A density matrix $ρ$ may be represented in many different ways as a mixture of pure states, $ρ= \sum_i p_i |ψ_i\ra \la ψ_i|$. This paper characterizes the class of probability distributions $(p_i)$ that may appear in such a decomposition, for a fixed density matrix $ρ$. Several illustrative applications of this result to quantum mechanics and quantum information theory are given.
采用Elitzur，Popescu和Rohrlich [A. Elitzur，S. Popescu和D. Rohrlich，Phys. Lett. A 162，25（1992）]。证明了局部权的下界与状态$p_L^{\max}= 1-\mathcal{C}(ρ)$的并发性有关。独立地构造了两族混合态的局域概率分布，它们都符合下界。
An algebraic analysis of Grover's quantum search algorithm is presented for the case in which the initial state is an arbitrary pure quantum state of n qubits. This approach reveals the geometrical structure of the quantum search process, which turns out to be confined to a four-dimensional subspace of the Hilbert space. This work unifies and generalizes earlier results on the time evolution of the amplitudes during the quantum search, the optimal number of iterations and the success probability. Furthermore, it enables a direct generalization to the case in which the initial state is a mixed …
Optimum measurement for unambiguously discriminating two mixed states: General considerations and special cases
根据我们以前的出版物[U。赫尔佐格和J. A. Bergou，Phys.Rev. A 71，050301（R）（2005）]，我们研究了用于以给定先验概率出现的两个混合量子态的明确区分的最佳测量。非正交态的非生物识别是可能的概率的方式，在不确定的结果，测量失败的非零概率为代价。随着一般问题的讨论，我们给出一个例子来说明我们的解决方法。我们还提供了一般的最小可实现的故障概率不等式，并讨论了更详细的必要条件，必须满足时，其绝对下限，比例的保真度的国家，可以达到。
The optimal (pure state) ensemble length of a separable state, A, is the minimum number of (pure) product states needed in convex combination to construct A. We study the set of all separable states with optimal (pure state) ensemble length equal to k or fewer. Lower bounds on k are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest, the lower bound for non-pure state ensembles is sharp. A consequence of our results is that for all two partic…
Preparation of entangled states of photons are useful for quantum computing and communication. In this paper, we present a simplistic protocol of entanglement generation using beam splitters with suitable reflectivity. The photons in an initial state with fully classical probability distribution pass through an optical network, made up of sequential beam splitters and are prepared in maximally entangled states. We also present the detailed theoretical analysis of entangled state generation, for an arbitrary number of photons, fed through the input ports of the beam splitters with equal probabi…
We investigate the discrimination of pure-mixed (quantum filtering) and mixed-mixed states and compare their optimal success probability with the one for discriminating other pairs of pure states superposed by the vectors included in the mixed states. We prove that under the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to quantum filtering, the coherence exists only in one pure state and is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case …
We discuss the possibility to formulate the dynamics of spin states described by the Schrodinger equation for pure states and the von Neumann equation (as well as the GKSL equation) for mixed states in the form of quantum kinetic equations for probability distributions. We review an approach to the spin-state description by means of the probability distributions of dichotomic random variables.
Majorization uncertainty relations are generalized for an arbitrary mixed quantum state $ρ$ of a finite size $N$. In particular, a lower bound for the sum of two entropies characterizing probability distributions corresponding to measurements with respect to arbitrary two orthogonal bases is derived in terms of the spectrum of $ρ$ and the entries of a unitary matrix $U$ relating both bases. The obtained results can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as uncertainty relation for the s…
We investigate sampling procedures that certify that an arbitrary quantum state on $n$ subsystems is close to an ideal mixed state $φ^{\otimes n}$ for a given reference state $φ$, up to errors on a few positions. This task makes no sense classically: it would correspond to certifying that a given bitstring was generated according to some desired probability distribution. However, in the quantum case, this is possible if one has access to a prover who can supply a purification of the mixed state. In this work, we introduce the concept of mixed-state certification, and we show that a natural sam…
We investigate the spectral statistics of the difference of two density matrices, each of which is independently obtained by partially tracing a random bipartite pure quantum state. We first show how a closed-form expression for the exact joint eigenvalue probability density function for arbitrary dimensions can be obtained from the joint probability density function of the diagonal elements of the difference matrix, which is straightforward to compute. Subsequently, we use standard results from free probability theory to derive a relatively simple analytic expression for the asymptotic eigenv…
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
We introduce the Gram-Hadamard Density Operator (GHDO), a new deep neural-network architecture that can encode positive semi-definite density operators of exponential rank with polynomial resources. We then show how to embed an autoregressive structure in the GHDO to allow direct sampling of the probability distribution. These properties are especially important when representing and variationally optimizing the mixed quantum state of a system interacting with an environment. Finally, we benchmark this architecture by simulating the steady state of the dissipative transverse-field Ising model.…
