubits are coupled to a collective motional mode. We show that a wide family of Dicke states and irradiant states can be generated by single global laser pulses, unitarily or helped with suitable postselection techniques.
We investigate the steady state photon transport in a nonequilibrium collective-qubit model. By adopting the noninteracting blip approximation, which is applicable in the strong photon-qubit coupling regime, we describe the essential contribution of indirect qubit-qubit interaction to the population distribution, mediated by the photonic baths. The linear relations of both the optimal flux and noise power with the qubits system size are obtained. Moreover, the inversed power-law style for the finite-size scaling of the optimal photon-qubit coupling strength is exhibited, which is proposed to b…
We investigate quantum correlations of the $N$-qubit states via a collective pseudo-spin interaction ($\propto J_y^2$) on arbitrary pure separable states for a given interval of time. Based on this dynamical generation of the $N$-qubit maximal entangled states, a quantum secret sharing protocol with $N$ continuous classical secrets is developed.
We propose a passively self-error-rejecting single-qubit transmission scheme for an arbitrary polarization state of a single qubit over a collective-noise channel, without resorting to additional qubits and entanglement. By splitting a single qubit into some wavepackets with some Mach-Zehnder interferometers, we can obtain an uncorrupted state with a success probability approaching 100% via postselection in different time bins, independent of the parameters of collective noise. It is simpler and more flexible than the schemes utilizing decoherence-free subspace and those with additional qubits…
We demonstrate the generation of heralded Fock-basis qubits $(a|0\rangle + b|1\rangle)$ from transient collective spin excitations in a hot atomic vapor. The collective spin excitation is heralded by Raman-scattered photons in a four-wave mixing process seeded by a weak coherent optical excitation. The amplitude and phase of the seed field allow arbitrary control over the qubit coefficients. The state of the ensemble is read out optically and then characterized using balanced homodyne tomography. This work represents a step towards quantum engineering of collective atomic spin states.
We describe an experimentally straightforward method for preparing an entangled W state of up to 100 qubits. Our repeat-until-success protocol relies on detection of single photons from collective spontaneous emission in free space. Our method allows entanglement preparation in a wide range of qubit implementations that lack entangling qubit-qubit interactions. We give detailed numerical examples for entanglement of neutral atoms in optical lattices and of nitrogen-vacancy centres in diamond. The simplicity of our method should enable preparation of mesoscopic entangled states in a number of p…
There is current interest in investigating which variables play an important role in the physical processes with an open composite quan- tum system that ranges from the foundational issues to the tasks of diverse applications in quantum physics and technology. In this paper we contrast the local versus the collective classical-like behavior of a pair of interacting qubits immersed in the mutually independent thermal baths. The qubits can be locally subjected to any of the standard, microscopically modelled, generalized amplitude damping and phase damping channel, as well as to the recently int…
We investigate full counting statistics of quantum heat transfer in a collective-qubit system, constructed by multi-qubits interacting with two thermal baths. The nonequilibrium polaron-transformed Redfield approach embedded with an auxiliary counting field is applied to obtain the steady state heat current and fluctuations, which enables us to study the impact of the qubit-bath interaction in a wide regime. The heat current, current noise and skewness are all found to clearly unify the limiting results in the weak and strong couplings, respectively. Moreover, the superradiant heat transfer is…
We present an analysis of transfer of quantum information between the collective spin degrees of freedom of a large ensemble of two-level systems and a single central qubit. The coupling between the central qubit and the individual ensemble members may be varied and thus provides access to more than a single storage mode. Means to store and manipulate several independent qubits are derived for the case where the variation in coupling strengths does not allow addressing of orthogonal modes of the ensemble. While our procedures and analysis may apply to a number of different physical systems, fo…
Entanglement between light and matter combines the advantage of long distance transmission of photonic qubits with the storage and processing capabilities of atomic qubits. To distribute photonic states efficiently over long distances several schemes to encode qubits have been investigated -- time-bin encoding being particularly promising due to its robustness against decoherence in optical fibers. Here, we demonstrate the generation of entanglement between a photonic time-bin qubit and a single collective atomic spin excitation (spin-wave) in a cold atomic ensemble, followed by the mapping of…
Quantum computers have now appeared in our society and are utilized for the investigation of science and engineering. At present, they have been built as intermediate-size computers containing about fifty qubits and are weak against noise effects. Hence, they are called noisy-intermediate scale quantum devices. In order to accomplish efficient quantum computation with using these machines, a key issue is going to be the coherent control of individual and collective quantum noises. In this work, we focus on a latter type and investigate formulations of the collective quantum noises represented …
We address the problem of optimal estimation of the relative phase for two-dimensional quantum systems in mixed states. In particular, we derive the optimal measurement procedures for an arbitrary number of qubits prepared in the same mixed state.
Quantum cloning of two identical mixed qubits $ρ\otimes ρ$ is studied. We propose the quantum cloning transformations not only for the triplet (symmetric) states but also for the singlet (antisymmetric) state. We can copy these two identical mixed qubits to M (M>2) copies. This quantum cloning machine is optimal in the sense that the shrinking factor between the input and the output single qubit achieves the upper bound. The result shows that we can copy two identical mixed qubits with the same quality as that of two identical pure states.
In this paper, we discuss the partial separability and its criteria problems of multipartite qubit mixed-states. First we strictly define what is the partial separability of a multipartite qubit system. Next we give a reduction way from N-partite qubit density matrixes to bipartite qubit density matrixes, and prove a necessary condition that a N-partite qubit mixed-state to be partially separable is its reduction to satisfy the PPT condition. PACC numbers: 03.67.Mn; 03.65.Ud; 03.67.Hk
We analyze the class of single qubit channels with the environment modeled by a one-qubit mixed state. The set of affine transformations for this class of channels is computed analytically, employing the canonical form for the two-qubit unitary operator. We demonstrate that, 3/8 of the generalized depolarizing channels can be simulated by the one-qubit mixed state environment by explicitly obtaining the shape of the volume occupied by this class of channels within the tetrahedron representing the generalized depolarizing channels. Further, as a special case, we show that the two-Pauli Channel …
The symmetric collective states of an atomic spin ensemble (i.e., many-body states that are invariant under particle exchange) are not preserved by decoherence that acts identically but individually on members of the ensemble. We develop a class of collective states in an ensemble of N spin-1/2 particles that is invariant under symmetric local decoherence and find that the dimension of the Hilbert space spanned by these collective states scales only as N^2. We then investigate the open system dynamics of experimentally relevant non-classical collective atomic states, including Schroedinger cat…
The separable mixed 2-qubit X-states are classified in accordance with degeneracies in the spectrum of density matrices. It is shown that there are four classes of separable X-states, among them: one 4D family, a pair of 2D family and a single, zero-dimensional maximally mixed state.
The longitudinal coupling of a system to the bath usually induces the pure dephasing of the system. In this paper, we study the collective dephasing induced dissipation and decoherence in a coupled-qubit system with a common bath. It is shown that, compared with the case of the same system with independent baths, the interference between the dephasing processes of different qubits induced by the common bath significantly changes the dissipation of the system. For the system of two coupled qubits, the interference leads to a faster decoherence in the non-single-excitation subspaces and a slower…
It is commonly assumed that Shor's quantum algorithm for the efficient factorization of a large number $N$ requires a pure initial state. Here we demonstrate that a single pure qubit together with a collection of $log_2 N$ qubits in an arbitrary mixed state is sufficient to implement Shor's factorization algorithm efficiently.
We analyze rigorously the dynamics of the entanglement between two qubits which interact only through collective and local environments. Our approach is based on the resonance perturbation theory which assumes a small interaction between the qubits and the environments. The main advantage of our approach is that the expressions for (i) characteristic time-scales, such as decoherence, disentanglement, and relaxation, and (ii) observables are not limited by finite times. We introduce a new classification of decoherence times based on clustering of the reduced density matrix elements. The charact…
We show that there does not exist any universal quantum cloning machine that can broadcast an arbitrary mixed qubit with a constant fidelity. Based on this result, we investigate the dependent quantum cloner in the sense that some parameter of the input qubit $ρ_s(θ,ω,λ)$ is regarded as constant in the fidelity. For the case of constant $ω$, we establish the $1\to2$ optimal symmetric dependent cloner with a fidelity 1/2. It is also shown that the $1\to M$ optimal quantum cloning machine for pure qubits is also optimal for mixed qubits, when $λ$ is the unique parameter in the fidelity. For gene…
We find the necessary and sufficient condition under which two two-qubit mixed states can be purified into a pure maximally entangled state by local operations and classical communication. The optimal protocol for such transformation is obtained. This result leads to a necessary and sufficient condition for the exact purification of $n$ copies of a two-qubit state.
In this work, we provide a complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities. For the complete anaysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for mixed four qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have non-zero elements. Using the condi…
We report the experimental observation, and a theoretical explanation, of collective suppression of linewidths for multiple superconducting qubits coupled to a good cavity. This demonstrates how strong qubit-cavity coupling can significantly modify the dephasing and dissipation processes that might be expected for individual qubits, and can potentially improve coherence times in many-body circuit QED.
A common environment acting on a pair of qubits gives rise to a plethora of different phenomena, such as the generation of qubit-qubit entanglement, quantum synchronization and subradiance. Here we define time-independent figures of merit for entanglement generation, quantum synchronization and subradiance, and perform an extensive analytical and numerical study of their dependence on model parameters. We also address a recently proposed measure of the collectiveness of the dynamics driven by the bath, and find that it almost perfectly witnesses the behavior of entanglement generation. Our res…
Samples with few hundred atoms within a few micrometer sized region of space are large enough to provide efficient cooperative absorption and emission of light, and small enough to ensure strong dipole-dipole interactions when atoms are excited into high-lying Rydberg states. Based on a recently proposed collective encoding scheme, we propose to build few-qubit quantum registers in such samples. The registers can store and transmit quantum information in the form of single photons, and they can employ entanglement pumping protocols to perform ideally in networks for scalable quantum computing …
We present two robust quantum key distribution protocols against two kinds of collective noise, following some ideas in quantum dense coding. Three-qubit entangled states are used as quantum information carriers, two of which forming the logical qubit which is invariant with a special type of collective noise. The information is encoded on logical qubits with four unitary operations, which can be read out faithfully with Bell-state analysis on two physical qubits and a single-photon measurement on the other physical qubit, not three-photon joint measurements. Two bits of information are exchan…
Mixed states of samples of spin s particles which are symmetric under permutations of the particles are described in terms of their total collective spin quantum numbers. We use this description to analyze the influence on spin squeezing due to imperfect initial state preparation.
We study a generalized amplitude damping channel when environment is initially in the single-qubit mixed state. Representing the affine transformation of the generalized amplitude damping by a three-dimensional volume, we plot explicitly the volume occupied by the channels simulatable by a single-qubit mixed-state environment. As expected, this volume is embedded in the total volume by the channels which is simulated by two-qubit enviroment. The volume ratio is approximately 0.08 which is much smaller than 3/8, the volume ratio for generalized depolarizing channels.
We investigate the collective radiance characteristics of qubits in the ultrastrong coupling regime, where the radiance witness is defined based on the resonator-qubit dressed basis. The ultrastrong hyperradiance effect is demonstrated when the dressed state of system is resonantly driven. Interestingly, we show that, besides the resonator-qubit coupling strength, the parity-symmetry-breaking induced cascade-transition can significantly enhance the collective radiance of qubits, which allows us to manipulate the transitions between subradiance, superradiance, and hyperradiance via adjusting th…
Estimating multiparamter simultaneously as precise as possible is an important goal of quantum metrolo- gy. As a first step to this end, here we give a condition determining whether two arbitrary parameters can be estimated simultaneously for a qubit in the mixed state. An application of this condition is shown.
We propose schemes that are efficient when each pair of qubits undergoes some imperfect collective decoherence with different baths. In the proposed scheme, each pair of qubits is first encoded in a decoherence-free subspace composed of two qubits. Leakage out of the encoding space generated by the imperfection is reduced by the quantum Zeno effect. Phase errors in the encoded bits generated by the imperfection are reduced by concatenation of the decoherence-free subspace with either a three-qubit quantum error correcting code that corrects only phase errors or a two-qubit quantum error detect…
We give a complete, hierarchic classification for arbitrary multi-qubit mixed states based on the separability properties of certain partitions. We introduce a family of N-qubit states to which any arbitrary state can be depolarized. This family can be viewed as the generalization of Werner states to multi-qubit systems. We fully classify those states with respect to their separability and distillability properties. This provides sufficient conditions for nonseparability and distillability for arbitrary states.
We consider the problem of minimum-error quantum state discrimination for single-qubit mixed states. We present a method which uses the Helstrom conditions constructively and analytically; this algebraic approach is complementary to existing geometric methods, and solves the problem for any number of arbitrary signal states with arbitrary prior probabilities.
We prove that universal quantum computation is possible using only (i) the physically natural measurement on two qubits which distinguishes the singlet from the triplet subspace, and (ii) qubits prepared in almost any three different (potentially highly mixed) states. In some sense this measurement is a `more universal' dynamical element than a universal 2-qubit unitary gate, since the latter must be supplemented by measurement. Because of the rotational invariance of the measurement used, our scheme is robust to collective decoherence in a manner very different to previous proposals - in effe…
We investigate the quantum correlation in a qutrit-qubit mixed spin chain based on measurement-induced disturbance (MID) [S. Luo, Phys. Rev. A, 77, (2008) 022301]. We also compare MID and thermal entanglement measured by negativity and illustrate their different characteristics.
A cogent theory of collective multipole-like quantum correlations in symmetric multiqubit states is presented by employing SO(3) irreducible spherical tensor representation. An arbitrary bipartite division of this system leads to a family of inequalities to detect entanglement involving averages of these tensors expressed in terms of the total system angular momentum operator. Implications of this theory to the quantum nature of multipole-like correlations of all orders in the Dicke states are deduced. A selected set of examples illustrate these collective tests. Such tests detect entanglement…
In this work we study numerically and analytically the interaction of two qubits in a one-dimensional waveguide, as mediated by the photons that propagate through the guide. We develop strategies to assert the Markovianity of the problem, the effective qubit-qubit interactions and their individual and collective spontaneous emission. We prove the existence of collective Lamb-shifts that affect the qubit-qubit interactions and the dependency of coherent and incoherent interactions on the qubit separation. We also develop the scattering theory associated to these models and prove single photon s…
We analyze the estimation of a qubit pure state by means of local measurements on $N$ identical copies and compare its averaged fidelity for an isotropic prior probability distribution to the absolute upper bound given by collective measurements. We discuss two situations: the first one, where the state is restricted to lie on the equator of the Bloch sphere, is formally equivalent to phase estimation; the second one, where there is no constrain on the state, can also be regarded as the estimation of a direction in space using a quantum arrow made out of $N$ parallel spins. We discuss various …
We revisit qubit-qutrit quantum systems under collective dephasing and answer some of the questions which have not been asked and addressed so far in the literature. In particular, we examine the possibilities of non-trivial phenomena of {\it time-invariant} entanglement and {\it freezing} dynamics of entanglement for this dimension of Hilbert space. Interestingly, we find that for qubit-qutrit systems both of these peculiar features coexist, that is, we observe not only time-invariant entanglement for certain quantum states but we find also find evidence that many quantum states freeze their …
Two-qubit states occupy a large and relatively unexplored Hilbert space. Such states can be succinctly characterized by their degree of entanglement and purity. In this letter we investigate entangled mixed states and present a class of states that have the maximum amount of entanglement for a given linear entropy.
We analyze a controllable generation of maximally entangled mixed states of a circuit containing two-coupled superconducting charge qubits. Each qubit is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Illustrative variational calculations were performed to demonstrate the effect on the two-qubits entanglement. At sufficiently deviation between the Josephson energies of the qubits and/or strong coupling regime, maximally entangled mixed states at certain instances of time is synthesized. We show that entanglement has an interesting subsequent time ev…
We experimentally demonstrate optimal entanglement distillation from two forms of two-qubit mixed states under local filtering operations according to the constructive method intruduced by F. Verstraete et al. [Phys. Rev. A 64, 010101(R) (2001)]. In principle, our set-up can be easily applied to distilling entanglement from arbitrary two-qubit partially mixed states. We also test the violation of the Clauser-Horne-Shinmony-Holt (CHSH) inequality for the distilled state from the first form of mixed state to show its "hidden non-locality".
Most studies of collective dephasing for bipartite as well as multipartite quantum systems focus on a very specific orientation of magnetic field, that is, z-orientation. However, in practical situations, there are always small fluctuations in stochastic field and it is necessary that more general orientations of fields should be considered. We extend this problem to qubit-qutrit systems and study correlation dynamics for entanglement and local quantum uncertainty for some specific quantum states. We find that certain quantum states exhibit freezing dynamics both for entanglement and local qua…
We consider the mixed states of the bipartite quantum system with the first party a qubit and the second a qutrit. The group of local unitary transformations of the system, ignoring the overall phase factor, is the direct product G of SU(2) and SU(3). We compute the simply graded Poincare series of the algebra of G-invariant polynomial functions on the set of mixed states of the system, and construct several low degree invariants.
We propose novel mixed states in two qubits, ``maximally entangled mixed states'', which have a property that the amount of entanglement of these states cannot be increased further by applying any unitary operations. The property is proven when the rank of the states is less than 4, and confirmed numerically in the other general cases. The corresponding entanglement of formation specified by its eigenvalues gives an upper bound of that for density matrices with same eigenvalues.
The paper [Phys. Lett. A 373 (2009) 3610] by D.-C. Li analyzes the transformation between two-qubit mixed states by local operations and classical communication. We show that the proof of the main theorem, Theorem 2.6 in [Phys. Lett. A 373 (2009) 3610] is not complete. Therefore the generalization of Nielsen's theorem to mixed states still remains an open problem.
We investigate the entanglement, CHSH nonlocality, fully entangled fraction and symmetric extendibility of two-qubit states that have a single maximally mixed marginal. Within this set of states, the steering ellipsoid formalism has recently highlighted an interesting family of so-called 'maximally obese' states. These are found to have extremal quantum correlation properties that are significant in the steering ellipsoid picture and for the study of two-qubit states in general.
Majorana Representation (MR) of symmetric $ N $-qubit pure states has been used successfully in entanglement classification. Generalization of this has been a long standing open problem due to the difficulties faced in the construction of a Majorana like geometric representation for symmetric mixed state. We have overcome this problem by developing a method of classifying local unitary (LU) equivalent classes of symmetric $N$-qubit mixed states based on the geometrical Multiaxial Representation (MAR) of the density matrix. In addition to the two parameters defined for the entanglement classifi…
We theoretically investigate resonant dipole-dipole interaction (RDDI) between artificial atoms in a 1D geometry, implemented by N transmon qubits coupled through a transmission line. Similarly to the atomic cases, RDDI comes from exchange of virtual photons of the unexcited modes, and causes the so-called collective Lamb shift (CLS). To probe the shift, we effectively set one end of the transmission line as a mirror, and examine the reflection spectrum of the probe field from the other end. Our calculation shows that when a qubit is placed at the node of the standing wave formed by the incide…
Collective decoherence is possible if the departure between quantum bits is smaller than the effective wave length of the noise field. Collectivity in the decoherence helps us to devise more efficient quantum codes. We present a class of optimal quantum codes for preventing collective amplitude damping to a reservoir at zero temperature. It is shown that two qubits are enough to protect one bit quantum information, and approximately $L+ 1/2 \log_2((πL)/2)$ qubits are enough to protect $L$ qubit information when $L$ is large. For preventing collective amplitude damping, these codes are much mor…
We introduce a classification of mixed three-qubit states, in which we define the classes of separable, biseparable, W- and GHZ-states. These classes are successively embedded into each other. We show that contrary to pure W-type states, the mixed W-class is not of measure zero. We construct witness operators that detect the class of a mixed state. We discuss the conjecture that all entangled states with positive partial transpose (PPTES) belong to the W-class. Finally, we present a new family of PPTES "edge" states with maximal ranks.
A simple scheme to prepare an entanglement between two separated qubits from a given mixed state is proposed. A single qubit (entanglement mediator) is repeatedly made to interact locally and consecutively with the two qubits through rotating-wave couplings and is then measured. It is shown that we need to repeat this kind of process only three times to establish a maximally entangled state directly from an arbitrary initial mixed state, with no need to prepare the state of the qubits in advance or to rearrange the setup step by step. Furthermore, the maximum yield realizable with this scheme …
