Given a single copy of a mixed state of the form ρ=λρ_1+(1-λ)ρ_2, what is the optimal measurement to estimate the parameter λ, if ρ_1 and ρ_2 are known? We present a general strategy to obtain the optimal measurements employing a Bayesian estimator. The measurements are chosen to minimize the deviation between the estimated- and the true value of λ. We explicitly determine the optimal measurements for a general two-dimensional system and for important higher dimensional cases.
We demonstrate an optical receiver that achieves the quantum Chernoff bound for discriminating coherent states from thermal states in the multi-copy scenario. In contrast, we find that repeated use of the receiver approaching the Helstrom bound for single-copy measurement is sub-optimal in this multi-copy case. Furthermore, for a large class of multi-copy discrimination tasks between a pure and a mixed state, we prove that any Helstrom-bound achieving single-copy receiver is suboptimal by a factor of at least two in error-probability exponent compared to the multi-copy quantum Chernoff bound. …
In this note we consider the problem of preparing a {\em single} copy of an arbitrary two-qubit mixed state $ρ$ starting from an entangled pure state $ψ$ and using only local operations assisted with classical communication. We present an analytical expression for the minimal amount of pure state entanglement required, and describe the corresponding local strategy. We also examine optimal probabilistic generalizations of the previous process.
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.
We give the optimal bounds on the phase-estimation precision for mixed Gaussian states in the single-copy and many-copy regimes. Specifically, we focus on displaced thermal and squeezed thermal states. We find that while for displaced thermal states an increase in temperature reduces the estimation fidelity, for squeezed thermal states a larger temperature can enhance the estimation fidelity. The many-copy optimal bounds are compared with the minimum variance achieved by three important single-shot measurement strategies. We show that the single-copy canonical phase measurement does not always…
We present a detailed theoretical analysis for the distillation of one copy of a mixed two-mode continuous-variable entangled state using beamsplitters and coherent photon-detection techniques, including conventional on-off detectors and photon number resolving detectors. The initial Gaussian mixed-entangled states are generated by transmitting a two-mode squeezed state through a lossy bosonic channel, corresponding to the primary source of errors in current approaches to optical quantum communication. We provide explicit formulas to calculate the entanglement in terms of logarithmic negativit…
A basic building block of many quantum algorithms is the Phase Estimation algorithm (PEA). It estimates an eigenphase $ϕ$ of a unitary operator $U$ using a copy of the corresponding eigenstate $|ϕ\rangle$. Suppose, in place of $|ϕ\rangle$, we have a copy of an approximate eigenstate $|ψ\rangle$ whose overlap magnitude with $|ϕ\rangle$ is at least $\sqrt{2/3}$. Then PEA fails with a constant probability. However, using multiple copies of $|ψ\rangle$, the failure probaility can be made to decrease exponentially with the number of copies. In this paper, we show that as long as we can perform a se…
We derive the optimal universal broadcasting for mixed states of qubits. We show that the nobroadcasting theorem cannot be generalized to more than a single input copy. Moreover, for four or more input copies it is even possible to purify the input states while broadcasting. We name such purifying broadcasting superbroadcasting.
Quantum cloning machine for arbitrary mixed states in symmetric subspace is proposed. This quantum cloning machine can be used to copy part of the output state of another quantum cloning machine and is useful in quantum computation and quantum information. The shrinking factor of this quantum cloning achieves the well-known upper bound. When the input is identical pure states, two different fidelities of this cloning machine are optimal.
We describe a general framework to study covariant symmetric broadcasting maps for mixed qubit states. We explicitly derive the optimal N to M superbroadcasting maps, achieving optimal purification of the single-site output copy, in both the universal and the phase covariant cases. We also study the bipartite entanglement properties of the superbroadcast states.
We consider a single copy of a mixed state of two qubits and show how its fidelity or maximal singlet fraction is related to the entanglement measures concurrence and negativity. We characterize the extreme points of the convex set of states with constant fidelity, and use this to prove tight lower and upper bounds on the fidelity for a given amount of entanglement.
As small quantum computers are becoming available on different physical platforms, a benchmarking task known as cross-platform verification has been proposed that aims to estimate the fidelity of states prepared on two quantum computers. This task is fundamentally distributed, as no quantum communication can be performed between the two physical platforms due to hardware constraints, which prohibits a joint SWAP test. In this paper we settle the sample complexity of this task across all measurement and communication settings. The essence of the task, which we call distributed quantum inner pro…
Copying information is an elementary operation in classical information processing. However, copying seems rather different in the quantum regime. Since the discovery of the universal quantum cloning machine, much has been found from the fundamental point of view about quantum copying. But a basic question as to the utility of universal quantum cloning remains. We have considered its application in quantum state restoration by using cloning circuit for state estimation. It might be expected that classical information from the state estimation might help restore the quantum state that was distu…
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.
We present observable upper bounds of squared concurrence, which are the dual inequalities of the observable lower bounds introduced in [F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98, 140505 (2007)] and [L. Aolita, A. Buchleitner and F. Mintert, Phys. Rev. A 78, 022308 (2008)]. These bounds can be used to estimate entanglement for arbitrary experimental unknown finite-dimensional states by few experimental measurements on a twofold copy $ρ\otimesρ$ of the mixed states. Furthermore, the degree of mixing for a mixed state and some properties of the linear entropy also have certain relations…
We consider a single copy of a mixed state of two qubits and derive the optimal trace-preserving local operations assisted by classical communication (LOCC) such as to maximize the fidelity of teleportation that can be achieved with this state. These optimal local operations turn out to be implementable by one-way communication, and always yields a teleportation fidelity larger than 2/3 if the original state is entangled. This maximal achievable fidelity is an entanglement measure and turns out to quantify the minimal amount of mixing required to destroy the entanglement in a quantum state.
We present theory and experiment for the task of discriminating two nonorthogonal states, given multiple copies. We implement several local measurement schemes, on both pure states and states mixed by depolarizing noise. We find that schemes which are optimal (or have optimal scaling) without noise perform worse with noise than simply repeating the optimal single-copy measurement. Applying optimal control theory, we derive the globally optimal local measurement strategy, which outperforms all other local schemes, and experimentally implement it for various levels of noise.
We present observable lower and upper bounds for the entanglement of formation (EOF) and quantum discord (QD), which facilitates estimates of EOF and QD for arbitrary experimental unknown states in finite-dimensional bipartite systems. These bounds can be easily obtained by a few experimental measurements on a twofold copy $\varrho\otimes\varrho$ of the mixed states. Based on our results, we use the experimental measurement data of the real experiment given by Schmid \textit{et al.} [Phys. Rev. Lett. \textbf{101}, 260505 (2008)] to obtain the lower and upper bounds of EOF and QD for the experi…
Various aspects of distillation of noisy entanglement and some associated effects in quantum error correction are considered. In particular we prove that if only one--way classical communication (from Alice to Bob) is allowed and the shared $d \otimes d$ state is not pure then there is a threshold for optimal entanglement fraction $F$ of the state (being an overlap between the shared state and symmetric maximally entangled state) which can be obtained in single copy distillation process. This implies that to get (probabilistically) arbitrary good conclusive teleportation via mixed state at lea…
Activation of Bell nonlocality refers to the phenomenon that some entangled mixed states that admit a local hidden variable model in the standard Bell scenario nevertheless reveal their nonlocal nature in more exotic measurement scenarios. We present such a scenario that involves broadcasting the local subsystems of a single-copy of a bipartite quantum state to multiple parties, and use the scenario to study the nonlocal properties of the two-qubit isotropic state: \begin{align} \nonumber ρ_α= α\,|Φ^+ \rangle\langle Φ^+|+(1-α)\frac{\mathbb{1}}{4}. \end{align} We present two main results, consi…
The optimal entanglement manipulation for a single copy of mixed states of two qubits is to transform it to a Bell diagonal state. In this paper we derive an explicit form of the local operation that can realize such a transformation. The result obtained is universal for arbitrary entangled two-qubit states and it discloses that the corresponding local filter is not unique for density matrices with rank $n=2$ and can be exclusively determined for that with $n=3$ and 4. As illustrations, a four-parameters family of mixed states are explored, the local filter as well as the transformation probab…
Only with the simultaneous estimation of multiple parameters are the quantum aspects of metrology fully revealed. This is due to the incompatibility of observables. The fundamental bound for multi-parameter quantum estimation is the Holevo Cramér-Rao bound (HCRB) whose evaluation has so far remained elusive. For finite-dimensional systems we recast its evaluation as a semidefinite program, with reduced size for rank-deficient states. We use this result to study phase and loss estimation in optical interferometry and three-dimensional magnetometry with noisy multi-qubit systems. For the former,…
Entanglement shared among multiple parties presents complex challenges for the characterisation of different types of entanglement. One of the most fundamental insights is the fact that some mixed states can feature entanglement across every possible cut of a multipartite system yet can be produced via a mixture of states separable with respect to different partitions. To distinguish states that genuinely cannot be produced from mixing such partition-separable states, the term genuine multipartite entanglement was coined. All these considerations originate in a paradigm where only a single cop…
We investigate the optimal estimation of quantum expectation value of a physical observable, which minimizes a mean error with respect to general measure of deviation, when a finite number of copies of a pure state are prepared. If pure sates are uniformly distributed, the minimum value of mean error for any measure of deviation is achieved by projective measurement on each copy.
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 consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on $d\times d\times n$-dimensional tripartite mixed states. We show that this {\em assisted entanglement}, when measured in terms of the generalized concurrence (named G-concurrence) is (tightly) bounded by an entanglement monotone, which we call the G-concurrence of assistance. The G-concurrence is one of the possible generalizations of the concurrence to higher dimensions, and for pure bipartite states it measures the {\em geometric mean} of t…
We present a quantum algorithm to estimate parameters at the quantum metrology limit using deterministic quantum computation with one bit. When the interactions occurring in a quantum system are described by a Hamiltonian $H= θH_0$, we estimate $θ$ by zooming in on previous estimations and by implementing an adaptive Bayesian procedure. The final result of the algorithm is an updated estimation of $θ$ whose variance has been decreased in proportion to the time of evolution under H. For the problem of estimating several parameters, we implement dynamical-decoupling techniques and use the result…
We discuss single adaptive measurements for the estimation of mixed quantum states of qubits. The results are compared to the optimal estimation schemes using collective measurements. We also demonstrate that the advantage of collective measurements increases when the degree of mixing of the quantum states increases.
Inria, Télécom Paris – LTCI, Institut Polytechnique de Paris（法国国家信息与自动化研究所，巴黎电信学院-信息与通信实验室，巴黎理工学院）；QuSoft & Institute for Logic, Language and Computation & Korteweg-de Vries Institute for Mathematics, University of Amsterdam（QuSoft、逻辑语言与计算研究所、科特维格-德弗里斯数学研究所，阿姆斯特丹大学）；UC Berkeley（加州大学伯克利分校）
The accuracy of any physical scheme used to estimate the parameter describing the strength of a single qubit Pauli channel can be quantified using standard techniques from quantum estimation theory. It is known that the optimal estimation scheme, with m channel invocations, uses initial states for the systems which are pure and unentangled and provides an uncertainty of O[1/m^(1/2)]. This protocol is analogous to a classical repetition and averaging scheme. We consider estimation schemes where the initial states available are not pure and compare a protocol involving quantum correlated states …
We revisit the basic problem of quantum state certification: given copies of unknown mixed state $ρ\in\mathbb{C}^{d\times d}$ and the description of a mixed state $σ$, decide whether $σ= ρ$ or $\|σ- ρ\|_{\mathsf{tr}} \ge ε$. When $σ$ is maximally mixed, this is mixedness testing, and it is known that $Ω(d^{Θ(1)}/ε^2)$ copies are necessary, where the exact exponent depends on the type of measurements the learner can make [OW15, BCL20], and in many of these settings there is a matching upper bound [OW15, BOW19, BCL20]. Can one avoid this $d^{Θ(1)}$ dependence for certain kinds of mixed states $σ…
We consider quantum enhanced measurements with initially mixed states. We show very generally that for any linear propagation of the initial state that depends smoothly on the parameter to be estimated, the sensitivity is bound by the maximal sensitivity that can be achieved for any of the pure states from which the initial density matrix is mixed. This provides a very general proof that purely classical correlations cannot improve the sensitivity of parameter estimation schemes in quantum enhanced measurement schemes.
We consider the problem of quantum state certification, where we are given the description of a mixed state $σ\in \mathbb{C}^{d \times d}$, $n$ copies of a mixed state $ρ\in \mathbb{C}^{d \times d}$, and $\varepsilon > 0$, and we are asked to determine whether $ρ= σ$ or whether $\| ρ- σ\|_1 > \varepsilon$. When $σ$ is the maximally mixed state $\frac{1}{d} I_d$, this is known as mixedness testing. We focus on algorithms which use incoherent measurements, i.e. which only measure one copy of $ρ$ at a time. Unlike those that use entangled, multi-copy measurements, these can be implemented without…
Quantifying mixed-state entanglement via partial transpose and realignment moments
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Technology Innovation Institute（技术创新研究所）
