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Information theory provides a framework for answering fundamental questions about the optimal performance of many important quantum communication and computational tasks. In many cases, the optimal rates of these tasks can be expressed in terms of regularized formulas that consist of linear combinations of von Neumann entropies optimized over state extensions. However, evaluation of regularized formulas is often intractable, since it involves computing a formula's value in the limit of infinitely many copies of a state. To find optimized, linear entropic functions of quantum states whose regularized versions are tractable to compute, we search for linear combinations of entropies on tripartite quantum states that are additive. We use the method of \cite{cross2017uniform}, which considers bipartite formulas, to identify convex polyhedral cones of uniformly additive \emph{tripartite} correlation measures. We rely only on strong subadditivity of the von Neumann entropy and use these cones to prove that three previously established tripartite optimized correlation measures are additive., Comment: main body: 12 pages, 3 tables appendix: 9 pages

Quantum communication and computation heavily rely on entanglement distillation protocols. There is a plethora of distillation protocols for Pauli channels and also for some non-Pauli channels. However, an effort to relate the effectiveness of these protocols has been missing. For most quantum channels, the gap between the existing lower and upper bounds on distillation rates is substantial, and improvements of achievable rates have been stagnant for decades. In this work, we improve the best known distillation lower bounds, for both the amplitude damping and depolarizing channels. We build on a key observation that distillation protocols reshape several uses of a very noisy channel into a better effective channel. We apply this channel processing in an adaptive and recurrent manner. For the amplitude damping channel, our suggested protocol reshapes the channel into an erasure channel, achieving rates exceeding the best known lower bound given by the channel's reverse coherent information. For the depolarizing channel, we introduce the Greedy recurrence protocol with proven performance guarantees and construct a combined protocol improving upon previously known distillation rates. Improved bounds on attainable distillation rates give insights for both practical implementations and theoretical understanding of quantum information processing., Comment: 20+11 pages, 11 figures. Comments welcome. v2: Changes in presentation throughout article, modified abstract, and some changes to proofs in the Appendix

Entanglement is a key resource for a wide range of tasks in quantum information and computing. Thus, verifying availability of this quantum resource is essential. Extensive research on entanglement detection has led to no-go theorems (Lu et al. [Phys. Rev. Lett., 116, 230501 (2016)]) that highlight the need for full state tomography (FST) in the absence of adaptive or joint measurements. Recent advancements, as proposed by Zhu, Teo, and Englert [Phys. Rev. A, 81, 052339, 2010], introduce a single-parameter family of entanglement witness measurements which are capable of conclusively detecting certain entangled states and only resort to FST when all witness measurements are inconclusive. We find a variety of realistic noisy two-qubit quantum states $\mathcal{F}$ that yield conclusive results under this witness family. We solve the problem of detecting entanglement among $K$ quantum states in $\mathcal{F}$, of which $m$ states are entangled, with $m$ potentially unknown. We recognize a structural connection of this problem to the Bad Arm Identification problem in stochastic Multi-Armed Bandits (MAB). In contrast to existing quantum bandit frameworks, we establish a new correspondence tailored for entanglement detection and term it the $(m,K)$-quantum Multi-Armed Bandit. We implement two well-known MAB policies for arbitrary states derived from $\mathcal{F}$, present theoretical guarantees on the measurement/sample complexity and demonstrate the practicality of the policies through numerical simulations. More broadly, this paper highlights the potential for employing classical machine learning techniques for quantum entanglement detection., Comment: 20 pages, 5 figures

Entanglement distillation is a fundamental information processing task whose implementation is key to quantum communication and modular quantum computing. Noise experienced by such communication and computing platforms occurs not only in the form of Pauli noise such as dephasing (sometimes called $T_2$) but also non-Pauli noise such as amplitude damping (sometimes called $T_1$). We initiate a study of practical and asymptotic distillation over what we call the joint damping-dephasing noise channel. In the practical setting, we propose a distillation scheme that completely isolates away the damping noise. In the asymptotic setting we derive lower bounds on the entanglement sharing capacities including the coherent and reverse coherent information. Like the protocol achieving the reverse coherent information, our scheme uses only backward classical communication. However, for realistic damping noise ($T_1 \neq 2T_2$) our strategy can exceed the reverse coherent strategy, which is the best known for pure damping. In the forward communication setting we numerically exceed the single-letter coherent information strategy by observing the channel displays non-additivity at the two-letter level. The work shows non-additivity can also be found in realistic noise models with magnitudes of non-additivity similar to those found in more idealized noise channels., Comment: 17 page, 8 figures, comments are welcome

Assessment of practical quantum information processing (QIP) remains partial without understanding limits imposed by noise. Unfortunately, mere description of noise grows exponentially with system size, becoming cumbersome even for modest sized systems of imminent practical interest. We fulfill the need for estimates on performing noisy quantum state preparation, verification, and observation. To do the estimation we propose fast numerical algorithms to maximize the expectation value of any $d$-dimensional observable over states of bounded purity. This bound on purity factors in noise in a measurable way. Our fastest algorithm takes $O(d)$ steps if the eigendecomposition of the observable is known, otherwise takes $O(d^3)$ steps at worst. The algorithms also solve maximum likelihood estimation for quantum state tomography with convex and even non-convex purity constraints. Numerics show performance of our key sub-routine (it finds in linear time a probability vector with bounded norm that most overlaps with a fixed vector) can be several orders of magnitude faster than a common state-of-the-art convex optimization solver. Our work fosters a practical way forward to asses limitations on QIP imposed by quantum noise. Along the way, we also give a simple but fundamental insight, noisy systems (equivalently noisy Hamiltonians) always give higher ground-state energy than their noiseless counterparts., Comment: 28 pages, 7 figures, comments are welcome

Sharing entanglement across quantum interconnects is fundamental for quantum information processing. We discuss a practical setting where this interconnect, modeled by a quantum channel, is used once with the aim of sharing high fidelity entanglement. For any channel, we provide methods to easily find both this maximum fidelity and optimal inputs that achieve it. Unlike most metrics for sharing entanglement, this maximum fidelity can be shown to be multiplicative. This ensures a complete understanding in the sense that the maximum fidelity and optimal inputs found in our one-shot setting extend even when the channel is used multiple times, possibly with other channels. Optimal inputs need not be fully entangled. We find the minimum entanglement in these optimal inputs can even vary discontinuously with channel noise. Generally, noise parameters are hard to identify and remain unknown for most channels. However, for all qubit channels with qubit environments, we provide a rigorous noise parametrization which we explain in-terms of no-cloning. This noise parametrization and a channel representation we call the standard Kraus decomposition have pleasing properties that make them both useful more generally., Comment: Essentially matches published version

The capacity of a channel characterizes the maximum rate at which information can be transmitted through the channel asymptotically faithfully. For a channel with multiple senders and a single receiver, computing its sum capacity is possible in theory, but challenging in practice because of the nonconvex optimization involved. To address this challenge, we investigate three topics in our study. In the first part, we study the sum capacity of a family of multiple access channels (MACs) obtained from nonlocal games. For any MAC in this family, we obtain an upper bound on the sum rate that depends only on the properties of the game when allowing assistance from an arbitrary set of correlations between the senders. This approach can be used to prove separations between sum capacities when the senders are allowed to share different sets of correlations, such as classical, quantum or no-signalling correlations. We also construct a specific nonlocal game to show that the approach of bounding the sum capacity by relaxing the nonconvex optimization can give arbitrarily loose bounds. Owing to this result, in the second part, we study algorithms for non-convex optimization of a class of functions we call Lipschitz-like functions. This class includes entropic quantities, and hence these results may be of independent interest in information theory. Subsequently, in the third part, we show that one can use these techniques to compute the sum capacity of an arbitrary two-sender MACs to a fixed additive precision in quasi-polynomial time. We showcase our method by efficiently computing the sum capacity of a family of two-sender MACs for which one of the input alphabets has size two. Furthermore, we demonstrate with an example that our algorithm may compute the sum capacity to a higher precision than using the convex relaxation., Comment: v3: 70 pages, 3 figures; to appear in IEEE Transactions on Information Theory

Understanding quantum channels and the strange behavior of their capacities is a key objective of quantum information theory. Here we study a remarkably simple, low-dimensional, single-parameter family of quantum channels with exotic quantum information-theoretic features. As the simplest example from this family, we focus on a qutrit-to-qutrit channel that is intuitively obtained by hybridizing together a simple degradable channel and a completely useless qubit channel. Such hybridizing makes this channel's capacities behave in a variety of interesting ways. For instance, the private and classical capacity of this channel coincide and can be explicitly calculated, even though the channel does not belong to any class for which the underlying information quantities are known to be additive. Moreover, the quantum capacity of the channel can be computed explicitly, given a clear and compelling conjecture is true. This "spin alignment conjecture," which may be of independent interest, is proved in certain special cases and additional numerical evidence for its validity is provided. Finally, we generalize the qutrit channel in two ways, and the resulting channels and their capacities display similarly rich behavior. In the companion paper [Phys. Rev. Lett. 130, 200801 (2023); arXiv:2202.08377], we further show that the qutrit channel demonstrates superadditivity when transmitting quantum information jointly with a variety of assisting channels, in a manner unknown before., Comment: Comments: 42 pages, 4 figures. v2: matches published version. See also the companion paper arXiv:2202.08377

Determining capacities of quantum channels is a fundamental question in quantum information theory. Despite having rigorous coding theorems quantifying the flow of information across quantum channels, their capacities are poorly understood due to super-additivity effects. Studying these phenomena is important for deepening our understanding of quantum information, yet simple and clean examples of super-additive channels are scarce. Here we study a family of channels called platypus channels. Its simplest member, a qutrit channel, is shown to display super-additivity of coherent information when used jointly with a variety of qubit channels. Higher-dimensional family members display super-additivity of quantum capacity together with an erasure channel. Subject to the "spin-alignment conjecture" introduced in the companion paper [IEEE Trans. Inf. Theory 69(6), pp. 3825-3849, 2023; arXiv:2202.08380], our results on super-additivity of quantum capacity extend to lower-dimensional channels as well as larger parameter ranges. In particular, super-additivity occurs between two weakly additive channels each with large capacity on their own, in stark contrast to previous results. Remarkably, a single, novel transmission strategy achieves super-additivity in all examples. Our results show that super-additivity is much more prevalent than previously thought. It can occur across a wide variety of channels, even when both participating channels have large quantum capacity., Comment: Comments: 25 pages, 9 figures. v2: matches published version. See also the companion paper arXiv:2202.08380

As the title indicates, this chapter presents a brief, self-contained introduction to five fundamental problems in Quantum Information Science (QIS) that are especially well-suited to be formulated as Semi-definite Programs (SDP). We have in mind two audiences. The primary audience comprises of Operations Research (and Computer Science) graduate students who have familiarity with SDPs, but have found it daunting to become even minimally conversant with pre-requisites of QIS. The second audience consists of Physicists (and Electrical Engineers) already knowledgeable with modeling of QIS via SDP but interested in computational tools that are applicable more generally. For both audiences, we strive for rapid access to the unfamiliar material. For the first, we provide just enough required background material (from Quantum Mechanics, treated via matrices, and mapping them in Dirac notation) and simultaneously for the second audience we recreate, computationally in Jupyter notebooks, known closed-form solutions. We hope you will enjoy this introduction and gain understanding of the marvelous connection between SDP and QIS by self-study, or as a short seminar course. Ultimately, we hope this disciplinary outreach will fuel advances in QIS through their fruitful study via SDPs., Comment: 38 pages, 4 figures, and some relevant code, see https://github.com/vsiddhu/SDP-Quantum-OR, included here. v3 essentially matches published version

Quantum queue-channels arise naturally in the context of buffering in quantum networks, wherein the noise suffered by the quantum states depends on the time spent waiting in the buffer. It has been shown that the upper-bound on the classical capacity of an additive queue-channel has a simple expression, and is achievable for the erasure and depolarizing channels [IEEE JSAIT, 1(2):432-444]. In this paper, we characterise the classical capacity for the class of unital qubit queue-channels, and show that a simple product (non-entangled) decoding strategy is capacity-achieving. As an intermediate result, we derive an explicit capacity achieving product decoding strategy for any i.i.d. unital qubit channel, which could be of independent interest. As an important special case, we also derive the capacity and optimal decoding strategies for a symmetric generalized amplitude damping (GAD) queue-channel. Our results provide useful insights towards designing practical quantum communication networks, and highlight the need to explicitly model the impact of buffering., Comment: 25 pages with 3 Figures. Merges and supersedes our prior submissions arXiv:2107.13486 and arXiv:2110.02800

Processing cores and the accompanying main memory working in tandem enable the modern processors. Dissipating heat produced from computation, memory access remains a significant problem for processors. Therefore, processor thermal management continues to be an active research topic. Most thermal management research takes place using simulations, given the challenges of measuring temperature in real processors. Since core and memory are fabricated on separate packages in most existing processors, with the memory having lower power densities, thermal management research in processors has primarily focused on the cores. Memory bandwidth limitations associated with 2D processors lead to high-density 2.5D and 3D packaging technology. 2.5D packaging places cores and memory on the same package. 3D packaging technology takes it further by stacking layers of memory on the top of cores themselves. Such packagings significantly increase the power density, making processors prone to heating. Therefore, mitigating thermal issues in high-density processors (packaged with stacked memory) becomes an even more pressing problem. However, given the lack of thermal modeling for memories in existing interval thermal simulation toolchains, they are unsuitable for studying thermal management for high-density processors. To address this issue, we present CoMeT, the first integrated Core and Memory interval Thermal simulation toolchain. CoMeT comprehensively supports thermal simulation of high- and low-density processors corresponding to four different core-memory configurations - off-chip DDR memory, off-chip 3D memory, 2.5D, and 3D. CoMeT supports several novel features that facilitate overlying system research. Compared to an equivalent state-of-the-art core-only toolchain, CoMeT adds only a ~5% simulation-time overhead. The source code of CoMeT has been made open for public use under the MIT license., Comment: https://github.com/marg-tools/CoMeT

The generalized amplitude damping channel (GADC) is considered an important model for quantum communications, especially over optical networks. We make two salient contributions in this paper apropos of this channel. First, we consider a symmetric GAD channel characterized by the parameter $n=1/2,$ and derive its exact classical capacity, by constructing a specific induced classical channel. We show that the Holevo quantity for the GAD channel equals the Shannon capacity of the induced binary symmetric channel, establishing at once the capacity result and that the GAD channel capacity can be achieved without the use of entanglement at the encoder or joint measurements at the decoder. Second, motivated by the inevitable buffering of qubits in quantum networks, we consider a generalized amplitude damping \emph{queue-channel} -- that is, a setting where qubits suffer a waiting time dependent GAD noise as they wait in a buffer to be transmitted. This GAD queue channel is characterized by non-i.i.d. noise due to correlated waiting times of consecutive qubits. We exploit a conditional independence property in conjunction with additivity of the channel model, to obtain a capacity expression for the GAD queue channel in terms of the stationary waiting time in the queue. Our results provide useful insights towards designing practical quantum communication networks, and highlight the need to explicitly model the impact of buffering., Comment: Comments welcome!

Entanglement lies at the root of quantum theory. It is a remarkable resource that is generally believed to diminish when entangled systems interact with their environment. On the contrary, we find that engaging a system with its environment increases its ability to retain entanglement. The maximum rate of retaining entanglement is given by the quantum channel capacity. We counter-intuitively boost the quantum capacity of a channel by leaking almost all quantum information to the channel's environment. This boost exploits two-letter level non-additivity in the channel's coherent information. The resulting non-additivity has a far larger magnitude and a qualitatively wider extent than previously known. Our findings have a surprising implication for quantum key distribution: maximum rates for key distribution can be boosted by allowing leakage of information to the eavesdropping environment., Comment: 14 pages, 3 figures, comments are welcome

When can noiseless quantum information be sent across noisy quantum devices? And at what maximum rate? These questions lie at the heart of quantum technology, but remain unanswered because of non-additivity -- a fundamental synergy which allows quantum devices (aka quantum channels) to send more information than expected. Previously, non-additivity was known to occur in very noisy channels with coherent information much smaller than that of a perfect channel; but, our work shows non-additivity in a simple low-noise channel. Our results extend even further. We prove a general theorem concerning positivity of a channel's coherent information. A corollary of this theorem gives a simple dimensional test for a channel's capacity. Applying this corollary solves an open problem by characterizing all qubit channels whose complement has non-zero capacity. Another application shows a wide class of zero quantum capacity qubit channels can assist an incomplete erasure channel in sending quantum information. These results arise from introducing and linking logarithmic singularities in the von-Neumann entropy with quantum transmission: changes in entropy caused by this singularity are a mechanism responsible for both positivity and non-additivity of the coherent information. Analysis of such singularities may be useful in other physics problems., Comment: 20 pages, 5 figures, comments are welcome, v2: matches published version

We consider various forms of a process, which we call {\em gluing}, for combining two or more complementary quantum channel pairs $(\mathcal{B},\mathcal{C})$ to form a composite. One type of gluing combines a perfect channel with a second channel to produce a \emph{generalized erasure channel} pair $(\mathcal{B}_g,\mathcal{C}_g)$. We consider two cases in which the second channel is (i) an amplitude-damping, or (ii) a phase-damping qubit channel; (ii) is the \emph{dephrasure channel} of Leditzky et al. For both (i) and (ii), $(\mathcal{B}_g,\mathcal{C}_g)$ depends on the damping parameter $0\leq p\leq 1$ and a parameter $0 \leq \lambda \leq 1$ that characterizes the gluing process. In both cases we study $Q^{(1)}(\mathcal{B}_g)$ and $Q^{(1)}(\mathcal{C}_g)$, where $Q^{(1)}$ is the channel coherent information, and determine the regions in the $(p,\lambda)$ plane where each is zero or positive, confirming previous results for (ii). A somewhat surprising result for which we lack any intuitive explanation is that $Q^{(1)}(\mathcal{C}_g)$ is zero for $\lambda \leq 1/2$ when $p=0$, but is strictly positive (though perhaps extremely small) for all values of $\lambda> 0$ when $p$ is positive by even the smallest amount. In addition we study the nonadditivity of $Q^{(1)}(\mathcal{B}_g)$ for two identical channels in parallel. It occurs in a well-defined region of the $(p,\lambda)$ plane in case (i). In case (ii) we have extended previous results for the dephrasure channel without, however, identifying the full range of $(p,\lambda)$ values where nonadditivity occurs. Again, an intuitive explanation is lacking., Comment: 13 pages (two-column), 6 figures, v2 matches published version

Using a Bayesian methodology, we introduce the maximum a posteriori~(MAP) estimator for quantum state and process tomography. The maximum likelihood, hedged maximum likelihood, maximum likelihood-maximum entropy estimator, and estimators of this general type, can be viewed as special cases of the MAP estimator. The MAP, like the Bayes mean estimator includes prior knowledge, however for cases of interest to tomography can take advantage of convex optimization tools making it numerically easier to compute. We show how the MAP and other Bayesian quantum state estimators can be corrected for noise produced if the quantum state passes through a noisy quantum channel prior to measurement., Comment: 10 pages. Version 2 has an additional Sec. 4.1 with application to qubit tomography under noise, and an Appendix on convex optimization over qubit density operators

We consider a family of quantum channels characterized by the fact that certain (in general nonorthogonal) Pure states at the channel entrance are mapped to (tensor) Products of Pure states (PPP, hence "pcubed") at the complementary outputs (the main output and the "environment") of the channel. The pcubed construction, a reformulation of the twisted-diagonal procedure by M. M Wolf and D. Perez-Garcia, [Phys. Rev. A 75, 012303 (2007)], can be used to produce a large class of degradable quantum channels; degradable channels are of interest because their quantum capacities are easy to calculate. Several known types of degradable channels are either pcubed channels, or subchannels (employing a subspace of the channel entrance), or continuous limits of pcubed channels. The pcubed construction also yields channels which are neither degradable nor antidegradable (i.e., the complement of a degradable channel); a particular example of a qutrit channel of this type is studied in some detail. Determining whether a pcubed channel is degradable or antidegradable or neither is quite straightforward given the pure input and output states that characterize the channel. Conjugate degradable pcubed channels are always degradable., Comment: 17 pages. Version 3: Corrections in Eqns. (54) and (66); otherwise minor changes make it close to the published version

Quantum Private Comparison (QPC) allows us to protect private information during its comparison. In the past various three-party quantum protocols have been proposed that claim to work well under noisy conditions. Here we tackle the problem of QPC under noise. We analyze the EPR-based protocol under depolarizing noise, bit flip and phase flip noise. We show how noise affects the robustness of the EPR-based protocol. We then present a straightforward protocol based on CSS codes to perform QPC which is robust against noise and secure under general attacks., Comment: 8 pages, 1 figure

India has witnessed a major expansion in participation in basic education in recent years, in the context of a major programme for its promotion: Sarva Shiksha Abhiyan (SSA). Despite a significantly improved picture at the basic level, a considerable proportion of children continue to drop out before reaching the secondary level. Over half of children who enrol in grade one do not make it to grade eight, and of those who do complete their basic education (grades one through eight), only 88 percent go on to participate in secondary schooling. Problems surrounding retention are more acute in rural areas of India, and as the country is a predominantly rural society, this study focuses on a farming district of Uttar Pradesh. It is in this dual context of drop out and rural villages that the present study investigates the factors influencing parents' decision whether or not to send their children to secondary schooling. The study draws upon an expanded vision of access to education, embodied in zone 4 of CREATE's framework for considering access and transition in education: looking at those children who complete the primary cycle but who do not make the transition to secondary education. The study focuses on four key aspects in parental school decision making: incentives leading parents to view investment in education as worthwhile; the cost of secondary education; distance to secondary schools; and the impact of health-related factors. The study is based on survey data collected by the author over the first two school years of a four-year longitudinal study: 2008-09 and 2009-10; and the data is analysed using multivariate analysis to identify factors in transition and drop out. Overall the effect of additional cost and distance to secondary schooling proves significant, with cost being the most consistent factor, while distance is found to have a more nuanced effect. The study is particularly relevant in the context of the Government of India's latest major initiative, Rashtriya Madhyamik Shiksha Abhiyan which aims to achieve universal retention of children in secondary education by 2015, and has implications for the direction that interventions under this programme should take. The following are appended: (1) Descriptive statistics for all study variables; (2) Correlation with dependent variable; (3) Odd ratio of likelihood of transition by gender and social groups (Scenario 1); (4) Odd ratio of likelihood of transition by gender and social groups (Scenario 2); and (5) CREATE's zones of exclusion model. (Contains 19 tables, 5 footnotes, and 2 figures.)