arXiv subjects

Motohisa Fukuda

Publications and source records attributed to Motohisa Fukuda.

At least 19 recordsLinked to original sources

arXiv2024

Concentration of quantum channels with random Kraus operators via matrix Bernstein inequality

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 $(\mathbb C^{d})^{\otimes t}$. 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.

quant-ph↗

arXiv2023

Symbolically integrating tensor networks over various random tensors by the second version of Python RTNI

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.

physics.comp-ph↗

arXiv2021

Additivity violation of quantum channels via strong convergence to semi-circular and circular elements

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.

math-ph↗

arXiv2021

Generating series and matrix models for meandric systems with one shallow side

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.

math.CO↗

arXiv2020

Central object segmentation by deep learning for fruits and other roundish objects

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.

cs.CV↗

arXiv2019

Typical entanglement for Gaussian states

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)].

quant-ph↗

arXiv2019

RTNI - A symbolic integrator for Haar-random tensor networks

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.

quant-ph↗

arXiv2017

On the minimum output entropy of random orthogonal quantum channels

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.

quant-ph↗

arXiv2017

Enumerating meandric systems with large number of loops

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 $r$.

math.CO↗

arXiv2015

Additivity rates and PPT property for random quantum channels

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 $p$-Rényi 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 $p$-Rényi entropy for all $p\geq30.95$.

math-ph↗

arXiv2015

Additive bounds of minimum output entropies for unital channels and an exact qubit formula

We investigate minimum output (Rényi) 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 Rényi $2$-entropy is additive.

quant-ph↗

arXiv2014

Revisiting additivity violation of quantum channels

We prove additivity violation of minimum output entropy of quantum channels by straightforward application of ε-net argument and Lévy'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.

math-ph↗

arXiv2014

Estimates for compression norms and additivity violation in quantum information

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.

math.OA↗

arXiv2014

Quantum channels with polytopic images and image additivity

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.

quant-ph↗

arXiv2013

On the convergence of output sets of 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 $Φ_n$ such that along with any other quantum channel $Ξ$, we have convergence of the output set of $Φ_n$ and $Φ_n\otimes Ξ$ simultaneously; the case when $Ξ$ is entanglement breaking is investigated in details.

math-ph↗

arXiv2013

Partial transpose of random quantum states: exact formulas and meanders

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.

math-ph↗

arXiv2012

Asymptotically well-behaved input states do not violate additivity for conjugate pairs of random quantum channels

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.

math-ph↗

arXiv2012

Low entropy output states for products of random unitary channels

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).

math-ph↗