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 generated by a local 2-outcome POVM and study under what conditions it is deterministic, i.e., both outcomes belong to the same orbit. We prove that for real states it is always possible to find such a POVM and we discuss analytical and numerical evidence that suggests that this result also holds for complex states. We study the transformation generated in the space of orbits when one or more parties apply several deterministic POVMs in succession and use these results to give a complete characterization of the real states that can be obtained from the GHZ state with probability 1.
We demonstrate a robust implementation of a deterministic linear-optical Controlled-NOT (CNOT) gate for single-photon two-qubit quantum logic. A polarization Sagnac interferometer with an embedded 45$^{\circ}$-oriented dove prism is used to enable the polarization control qubit to act on the momentum (spatial) target qubit of the same photon. The CNOT gate requires no active stabilization because the two spatial modes share a common path, and it is used to entangle the polarization and momentum qubits.
We propose deterministic realizations of nonlinear phase gates by repeating non-commuting Rabi interactions feasible between a harmonic oscillator and {\em only} a single two-level ancillary qubit. We show explicitly that the key nonclassical features of the states after the ideal cubic phase gate and the quartic phase gate are reproduced faithfully by the engineered operators. This theoretical proposal completes the universal set of operators in continuous variable quantum computation.
We present a simple quantum circuit that allows for the universal and deterministic manipulation of the quantum state of confined harmonic oscillators. The scheme is based on the selective interactions of the referred oscillator with an auxiliary three-level system and a classical external driving source, and enables any unitary operations on Fock states, two-by-two. One circuit is equivalent to a single qubit unitary logical gate on Fock states qubits. Sequences of similar protocols allow for complete, deterministic and state-independent manipulation of the harmonic oscillator quantum state.
We propose the implementation of a deterministic Hadamard gate for logical photonic qubits encoded in superpositions of coherent states of a harmonic oscillator. The proposed scheme builds on a recently introduced set of conditional operations in the strong dispersive regime of circuit QED [Z. Leghtas et al. Phys. Rev. A. {\bf 87} (2013)]. We further propose an architecture for coupling two such logical qubits and provide a universal set of deterministic quantum gates. Based on parameter values taken from the current state of the art, we give estimates for the achievable gate fidelities accoun…
A necessary and sufficient condition of the possibility of a deterministic local operations and classical communication (LOCC) transformation of three-qubit pure states is given. The condition shows that the three-qubit pure states are a partially ordered set parametrized by five well-known entanglement parameters and a novel parameter; the five are the concurrences $C_{AB}$, $C_{AC}$, $C_{BC}$, the tangle $τ_{ABC}$ and the fifth parameter $J_{5}$ of Ref. [18], while the other new one is the entanglement charge $Q_{e}$. The order of the partially ordered set is defined by the possibility of a …
We propose a scheme to realize deterministic quantum teleportation using linear optics and hybrid qubits. It enables one to efficiently perform teleportation and universal linear-optical gate operations in a simple and near-deterministic manner using all-optical hybrid entanglement as off-line resources. Our analysis shows that our new approach can outperforms major previous ones when considering both the resource requirements and fault tolerance limits.
Here, we numerically simulate probabilistic elementary entangling operations between rail-encoded photons for the purpose of scalable universal quantum computation or communication. We propose grouping logical qubits into single-photon blocks wherein single-qubit rotations and the CNOT gate are fully deterministic and simple to implement. Inter-block communication is then allowed through said probabilistic entangling operations. We find a promising trend in the increasing probability of successful inter-block communication as we increase the number of optical modes operated on by our elementar…
This paper studies the stochastic differential equation (SDE) associated to a two-level quantum system (qubit) subject to Hamiltonian evolution as well as unmonitored and monitored decoherence channels. The latter imply a stochastic evolution of the quantum state (density operator), whose associated probability distribution we characterize. We first show that for two sets of typical experimental settings, corresponding either to weak quantum non demolition measurements or to weak fluorescence measurements, the three Bloch coordinates of the qubit remain confined to a deterministically evolving…
We propose a scheme to construct a deterministic CNOT gate on static electron-spin qubits, allowing for deterministic scalable quantum computing in solid-state systems.The excess electron confined in a charged quantum dot inside a double-sided optical microcavity represents the qubit, and the single photons play a medium role. Moreover, our device can work in both the weak coupling and the strong coupling regimes, but high fidelities are achieved only when the ratio of the side leakage to the cavity loss is low. Finally, we assess the feasibility of this device and show it can be implemented w…
We introduce a new gate that transfers an arbitrary state of a qubit into a superposition of two quasi-orthogonal coherent states of a cavity mode, with opposite phases. This qcMAP gate is based on conditional qubit and cavity operations exploiting the energy level dispersive shifts, in the regime where they are much stronger than the cavity and qubit linewidths. The generation of multi-component superpositions of quasi-orthogonal coherent states, non-local entangled states of two resonators and multi-qubit GHZ states can be efficiently achieved by this gate.
We propose a simple interaction protocol to be implemented on a scalable quantum network, in which the quantum nodes consist of qubit systems confined in cavities. The nodes are deterministically coupled by transmission and reflection of a single photon, which is disentangled from the qubits at the end of the coupling operation. This single photon can generate an entangling controlled phase (C-PHASE) gate between any selected number of qubits in the network. Our multi-qubit gate reaches a much higher fidelity compared to schemes concatenating one-qubit and two-qubit gates; thus it forms an eff…
We study the possible limitations and sources of decoherence in the scheme for the deterministic generation of polarization-entangled photons, recently proposed by Gheri et al. [K. M. Gheri et al., Phys. Rev. A 58, R2627 (1998)], based on an appropriately driven single atom trapped within an optical cavity. We consider in particular the effects of laser intensity fluctuations, photon losses, and atomic motion.
Quantum entanglement is a key resource for quantum computation and quantum communication \cite{Nielsen2010}. Scaling to large quantum communication or computation networks further requires the deterministic generation of multi-qubit entanglement \cite{Gottesman1999,Duan2001,Jiang2007}. The deterministic entanglement of two remote qubits has recently been demonstrated with microwave photons \cite{Kurpiers2018,Axline2018,Campagne2018,Leung2019,Zhong2019}, optical photons \cite{Humphreys2018} and surface acoustic wave phonons \cite{Bienfait2019}. However, the deterministic generation and transmis…
We define an entanglement measure, called the partial tangle, which represents the residual two-qubit entanglement of a three-qubit pure state. By its explicit calculations for three-qubit pure states, we show that the partial tangle is closely related to the faithfulness of a teleportation scheme over a three-qubit pure state.
A scheme for the deterministic joint remote preparation of a four- qubit cluster-type state using only two Greenberger-Horne-Zeilinger (GHZ) states as quantum channels is presented. In this scheme, the first sender per- forms a two-qubit projective measurement according to the real coefficient of the desired state. Then, the other sender utilizes the measurement result and the complex coefficient to perform another projective measurement. To obtain the desired state, the receiver applies appropriate unitary operations to his/her own two qubits and two CNOT operations to the two ancillary ones.…
We show that deterministic quantum computing with one qubit (DQC1) can be experimentally implemented with a spatial light modulator, using the polarization and the transverse spatial degrees of freedom of light. The scheme allows the computation of the trace of a high dimension matrix, being limited by the resolution of the modulator panel, and the technical imperfections. In order to illustrate the method, we compute the normalized trace of unitary matrices, and implement the Deutsch-Jozsa algorithm. The largest matrix that can be manipulated with our set-up is 1080$\times$1920, which is able…
We show that a necessary and sufficient condition for a set of $n$ one-qubit mixed states to be the reduced states of a pure $n$-qubit state is that their smaller eigenvalues should satisfy polygon inequalities: no one of them can exceed the sum of the others.
We propose a deterministic remote state preparation (RSP) scheme for preparing an arbitrary (including pure and mixed) qubit, where a partially entangled state and finite classical communication are used. To our knowledge, our scheme is the first RSP scheme that fits into this category. One other RSP scheme proposed by Berry shares close features, but can only be used to prepare an arbitrary pure qubit. Even so, our scheme saves classical communication by approximate 1 bit per prepared qubit under equal conditions. When using a maximally entangled state, the classical communication for our sch…
The Majorana representation of symmetric N-qubit states is employed here to investigate how correlation information of the whole pure symmetric state gets imprinted in its parts. It is shown that reduced states of (N - 1) qubits uniquely specify the entire class of pure N qubit states containing two distinct spinors.
We present a mapping which associates pure N-qubit states with a polynomial. The roots of the polynomial characterize the state completely. Using the properties of the polynomial we construct a way to determine the separability and the number of unentangled qubits of pure N-qubit states.
First, I show explicitly a scheme to {\it faithfully} and {\it deterministically} teleport an arbitrary 2-qubit state from Alice to Bob. In this scheme two same Bell states are sufficient for use. Bob can recover the 2-qubit state by performing at most 4 single-qubit unitary operations conditioned on Alice's 4-bit classical public message corresponding to her two Bell-state measurement outcomes. Then I generalize the 2-qubit teleportation scheme to an aribitrary $N(N\ge 3)$-qubit state teleportation case by using $N$ same Bell states. In the generalized scheme, Alice only needs to identify $N$…
We propose a scheme for the detection of qubit-environment entanglement at time $τ$ which requires only operations and measurements on the qubit, all within reach of current experimental state-of-the-art. The scheme works for any type of interaction which leads to pure dephasing of the qubit as long as the initial qubit state is pure. It becomes particularly simple when one of the qubit states is neutral with respect to the environment, such as in case of the most common choice of the NV center spin qubit or for excitonic charge qubits, when the environment is initially at thermal equilibrium.
We have implemented a universal quantum logic gate between qubits stored in the spin state of a pair of trapped calcium 40 ions. An initial product state was driven to a maximally entangled state deterministically, with 83% fidelity. We present a general approach to quantum state tomography which achieves good robustness to experimental noise and drift, and use it to measure the spin state of the ions. We find the entanglement of formation is 0.54.
We propose the scheme implementing partial deterministic non-demolition Bell measurement. When it is used in quantum teleportation the information about an unknown input state is optimally distributed among three outgoing qubits. The optimality means that the output fidelities saturate the asymmetric cloning inequality. The flow of information among the qubits is controlled by the preparation of a pair of ancillary qubits used in the Bell measurement. It is also demonstrated that the measurement is optimal two-qubit operation in the sense of the trade-off between the state disturbance and the …
We consider a four-state pure bipartite system consisting of four qubits shared among two parties using the same Schmidt basis, two qubits per party. In some cases, transformation between two known pure states may not be possible using LOCC transformations but may be possible with the addition of a two-qubit catalyst. We provide a necessary and sufficient condition for this to occur.
We propose a deterministic and scalable scheme to construct a two-qubit controlled-NOT (CNOT) gate and realize entanglement swapping between photonic qubits using a quantum-dot (QD) spin in a double-sided optical microcavity. The scheme is based on spin selective photon reflection from the cavity and can be achieved in a nondestructive and heralded way. We assess the feasibility of the scheme and show that the scheme can work in both the weak coupling and the strong coupling regimes. The scheme opens promising perspectives for long-distance photonic quantum communication and distributed quantu…
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 study the trade-off relations on the maximal violation of CHSH tests for the multi-qubit pure states. Firstly, according to the classification of 3-qubit pure states under stochastic local operations and classical communication, four trade-off relations on CHSH tests are provided. The process of proof also provides the method of calculating its exact value. Then we consider the multi-qubit quantum systems, and we prove that the conjecture in [PRA 94, 042105 (2016)] is right for some special 4-qubit or more qubit pure states.
We propose the use of Rydberg interactions and ensembles of cold atoms in mixed state for the implementation of a protocol for deterministic quantum computation with one quantum bit (DQC1) that can be readily operated in high dimensional Hilbert spaces. We propose an experimental test for the scalability of the protocol and to study the physics of discord. Furthermore we develop a scheme to add control to non-trivial unitaries that will enable the study of many-body physics with ensembles in mixed states.
We propose two deterministic secure quantum communication (DSQC) protocols employing three-qubit GHZ-like states and five-qubit Brown states as quantum channels for secure transmission of information in units of two bits and three bits using multipartite teleportation schemes developed here. In these schemes, the sender's capability in selecting quantum channels and the measuring bases leads to improved qubit efficiency of the protocols.
We study a relation between the concurrence of assistance and the Mermin inequality on three-qubit pure states. We find that if a given three-qubit pure state has the minimal concurrence of assistance greater than 1/2 then the state violates some Mermin inequality.
We propose a class of preparation schemes for orbital angular momentum and polarisation qubits carried by single photons or classical states of light based on coherent feedback control by an ancillary degree of freedom of light. The preparation methods use linear optics and include the transcription of an arbitrary polarisation state onto a two-level OAM system (swap) for arbitrary OAM values plus/minus "l" within a light beam, i.e. without spatial interferometer. The preparations can be carried out with unit efficiency independent from the potentially unknown initial state of the system. More…
Mixed state quantum computation can perform certain tasks which are believed to be efficiently intractable on a classical computer. For a specific model of mixed state quantum computation, namely, {\it deterministic quantum computation with a single qubit} (DQC1), recent investigations suggest that quantum correlations other than entanglement might be responsible for the power of DQC1 model. However, strictly speaking, the role of entanglement in this model of computation was not entirely clear. We provide conclusive evidence that there are instances where quantum entanglement is not present i…
We propose a way for transferring Greenberger-Horne-Zeilinger (GHZ) entangled states from $n$ qubits in one cavity onto another $n$ qubits in the other cavity. It is shown that $n$-qubit GHZ states $α\left\vert 00...0\right\rangle +β\left\vert 11...1\right\rangle $ with arbitrary degree of entanglement can be transferred deterministically. Both of the GHZ state transfer and the operation time are not dependent on the number of qubits, and there is no need of measurement. This proposal is quite general and can be applied to accomplish the same task for a wide range of physical qubits. Furthermo…
Based on an EPR pair of qubits and allowing asymptotically secure key distribution, a secure communication protocol is presented. Bob sends either of the EPR pair qubits to Alice. Alice receives the travel qubit. Then she can encode classical information by local unitary operations on this travel qubit. Alice send the qubit back to Bob. Bob can get Alice's information by measurement on the two photons in Bell operator basis. If Eve in line, she has no access to Bob's home qubit. All her operations are restricted to the travel qubit. In order to find out which opeartion Alice performs, Eve's op…
We present a decomposition technique that uses non-deterministic circuits to approximate an arbitrary single-qubit unitary to within distance $ε$ and requires significantly fewer non-Clifford gates than existing techniques. We develop "Repeat-Until-Success" (RUS) circuits and characterize unitaries that can be exactly represented as an RUS circuit. Our RUS circuits operate by conditioning on a given measurement outcome and using only a small number of non-Clifford gates and ancilla qubits. We construct an algorithm based on RUS circuits that approximates an arbitrary single-qubit $Z$-axis rota…
The loss of qubits - the elementary carriers of quantum information - poses one of the fundamental obstacles towards large-scale and fault-tolerant quantum information processors. In this work, we experimentally demonstrate a complete toolbox and the implementation of a full cycle of qubit loss detection and correction on a minimal instance of a topological surface code. This includes a quantum non-demolition measurement of a qubit loss event that conditionally triggers a restoration procedure, mapping the logical qubit onto a new encoding on the remaining qubits. The demonstrated methods, imp…
We propose a method to detect genuine quantum correlation for multi-qubit pure states. We then derive a complementarity relations for pure quantum states of N qubits. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. On the other hand, it is known that the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters is in general not true for multiqubit states. Inducing from the CKW equality, we find a proper form of entanglement monogamy equality for arbitrary quantum states. The total quantum correlation of qubit k with the remainin…
We propose a criterion for the detection of genuine entanglement of pure multiqubit states. To this aim, we define an operator called the losing one qubit operator, which is different from the reduced density operator. The states obtained from a multiqubit state by applying the losing one qubit operator are referred to as its projected states. We show that all of the projected states of a pure product n-qubit state are pure product states provided that it cannot be written as a product of a single qubit state and a genuinely entangled (n-1)-qubit state. We also show that a pure n-qubit state i…
We introduce a set of Bell inequalities for a three-qubit system. Each inequality within this set is violated by all generalized GHZ states. More entangled a generalized GHZ state is, more will be the violation. This establishes a relation between nonlocality and entanglement for this class of states. Certain inequalities within this set are violated by pure biseparable states. We also provide numerical evidence that at least one of these Bell inequalities is violated by a pure genuinely entangled state. These Bell inequalities can distinguish between separable, biseparable and genuinely entan…
In this work we study the entanglement of pure fourpartite of qubit states. The analysis is realized through the comparison between two different entanglement measures: the Groverian entanglement measure and the residual entanglement measured with negativities. Furthemore, we discuss some applications of four-way entangled fourpartite states.
The deterministic quantum computing with one qubit (DQC1) is a mixed-state quantum computation algorithm that evaluates the normalized trace of a unitary matrix and is more powerful than the classical counterpart. We find that the normalized trace of the unitary matrix can be directly described by the entangling power of the quantum circuit of the DQC1, so the nontrivial DQC1 is always accompanied with the non-vanishing entangling power. In addition, it is shown that the entangling power also determines the intrinsic complexity of this quantum computation algorithm, i.e., the larger entangling…
We construct a class of algebraic invariants for N-qubit pure states based on bipartite decompositions of the system. We show that they are entanglement monotones, and that they differ from the well know linear entropies of the sub-systems. They therefore capture new information on the non-local properties of multipartite systems.
We present an efficient scheme for the controlled generation of pure two-qubit states possessing {\em any} desired degree of entanglement and a {\em prescribed} symmetry in two cavity QED based systems, namely, cold trapped ions and flying atoms. This is achieved via on-resonance ion/atom-cavity couplings which are time-dependent and asymmetric, leading to a trapping vacuum state condition which does not arise for identical couplings. A duality in the role of the coupling ratio yields states with a given concurrence but opposing symmetries. The experimental feasibility of the proposed scheme i…
In this paper we analyze the canonical forms into which any pure three-qubit state can be cast. The minimal forms, i.e. the ones with the minimal number of product states built from local bases, are also presented and lead to a complete classification of pure three-qubit states. This classification is related to the values of the polynomial invariants under local unitary transformations by a one-to-one correspondence.
We present an efficient method to generate a Greenberger-Horne-Zeilinger (GHZ) entangled state of three cat-state qubits (cqubits) via circuit QED. The GHZ state is prepared with three microwave cavities coupled to a superconducting transmon qutrit. Because the qutrit remains in the ground state during the operation, decoherence caused by the energy relaxation and dephasing of the qutrit is greatly suppressed. The GHZ state is created deterministically because no measurement is involved. Numerical simulations show that high-fidelity generation of a three-cqubit GHZ state is feasible with prese…
A new form of local unitary (LU) transformation invariant is given for multi-qubit states . The general relation between tangle and the LU transformation invariant of pure three and four-qubit states is given. We find that the tangle actually is a special LU transformation invariant. Furthermore, the tangle is a important local unitary (LU) transformation invariant which is a nessesary condition for maximally multi-qubit entangled state.
We propose a qubit-environment entanglement measure which is tailored for evolutions that lead to pure dephasing of the qubit, such as are abundant in solid state scenarios. The measure can be calculated directly form the density matrix without minimization of any kind. In fact it does not require the knowledge of the full density matrix, and it is enough to know the initial qubit state and the states of the environment conditional on qubit pointer states. This yields a computational advantage over standard entanglement measures, which becomes large when there are no correlations between envir…
The necessary and sufficient conditions for minimization of the generalized rate error for discriminating among $N$ pure qubit states are reformulated in terms of Bloch vectors representing the states. For the direct optimization problem an algorithmic solution to these conditions is indicated. A solution to the inverse optimization problem is given. General results are widely illustrated by particular cases of equiprobable states and $N=2,3,4$ pure qubit states given with different prior probabilities.
Quantum correlations have been pointed out as the most likely source of the speed-up in quantum computation. Here we analyzed the presence of quantum correlations in the implementation of Deutsch-Jozsa algorithm running in the DQC1 and DQCp models of quantum computing. For some balanced functions, the qubits in DQC1 model are quantum correlated just in the intermediate steps of the algorithm for a given decomposition into one and two qubits gates. In the DQCp model the final state is strongly quantum correlated for some balanced functions, so that the pairwise entanglement between blocks scale…
