We consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states. This is motivated in part by results of Hayden et al., which show that in large d x d--dimensional systems there exist random subspaces of dimension almost d^2, all of whose states have entropy of entanglement at least log d - O(1). It is also related to results due to Parthasarathy on the dimension of completely entangled subspaces, which have connections with the construction of unextendible product bases. Here we take as entanglement measure the Schmidt rank, and determine, for every pair of local dimensions dA and dB, and every r, the largest dimension of a subspace consisting only of entangled states of Schmidt rank r or larger. This exact answer is a significant improvement on the best bounds that can be obtained using random subspace techniques. We also determine the converse: the largest dimension of a subspace with an upper bound on the Schmidt rank. Finally, we discuss the question of subspaces containing only states with Schmidt equal to r.
Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation $I_2\otimes V$ and $I_2\otimes W$. We show that $V$ and $W$ have no zero entry, and apply it to exclude examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is $\log_2 3$ ebits. …
Can vectors with low Schmidt rank form mutually unbiased bases? Can vectors with high Schmidt rank form positive under partial transpose states? In this work, we address these questions by presenting several new results related to Schmidt rank constraints and their compatibility with other properties. We provide an upper bound on the number of mutually unbiased bases of $\mathbb{C}^m\otimes\mathbb{C}^n$ $(m\leq n)$ formed by vectors with low Schmidt rank. In particular, the number of mutually unbiased product bases of $\mathbb{C}^m\otimes\mathbb{C}^n$ cannot exceed $m+1$, which solves a conjec…
It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we …
Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, we show that $2(N-1)$ generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the $y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types which are defined in this paper are sufficient to simulate a controlled-unitary gate $\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling on $\mathbb{C}^2\otimes \mathbb{C}^N$. In the scenario of the unitary gate $\mathcal{U}_{cd(M\otimes N)}$ with $M\geq3$ that is locally equivalent to a diagonal uni…
Localization of joint quantum measurements on $\mathbb{C}^d \otimes \mathbb{C}^d$ by entangled resources with Schmidt number at most $d$
Communication Science Laboratories, NTT, Inc.（NTT株式会社通信科学实验室）；NTT Research Center for Theoretical Quantum Information（NTT理论量子信息研究中心）；NTT Institute for Fundamental Mathematics（NTT基础数学研究所）；Graduate School of Science, The University of Tokyo（东京大学理学研究科）；Ritsumeikan University BKC Research Organization of Social Sciences（立命馆大学BKC社会科学研究机构）
The trace distance between two quantum states, $ρ$ and $σ$, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of $ρ- σ$. In contrast, the Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt distance with a bound that is particularly strong when either $ρ$ or $σ$ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace…
克服我们最近的计算密集型（准蒙特卡罗）工作的情况下，N=4（quant-ph/0308037），我们承担了N=6的高数值精度计算任务，Sommers和Zyczkowski的公式（quant-ph/0304041）的（N^2-1）维体积和（N^2-2）维超面积，（可分和不可分）N x N密度矩阵，基于Bures（最小单调）度量-以及（非单调）希尔伯特-施密特度量的类似公式（quant-ph/0302197）。使用相同的70亿个分布良好（“低差异”）的样本点，我们基于五个额外的（单调）感兴趣的度量（包括Kubo-Mori和Wigner-Yanase）来估计未知体积和超面积。此外，我们估计所有这七个体积和七个超面积（未知）的数量时，限制可分离的密度矩阵。可分离体积（超区域）与可分离体积加不可分离体积（超区域）的比值产生一般秩六（秩五）密度矩阵的可分离概率的估计。基于35维体积获得的（秩6）可分性概率似乎是--独立于所采用的度量（7个诱导Haar度量中的每一个）--是基于34维超面积的（秩5）可分性概率的两倍。接受这样的关系，我们适合确切的公式估计的Bures和希尔伯特-施密特可分离的体积和超面积。(An秩6的希尔伯特-施密特可分性概率与秩4的希尔伯特-施密特可分性概率之比的额外估计-- 33.9982 --也非常明显地接近于积分。对于N=4的希尔伯特-施密特度规，加倍关系似乎也成立，但其…
We introduce a hierarchy of linear systems for showing that a given subspace of pure quantum states is entangled (i.e., contains no product states). This hierarchy outperforms known methods already at the first level, and it is complete in the sense that every entangled subspace is shown to be so at some finite level of the hierarchy. It generalizes straightforwardly to the case of higher Schmidt rank, as well as the multipartite cases of completely and genuinely entangled subspaces. These hierarchies work extremely well in practice even in very large quantum systems, as they can be implemente…
Entanglement in the symmetric subspace: mapping multipartite to bipartite states
Istituto Nazionale di Ottica - Consiglio Nazionale delle Ricerche (INO-CNR)（意大利国家光学研究所-意大利国家研究委员会）；University of Tsukuba（筑波大学）；Center for Artificial Intelligence Research (C-AIR)（人工智能研究中心）；LENS（欧洲纳米科学实验室）
The operator Schmidt rank is the minimum number of terms required to express a state as a sum of elementary tensor factors. Here we provide a new proof of the fact that any bipartite mixed state with operator Schmidt rank two is separable, and can be written as a sum of two positive semidefinite matrices per site. Our proof uses results from the theory of free spectrahedra and operator systems, and illustrates the use of a connection between decompositions of quantum states and decompositions of nonnegative matrices. In the multipartite case, we prove that any Hermitian Matrix Product Density …
In this short note we show two completely opposite methods of constructing entangled states. Given a bipartite state $γ\in M_k\otimes M_k$, define $γ_S=(Id+F)γ(Id+F)$, $γ_A=(Id-F)γ(Id-F)$, where $F\in M_k\otimes M_k$ is the flip operator. In the first method, entanglement is a consequence of the inequality $\text{rank}(γ_S)<\sqrt{\text{rank}(γ_A)}$. In the second method, there is no correlation between $γ_S$ and $γ_A$. These two methods show how diverse is quantum entanglement. We prove that any bipartite state $γ\in M_k\otimes M_k$ satisfies $\displaystyle SN(γ)\geq\max \left\{ \frac{\text{ra…
We implement a procedure-based on the Wishart-Laguerre distribution-recently outlined by {Ż}yczkowski and Khvedelidze, Rogojin and Abgaryan, for the generation of random (complex or real) $N \times N$ density matrices of rank $k \leq N$ with respect to Hilbert-Schmidt (HS) measure. In the complex case, one commences with a Ginibre matrix $A$ of dimensions $k \times k+ 2 (N-k)$, while for a real scenario, one employs a Ginibre matrix $B$ of dimensions $k \times k+1+ 2 (N-k)$. Then, the $k \times k$ product $A A^{\dagger}$ or $B B^T$ is diagonalized-padded with zeros to size $N \times N$-and rot…
Schmidt Decomposition-Based Methods for Efficient Quantum Image Encoding
Fraunhofer Institute for Industrial Mathematics ITWM（弗劳恩霍夫工业数学研究所ITWM）；Technical University of Sofia（索非亚技术大学）；Institut d’Optique Graduate School - Université Paris-Saclay（巴黎萨克雷大学光学研究所）；Algorithms for Computer Vision, Imaging and Data Analysis Lab, Technical University of Applied Sciences Schweinfurt（施韦因富特应用技术大学计算机视觉、成像与数据分析实验室）
