The Critical Points of Coherent Information on the Manifold of Positive Definite Matrices by Alireza Tehrani A Thesis presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Mathematics Guelph, Ontario, Canada c Alireza Tehrani, January, 2020 ABSTRACT THE CRITICAL POINTS OF COHERENT INFORMATION ON THE MANIFOLD OF POSITIVE DEFINITE MATRICES Alireza Tehrani Advisors: University of Guelph, 2020 Dr. Bei Zeng Dr. Rajesh Pereira The coherent information of quantum channels plays a important role in quantum infor- mation theory as it can be used to calculate the quantum capacity of a channel. However, it is a non-linear, non-differentiable optimization problem. This thesis discusses that by restricting to the space of positive definite density matrices and restricting the class of quan- tum channels to be strictly positive, the coherent information becomes differentiable. This allows the computation of the Riemannian gradient and Hessian of the coherent information. It will be shown that the maximally mixed state is a critical point for the n-shot coherent information of the Pauli, dephrasure and Pauli-erasure channels. In addition, the classifica- tion of the maximally mixed state as a local maxima/minima and saddle-point will be solved for the one shot coherent information. The hope of this work is to provide a new avenue to explore the quantum capacity problem. Dedication To those who have inspired or encouraged me: Brother, Father, Randy, Farnaz and Paul. iii Acknowledgements I want to express my gratitude and thanks to Dr. Zeng for the guidance, advice and direction that was provided and for giving me the opportunity to work in quantum information theory and to be part of IQC and the Peng Cheng Laborotory. It has been a inspiring experience to witness the level of enthusiasm with respect to science that a person could have. In addition to Dr.Pereira, whom made sure I was on the right path, answering any questions and providing new suggestions. I'm thankful for the time spent doing research and words can never be enough to express the kindness that was shown. Couldn't ask for better office mates, David, Eric, Katrina, Momina and Ningping thanks for the time spent together. iv Contents Abstract ii Dedication iii Acknowledgements iv List of Figures viii 1 Introduction 1 1.1 Introduction to Problem . .1 1.1.1 Contributions . .2 1.2 Overview of Chapters . .3 2 Coherent Information 5 2.1 von Neumann Entropy . .5 2.1.1 Classical Entropy . .6 2.1.2 Density Matrices . .7 2.1.3 Quantum/von Neumann Entropy . .8 2.2 Coherent Information . 11 2.2.1 Quantum Channels . 11 2.2.2 Motivation . 15 2.2.3 Coherent Information of a channel . 17 2.3 Properties . 19 2.3.1 Data Processing and Error-Correction . 19 2.3.2 Quantum Capacity . 20 2.3.3 Lipschitz . 21 2.3.4 Conjecture: Optima on Positive Definite Density Matrices . 22 2.4 Known Results . 22 2.4.1 Amplitude-Damping Channel . 23 2.4.2 Pauli Channel . 23 2.4.3 Dephrasure Channel . 24 2.4.4 Degradable and Antidegradable Quantum Channels . 25 v 3 Introduction to Manifold Theory and The Gradient/Hessian 27 3.1 Preliminaries to Manifold Theory . 28 3.1.1 Manifolds . 28 3.1.2 Smooth Maps . 33 3.1.3 Tangent Space . 36 3.1.4 Differential . 37 3.1.5 Tangent Bundle . 43 3.1.6 Submanifolds . 44 3.1.7 Riemannian Manifolds . 46 3.2 Riemannian Gradient and Hessian . 48 3.2.1 Euclidean Gradient . 49 3.2.2 Riemannian Gradient . 50 3.2.3 Hessian . 53 4 Gradient/Hessian of Coherent Information on Positive Definite Matrices 59 4.1 Strictly Positive Quantum Channels . 60 4.1.1 Definition . 60 4.1.2 Dense . 63 4.1.3 Examples . 64 4.2 Differentiability of Coherent Information . 64 4.2.1 Differentiability of Entropy Exchange and Coherent Information . 65 4.3 Gradient and Hessian . 66 4.3.1 Gradient . 66 4.3.2 Hessian . 71 5 Critical Points and Local Maximas/Minimas 76 5.1 n-shot Coherent Information . 77 5.1.1 Critical Points of Product States . 77 5.2 Unital Channels . 80 5.3 Pauli Channel . 83 5.3.1 Dephasing Channel . 87 5.3.2 Depolarizing Channel . 88 5.4 Dephrasure Channel . 95 5.5 Pauli Erasure Channel . 103 6 Conclusion And Further Work 107 6.1 Further Work . 109 Bibliography 112 vi A Matrix Calculus 115 A.1 Matrix Functions . 115 A.2 Frechet and Gateux Derivatives . 117 A.3 Matrix Exponential and Logarithm . 118 A.4 Power Series . 120 vii List of Figures 2.1 The initial model and assumptions of the coherent information. 16 5.1 Coherent Information of Depolarizing Channel Evaluated at the Maximally Mixed State. 91 5.2 Maximum of Coherent Information of Depolarizing Channel. 92 5.3 von Neumann entropy of the optimal solution of maximum of coherent infor- mation . 93 5.4 The eigenvalue of the Hessian at the Maximally Mixed state of coherent in- formation. 94 viii Chapter 1 Introduction 1.1 Introduction to Problem One of the most challenging problems of Quantum Information is the evaluation of quantum capacity Q(N ) of a quantum channel N , the maximal amount of quantum information that can be sent through N . It was shown in [22], [11], and [28], known as the LSD theorem, that it can be calculated as I (N ⊗n) Q(N ) = lim c ; n!1 n c where Ic(N ) = maxρ2D S(N (ρ)) − S(N (ρ)) is the coherent information of the quantum ⊗n channel N and D is the space of all density matrices. The term Ic(N ) is called the n-shot coherent information. It is the quantum analogue of mutual information found in classical information theory. Unlike the classical case, quantum capacity does not have a single letter expression [27]. This is due to the existence of superadditivity ⊗n Ic(N ) > nIc(N ); for some channel use n. Note that in general, for all n the following inequality holds ⊗n Ic(N ) ≥ nIc(N ). It was shown in 2005 by Devetak and Shor [10], that if the quantum channel is degradable then the quantum capacity has a single letter formula as Q(N ) = Ic(N ). This is due to ⊗n the fact that for all n, the coherent information is subadditive Ic(N ) = nIc(N ): If the quantum channel is antidegradable, then the quantum capacity is known to be zero. The notion of approximate degradable quantum channels were introduced in 2017 and bounds 1 on the quantum capacities were found [32] which were further studied in [18]. The quantum capacity is generally an extremely difficult problem. It is shown in 2015, that an unbounded number of channel uses of optimizing the coherent information may be required to find non-zero quantum capacity [7]. Due to this result, the problem shifted to understanding examples of when superadditivity occurs. This is further studied in the following papers [21] (2017) and [19] (2018), respectively. The overall goal of this thesis is to investigate the critical points (ie when its gradient is zero) of the n-shot coherent information in a coordinate invariant way. This is done by first investigating when the coherent information becomes a smooth map on the space of positive definite density matrices D++. The reasoning behind this restriction is that the global optima is conjectured to generically be positive definite. Additionally, the set of positive definite density matrices is dense inside the space of all density matrices and it has a smooth manifold structure. This allows the calculation of the gradient and Hessian of the coherent information. It will have the advantage of being coordinate invariant and will be shown that critical points can be completely studied on the space of positive definite matrices M++ (rather than D++). 1.1.1 Contributions The contributions of this thesis is as follows. • The restriction of the coherent information to the manifold of postive definite matrices and the conditions on the quantum channel N and the complementary channel N c to insure smoothness of coherent information is explored. This is heavily tied to the notion of a strictly positive map [4]. This is discussed in chapter four. • The Riemannian Gradient of the coherent information has been solved within the dense class of strictly positive quantum channels. It will be shown that a positive definite density matrix ρ is a critical point of the coherent information Ic of a strictly positive quantum channel N iff −N y log(N (ρ)) + (N c)y log(N c(ρ)) 2 spanfIg; where log is the matrix logarithm, N c is the complementary channel, N y is the adjoint of a channel and I is the identity matrix. This is the statement of Theorem 4.3.2. 2 • It will be shown that the coherent information Ic(N ; ρ) can be written using the gra- dient term above, ie Ic(N ; ρ) = hgrad(Ic(N ; ρ))jρi; y where hAjBi = T r(A B) is the Frobenius inner product and the gradient grad(Ic(N ; ρ)) is −N y(log(N (ρ))) + N cy(log(N c(ρ))) on the manifold of positive definite matrices. It will be shown that the coherent information is a positively homogeneous function of degree zero. • In addition, the Riemannian Hessian Hess(Ic(N ; ρ)) of the coherent information Ic at a positive definite density matrix ρ is solved. It is a linear map from trace zero Hermitian matrices H0 to itself such that y c y c Hess(Ic(N ; ρ))[V ] = P − N d logN (ρ) N (V ) + (N ) d logN c(ρ) N (V ) ; where V is a trace zero Hermitian matrix, P is the orthogonal projection from Her- 0 mitian matrices H to trace zero, Hermitian matrices H and d logX : H ! H is the differential of the matrix logarithm.