We propose an extended quantum mechanical formalism that is based on a wave operator $\vr$, which is related to the ordinary density matrix via $ρ=\vr\vr^\dagger$. This formalism allows a (generalized) unitary evolution between pure and mixed states. It also preserves much of the connection between symmetries and conservation laws. The new formalism is illustrated for the case of a two level system.
In the theory of quantum information, the mixed-unitary quantum channels, for any positive integer dimension $n$, are those linear maps that can be expressed as a convex combination of conjugations by $n\times n$ complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank~$N$ and the Choi rank~$r$ of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a K…
A quantum channel is said to be a mixed-unitary channel if it can be expressed as a convex combination of unitary channels. We prove that, given the Choi representation of a quantum channel, it is NP-hard with respect to polynomial-time Turing reductions to determine whether or not that channel is a mixed-unitary channel. This hardness result holds even under the assumption that the channel is not within an inverse-polynomial distance (in the dimension of the space upon which it acts) of the boundary of the mixed-unitary channels.
For a positive integer $d$ and a unitary representation $ρ:G\rightarrow\mathrm{U}(d)$ of a compact group $G$, the twirling channel for this representation is the linear mapping $Φ: M_d\rightarrow M_d$ defined as $Φ(X)=\int_{G}\mathrm{d}μ(g)\,ρ(g)Xρ(g^{-1})$ for every $X\in M_d$, where $μ$ is the Haar measure on $G$. Such channels are examples of mixed-unitary channels, as they are in the convex hull of the set of unitary channels of a fixed size. By Carathéodory's theorem, these channels can always be expressed as a finite linear combination of unitary channels. We consider the mixed-unitary r…
In the framework of finite-dimensional Fock space models, for a predefined fixed mean number of particles $\bar{n}_{k}$, it is shown that there is a ``large'' multi-dimensional subspace $s_{\bar{n}_{k}}$ of initial pure states, in the space $S$ of all pure states, unitarily evolving to a subspace $S_{\bar{n}_{k}}$ of final pure states which yield $\bar{n}_{k}$. As an example, in particular it follows that the blackbody form of the mean number of particles $\bar{n}_{k}$ does not by itself contradict unitarity of black hole evaporation.
We consider the tensor product of the completely depolarising channel on $d\times d$ matrices with the map of Schur multiplication by a $k \times k$ correlation matrix and characterise, via matrix theory methods, when such a map is a mixed (random) unitary channel. When $d=1$, this recovers a result of O'Meara and Pereira, and for larger $d$ is equivalent to a result of Haagerup and Musat that was originally obtained via the theory of factorisation through von Neumann algebras. We obtain a bound on the distance between a given correlation matrix for which this tensor product is nearly mixed un…
Local unitary equivalence and entanglement of multipartite pure states
在[B]中给出了任意n比特纯量子态在局域幺正（LU）运算下等价的充要条件。Kraus Phys. Rev. Lett. 104，020504（2010）]用于确定多达五个量子比特状态的不同LU等价类。由于这种分类，发现了表征多体纠缠的新参数，并给出了它们的物理解释。此外，我们还用这种方法导出了两个n比特态（n>2）的例子，它们具有任意子系统的熵都一致的性质，但它们既不是LU等价的，也不能通过一般的局域操作和经典通信相互映射.
Local Unitary Equivalent Classes of Symmetric N-Qubit Mixed States
对称$ N $ -量子比特纯态的Majorana表示已成功地用于纠缠分类。由于在构造对称混合态的Majorana几何表示时所面临的困难，这一问题的推广一直是一个悬而未决的问题。我们已经克服了这个问题，通过开发的方法分类本地酉（LU）等价类的对称$N$ -量子比特混合态的基础上的几何多轴表示（MAR）的密度矩阵。除了基于MR的对称纯态纠缠分类所定义的两个参数，即多样度和简并构型外，我们还证明了对称混合态分类还需要引入另一个参数秩.我们的分类方案是更一般的，因为它可以适用于纯态和混合态。为了揭示MR和MAR之间的异同，我们选取了$ N $ -量子比特GHZ态进行了详细的研究。我们的结论是基于MR的纯态分类不是我们基于MAR的分类方案的特例。我们还给出了识别最一般的对称$N$ -量子比特纯可分态的方法。我们的方法的权力证明使用几个著名的对称两个量子比特纯态和混合态以及三个量子比特纯态的例子。详细研究了实验室中可以制备的单轴、双轴和三轴对称两量子比特混合态的分类。
We consider the statistics of overlaps between a mixed state and its image under random unitary transformations. Choosing the transformations from the unitary group with its invariant (Haar) measure, the distribution of overlaps depends only on the eigenvalues of the mixed state. This allows one to estimate these eigenvalues from the overlap statistics. In the first part of this work, we present explicit results for qutrits, including a discussion of the expected uncertainties in the eigenvalue estimation. In the second part, we assume that the set of available unitary transformations is restr…
Non-unitary evolution of a pure state into a mixed state in the measurement problem from standard Quantum Mechanics and its impact on complex space-time, no-boundary proposal and information loss paradox of singularity-free Quantum Cosmology
Pure quantum states are often approximately encoded as classical bit strings such as those representing probability amplitudes and those describing circuits that generate the quantum states. The crucial quantity is the minimum length of classical bit strings from which the original pure states are approximately reconstructible. We derive asymptotically tight bounds on the minimum bit length required for probabilistic encodings with which one can approximately reconstruct the original pure state as an ensemble of the quantum states encoded in classical strings. We also show that such a probabil…
