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In this study, we generate quantum channels with random Kraus operators to typically obtain almost twirling quantum channels and quantum expanders. To prove the concentration phenomena, we use matrix Bernstein's inequality. In this way, our random models do not utilize Haar-distributed unitary matrices or Gaussian matrices. Rather, as in the preceding research, we use unitary $t$-designs to generate mixed tenor-product unitary channels acting on $\mathbb C^{d^t}$. Although our bounds in Schatten $p$-norm are valid only for $1\leq p \leq 2$, we show that they are typically almost twirling quantum channels with the tail bound proportional to $1/\mathrm{poly}(d^t)$, while such bounds were previously constants. The number of required Kraus operators was also improved by powers of $\log d$ and $t$. Such random quantum channels are also typically quantum expanders, but the number of Kraus operators must grow proportionally to $\log d$ in our case. Finally, a new non-unital model of super-operators generated by bounded and isotropic random Kraus operators was introduced, which can be typically rectified to give almost randomizing quantum channels and quantum expanders.

We are upgrading the Python-version of RTNI, which symbolically integrates tensor networks over the Haar-distributed unitary matrices. Now, PyRTNI2 can treat the Haar-distributed orthogonal matrices and the real and complex normal Gaussian tensors as well. Moreover, it can export tensor networks in the format of TensorNetwork so that one can make further calculations with concrete tensors, even for low dimensions, where the Weingarten functions differ from the ones for high dimensions. The tutorial notebooks are found at GitHub: https://github.com/MotohisaFukuda/PyRTNI2. In this paper, we explain maths behind the program and show what kind of tensor network calculations can be made with it. For the former, we interpret the element-wise moment calculus of the above random matrices and tensors in terms of tensor network diagrams, and argue that the view is natural, relating delta functions in the calculus to edges in tensor network diagrams., Comment: The title was slitely changed, typos were fixed. PyRTNI2 is available at https://github.com/MotohisaFukuda/PyRTNI2

In this article, we investigate meandric systems having one shallow side: the arch configuration on that side has depth at most two. This class of meandric systems was introduced and extensively examined by I. P. Goulden, A. Nica, and D. Puder in 2020. Shallow arch configurations are in bijection with the set of interval partitions. We study meandric systems by using moment-cumulant transforms for non-crossing and interval partitions, corresponding to the notions of free and boolean independence, respectively, in non-commutative probability. We obtain formulas for the generating series of different classes of meandric systems with one shallow side, by explicitly enumerating the simpler, irreducible objects. In addition, we propose random matrix models for the corresponding meandric polynomials, which can be described in the language of quantum information theory, in particular that of quantum channels., Comment: 22 pages

Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems of free probability, we not only show the multiplicativity violation of maximum output norms in the asymptotic regimes but also prove the additivity violation via Haagerup inequality for a new class of random quantum channels constructed by rectifying the above completely positive maps based on strong convergence., Comment: 24 pages. Appendix D was added and minor corrections and changes were made

We present CROP (Central Roundish Object Painter), which identifies and paints the object at the center of an RGB image. Primarily CROP works for roundish fruits in various illumination conditions, but surprisingly, it could also deal with images of other organic or inorganic materials, or ones by optical and electron microscopes, although CROP was trained solely by 172 images of fruits. The method involves image segmentation by deep learning, and the architecture of the neural network is a deeper version of the original U-Net. This technique could provide us with a means of automatically collecting statistical data of fruit growth in farms. As an example, we describe our experiment of processing 510 time series photos automatically to collect the data on the size and the position of the target fruit. Our trained neural network CROP and the above automatic programs are available on GitHub with user-friendly interface programs., Comment: The version 2 contains a new section about the automatic processing of time series photos. All the programs are available at https://github.com/MotohisaFukuda/CROP

We consider ensembles of bipartite states resulting from a random passive Gaussian unitary applied to a fiducial pure Gaussian state. We show that the symplectic spectra of the reduced density operators concentrate around that of a thermal state with the same energy. This implies, in particular, concentration of the entanglement entropy as well as other entropy measures. Our work extends earlier results on the typicality of entanglement beyond the two ensembles and the reduced purity measure considered in [A. Serafini, O. C. O. Dahlsten, D. Gross, and M. B. Plenio, J. Phys. A: Math. Theor., 40(31):9551 (2007)]., Comment: 19pages. Other materials: one Mathematica notebook, two Python files and one PDF file for the manual. V2: ancillary files uploaded, reference corrected

We provide a computer algebra package called Random Tensor Network Integrator (RTNI). It allows to compute averages of tensor networks containing multiple Haar-distributed random unitary matrices and deterministic symbolic tensors. Such tensor networks are represented as multigraphs, with vertices corresponding to tensors or random unitaries and edges corresponding to tensor contractions. Input and output spaces of random unitaries may be subdivided into arbitrary tensor factors, with dimensions treated symbolically. The algorithm implements the graphical Weingarten calculus and produces a weighted sum of tensor networks representing the average over the unitary group. We illustrate the use of this algorithmic tool on some examples from quantum information theory, including entropy calculations for random tensor network states as considered in toy models for holographic duality. Mathematica and Python implementations are supplied., Comment: Code available (for Mathematica and python) at https://github.com/MotohisaFukuda/RTNI

We consider sequences of random quantum channels defined using the Stinespring formula with Haar-distributed random orthogonal matrices. For any fixed sequence of input states, we study the asymptotic eigenvalue distribution of the outputs through tensor powers of random channels. We show that the input states achieving minimum output entropy are tensor products of maximally entangled states (Bell states) when the tensor power is even. This phenomenon is completely different from the one for random quantum channels constructed from Haar-distributed random unitary matrices, which leads us to formulate some conjectures about the regularized minimum output entropy., Comment: minor modifications

We investigate meandric systems with a large number of loops using tools inspired by free probability. For any fixed integer $r$, we express the generating function of meandric systems on $2n$ points with $n-r$ loops in terms of a finite (the size depends on $r$) subclass of irreducible meandric systems, via the moment-cumulant formula from free probability theory. We show that the generating function, after an appropriate change of variable, is a rational function, and we bound its degree. Exact expressions for the generating functions are obtained for $r \leq 6$, as well as the asymptotic behavior of the meandric numbers for general $r$., Comment: Minor revision. 22 pages. To view supplementary material, please download and extract the file listed under "Other formats"

We investigate minimum output (R\'enyi) entropy of qubit channels and unital quantum channels. We obtain an exact formula for the minimum output entropy of qubit channels, and bounds for unital quantum channels. Interestingly, our bounds depend only on the operator norm of the matrix representation of the channels on the space of trace-less Hermitian operators. Moreover, since these bounds respect tensor products, we get bounds for the capacity of unital quantum channels, which is saturated by the Werner-Holevo channel. Furthermore, we construct an orthonormal basis, besides the Gell-Mann basis, for the space of trace-less Hermitian operators by using discrete Weyl operators. We apply our bounds to discrete Weyl covariant channels with this basis, and find new examples in which the minimum output R\'enyi $2$-entropy is additive., Comment: 15 pages, 1 figure

Inspired by Montanaro's work, we introduce the concept of additivity rates of a quantum channel $L$, which give the first order (linear) term of the minimum output $p$-R\'enyi entropies of $L^{\otimes r}$ as functions of $r$. We lower bound the additivity rates of arbitrary quantum channels using the operator norms of several interesting matrices including partially transposed Choi matrices. As a direct consequence, we obtain upper bounds for the classical capacity of the channels. We study these matrices for random quantum channels defined by random subspaces of a bipartite tensor product space. A detailed spectral analysis of the relevant random matrix models is performed, and strong convergence towards free probabilistic limits is showed. As a corollary, we compute the threshold for random quantum channels to have the positive partial transpose (PPT) property. We then show that a class of random PPT channels violate generically additivity of the $p$-R\'enyi entropy for all $p\geq30.95$., Comment: minor changes; a typo in the statement of Proposition 4.8 was corrected

The free contraction norm (or the (t)-norm) was introduced by Belinschi, Collins and Nechita as a tool to compute the typical location of the collection of singular values associated to a random subspace of the tensor product of two Hilbert spaces. In turn, it was used in by them in order to obtain sharp bounds for the violation of the additivity of the minimum output entropy for random quantum channels with Bell states. This free contraction norm, however, is difficult to compute explicitly. The purpose of this note is to give a good estimate for this norm. Our technique is based on results of super convergence in the context of free probability theory. As an application, we give a new, simple and conceptual proof of the violation of the additivity of the minimum output entropy., Comment: 18 pages with 2 figures

We study quantum channels with respect to their image, i.e., the image of the set of density operators under the action of the channel. We first characterize the set of quantum channels having polytopic images and show that additivity of the minimal output entropy can be violated in this class. We then provide a complete characterization of quantum channels $T$ that are universally image additive in the sense that for any quantum channel $S$, the image of $T \otimes S$ is the convex hull of the tensor product of the images of $T$ and $S$. These channels turn out to form a strict subset of entanglement breaking channels with polytopic images and a strict superset of classical-quantum channels.

We study the asymptotic behavior of the output states of sequences of quantum channels. Under a natural assumption, we show that the output set converges to a compact convex set, clarifying and substantially generalizing results in [BCN13]. Random mixed unitary channels satisfy the assumption; we give a formula for the asymptotic maximum output infinity norm and we show that the minimum output entropy and the Holevo capacity have a simple relation for the complementary channels. We also give non-trivial examples of sequences $\Phi_n$ such that along with any other quantum channel $\Xi$, we have convergence of the output set of $\Phi_n$ and $\Phi_n\otimes \Xi$ simultaneously; the case when $\Xi$ is entanglement breaking is investigated in details.

We prove additivity violation of minimum output entropy of quantum channels by straightforward application of \epsilon-net argument and L\'evy's lemma. The additivity conjecture was disproved initially by Hastings. Later, a proof via asymptotic geometric analysis was presented by Aubrun, Szarek and Werner, which uses Dudley's bound on Gaussian process (or Dvoretzky's theorem with Schechtman's improvement). In this paper, we develop another proof along Dvoretzky's theorem in Milman's view showing additivity violation in broader regimes than the existing proofs. Importantly, Dvoretzky's theorem works well with norms to give strong statements but these techniques can be extended to functions which have norm-like structures - positive homogeneity and triangle inequality. Then, a connection between Hastings' method and ours is also discussed. Besides, we make some comments on relations between regularized minimum output entropy and classical capacity of quantum channels., Comment: 16 pages

It is now well-known that, with high probability, the additivity of minimum output entropy does not hold for a pair of a random quantum channel and its complex conjugate. We investigate asymptotic behavior of output states of $r$-tensor powers of such pairs, as the dimension of inputs grows. We compute the limit output states for any sequence of well-behaved inputs, which consist of a large class of input states having a nice set of parameters. Then, we show that among these input states tensor products of Bell states give asymptotically the least output entropy, giving positive mathematical evidence towards additivity of above pairs of channels.

We investigate the asymptotic behavior of the empirical eigenvalues distribution of the partial transpose of a random quantum state. The limiting distribution was previously investigated via Wishart random matrices indirectly (by approximating the matrix of trace 1 by the Wishart matrix of random trace) and shown to be the semicircular distribution or the free difference of two free Poisson distributions, depending on how dimensions of the concerned spaces grow. Our use of Wishart matrices gives exact combinatorial formulas for the moments of the partial transpose of the random state. We find three natural asymptotic regimes in terms of geodesics on the permutation groups. Two of them correspond to the above two cases; the third one turns out to be a new matrix model for the meander polynomials. Moreover, we prove the convergence to the semicircular distribution together with its extreme eigenvalues under weaker assumptions, and show large deviation bound for the latter., Comment: v2: change of title, change of some methods of proofs

In this paper, we study the behaviour of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings and Hayden-Winter to the additivity problems. In particular, we study in depth the difference of behaviour between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in Collins and Nechita (2010).

We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs' eigenvalues between a quantum channel and its complementary version in the case of a non-pure input., Comment: 21 pages, 9 figures

Recently Hastings proved the existence of random unitary channels which violate the additivity conjecture. In this paper we use Hastings' method to derive new bounds for the entanglement of random subspaces of bipartite systems. As an application we use these bounds to prove the existence of non-unital channels which violate additivity of minimal output entropy., Comment: 25 pages

In this paper we present a class of maps for which the multiplicativity of the maximal output p-norm holds when p is 2 and p is larger than or equal to 4. The class includes all positive trace-preserving maps from the matrix algebra on the three-dimensional space to that on the two-dimensional., Comment: 9 pages

Hastings recently provided a proof of the existence of channels which violate the additivity conjecture for minimal output entropy. In this paper we present an expanded version of Hastings' proof. In addition to a careful elucidation of the details of the proof, we also present bounds for the minimal dimensions needed to obtain a counterexample., Comment: 38 pages

We study the additivity problems for the classical capacity of quantum channels, the minimal output entropy and its convex closure. We show for each of them that additivity for arbitrary pairs of channels holds iff it holds for arbitrary equal pairs, which in turn can be taken to be unital. In a similar sense, weak additivity is shown to imply strong additivity for any convex entanglement monotone. The implications are obtained by considering direct sums of channels (or states) for which we show how to obtain several information theoretic quantities from their values on the summands. This provides a simple and general tool for lifting additivity results., Comment: 5 pages

We simplify some conjectures in quantum information theory; the additivity of minimal output entropy, the multiplicativity of maximal output p-norm and the superadditivity of convex closure of output entropy. We construct a unital channel for a given channel so that they share the above additivity properties; we can reduce the conjectures for all channels to those for unital channels., Comment: 8 pages

We prove a lemma which allows one to extend results about the additivity of the minimal output entropy from highly symmetric channels to a much larger class. A similar result holds for the maximal output $p$-norm. Examples are given showing its use in a variety of situations. In particular, we prove the additivity and the multiplicativity for the shifted depolarising channel., Comment: 8 pages. This is the latest version of the first half of the original paper. The other half will appear in another paper