We develop a general scheme for an analysis of macroscopic qudit systems: a) introduce a set of collective observables, which characterizes the macroscopic properties of qudits in an optimal way; b) construct projected $\tilde{Q}$-functions for $N$ qudit systems, containing full macroscopic information; c) propose a collective tomographic protocol both for a general and symmetric $N$-qudit states. The example of $N$-qutrit is analyzed in details and compared to $N$-qubit case
Unitary control of qudits can improve the collective spin squeezing of an atomic ensemble. Preparing the atoms in a state with large quantum fluctuations in magnetization strengthens the entangling Faraday interaction. The resulting increase in interatomic entanglement can be converted into metrologically useful spin squeezing. Further control can squeeze the internal atomic spin without compromising entanglement, providing an overall multiplicative factor in the collective squeezing. We model the effects of optical pumping and study the tradeoffs between enhanced entanglement and decoherence.…
We study noiseless subsystems on collective rotation channels of qudits, i.e., quantum channels with operators in the set ${\mathcal E}(d,n) = \{ U^{\otimes n}: U \in {\mathrm{SU}}(d)\}.$ This is done by analyzing the decomposition of the algebra ${\mathcal A}(d,n)$ generated by ${\mathcal E}(d,n)$. We summarize the results for the channels on qubits ($d=2$), and obtain the maximum dimension of the noiseless subsystem that can be used as the quantum error correction code for the channel. Then we extend our results to general $d$. In particular, it is shown that the code rate, i.e., the number …
We give a full explanation of the noiseless subsystem that protects a single-qubit against collective errors and the corresponding recursive scheme described by C.-K. Li et. al. [Phys. Rev. A 84, 044301 (2011)] from a representation theory point of view. Furthermore, we extend the construction to qudits under the influence of collective SU($d$) errors. We find that under this recursive scheme, the asymptotic encoding rate is $1/d$.
We introduce a new quantum communication protocol for the transmission of quantum information under collective noise. Our protocol utilizes a decoherence-free subspace in such a way that an optimal asymptotic transmission rate is achieved, while at the same time encoding and decoding operations can be efficiently implemented. The encoding and decoding circuit requires a number of elementary gates that scale linearly with the number of transmitted qudits, m. The logical depth of our encoding and decoding operations is constant and depends only on the channel in question. For channels described …
We address the trade-off between information gain and state disturbance in measurement performed on qudit systems and devise a class of optimal measurement schemes that saturate the ultimate bound imposed by quantum mechanics to estimation and transmission fidelities. The schemes are minimal, i.e. they involve a single additional probe qudit, and optimal, i.e. they provide the maximum amount of information compatible with a given level of disturbance. The performances of optimal single-user schemes in extracting information by sequential measurements in a N -user transmission line are also inv…
Collective spin operators for symmetric multi-quDit (namely, identical $D$-level atom) systems generate a U$(D)$ symmetry. We explore generalizations to arbitrary $D$ of SU(2)-spin coherent states and their adaptation to parity (multicomponent Schrödinger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric $N$-quDit states, expressed in the last two cases in terms of collective U$(D)$-spin operator expectation values. Then we evaluate level and particle entanglement for symmetric multi-quDit states with linear and …
We investigate entanglement between collective operators of two wave-packets of finite spectral bandwidth, in two different approximations of the Multimode Parametric-Down Conversion (MPDC) process: the pairwise and the one-to-all interaction patterns. For collective operators we choose the macroscopic amplitudes of each wave-packet defined by the Fourier Transform of their microscopic mode amplitudes. This approach intends, to respond to realistic experimental conditions, where measurements apparatuses may not resolve single microscopic mode amplitudes but rather the collective amplitude of t…
We introduce a discrete Q-function of N qubit system projected into the space of symmetric measurements as a tool for analyzing general properties of quantum systems in the macroscopic limit. For known states the projected Q-function helps to visualize the results of collective measurements, and for unknown states it can be approximately reconstructed by measuring lowest moments of the of collective variables.
The hybridization of distinct quantum systems is now seen as an effective way to engineer the properties of an entire system leading to applications in quantum metamaterials, quantum simulation, and quantum metrology. One well known example is superconducting circuits coupled to ensembles of microscopic natural atoms. In such cases, the properties of the individual atom are intrinsic, and so are unchangeable. However, current technology allows us to fabricate large ensembles of macroscopic artificial atoms such as superconducting flux qubits, where we can really tailor and control the properti…
We study how the decoherence of macroscopic objects is induced intrisinically by relativistic effect. With the degree of freedom of center of mass (CM) characterizing the collective quantum state of a macroscopic object (MO), it is found that a MO consisting of N particles can decohere with time scale no more than sqrt(N). Here, the special relativity can induce the coupling of the collective motion mode and the relative motion modes in an order of 1/c2, which intrinsically results in the above minimum decoherence.
What can we learn about entanglement between individual particles in macroscopic samples by observing only the collective properties of the ensembles? Using only a few experimentally feasible collective properties, we establish an entanglement measure between two samples of spin-1/2 particles (as representatives of two-dimensional quantum systems). This is a tight lower bound for the average entanglement between all pairs of spins in general and is equal to the average entanglement for a certain class of systems. We compute the entanglement measures for explicit examples and show how to genera…
We consider macroscopic correlations in a bipartite system consisting of 2N particles described by a generalised probabilistic theory. In particular, we discuss a case of N PR-boxes shared between two parties. We characterise macroscopic measurements as collective measurements of the same property on all the boxes in the same region. Such measurements are assumed to reveal only the average value of the measured collective property. We show that for two measurements per observer and N greater or equal 2 there always exist a joint probability distribution explaining all observable data and there…
We report on the observation of quantum interference of a collective single spin excitation with a spin ensemble of $N_{\text{atom}} =10^5$ atoms. Detection of a single photon scattered from the atoms creates the single spin excitation, a Fock state embedded in the collective spin of the ensemble. The state of the atomic ensemble is then detected by tomography via a quantum non-demolition measurement of the collective spin. A macroscopic difference of the order of $\sqrt{N_{\text{atom}}}$ in the marginal distribution of the collective spin state arises from the interference between the single …
I present a general scheme through which the evidence of a superposition involving distinct classical-like states of a macroscopic system can be probed. The scheme relies on a qubit being coupled to a macroscopic harmonic oscillator in such a way that it can be used to both prepare and probe a macroscopic superposition. Two potentially realizable implementations, one with a flux qubit coupled to a LC circuit, and the other with an ion-trap qubit coupled to the collective motion of several ions, are proposed.
Macroscopic Spin GHZ States with a Levitated Ferromagnet
National University of Singapore（新加坡国立大学）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；A*STAR Quantum Innovation Centre (Q.InC), Agency for Science, Technology and Research (A*STAR)（A*STAR量子创新中心（Q.InC），科学技术研究局（A*STAR））
We report the observation of strong coupling of a macroscopic ensemble of ~10^{16} Fe_8 molecular nanomagnets to the resonant mode of a microwave cavity. We use millimeter-wave spectroscopy to measure the splitting of the system's resonant frequency induced by the coupling between the spins and the cavity mode. The magnitude of this splitting is found to scale with Sqrt[N], where N is the number of collectively coupled spins. We control N by changing the system's temperature and, thereby, the populations of the relevant spin energy levels. Strong coupling is observed for two distinct transitio…
We consider an open quantum system of N not directly interacting spins (qubits) in contact with both local and collective thermal environments. The qubit-environment interactions are energy conserving. We trace out the variables of the thermal environments and N-2 qubits to obtain the time-dependent reduced density matrix for two arbitrary qubits. We numerically simulate the reduced dynamics and the creation of entanglement (concurrence) as a function of the parameters of the thermal environments and the number of qubits, N. Our results demonstrate that the two-qubit entanglement generally dec…
We demonstrate that for an ensemble of qudits, subjected to collective decoherence in the form of perfectly correlated random SU(d) unitaries, quantum superpositions stored in the decoherence free subspace are fully immune against the removal of one particle. This provides a feasible scheme to protect quantum information encoded in the polarization state of a sequence of photons against both collective depolarization and one photon loss, and can be demonstrated with photon quadruplets using currently available technology.
We demonstrate through an exactly solvable model that collective coupling to any thermal bath induces effectively nonlinear couplings in a quantum many-body (multi-spin) system. The resulting evolution can drive an uncorrelated large-spin system with high probability into a macroscopic quantum-superposition state. We discuss possible experimental realizations.
