General Strong Polarization

Arikan's exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix $M$, a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the {\em polarization} of an associated $[0,1]$-bounded martingale, namely its convergence in the limit to either $0$ or $1$. Arikan showed polarization of the martingale associated with the matrix $G_2 = \left(\begin{matrix} 1& 0 1& 1\end{matrix}\right)$ to get capacity achieving codes. His analysis was later extended to all matrices $M$ that satisfy an obvious necessary condition for polarization. While Arikan's theorem does not guarantee that the codes achieve capacity at small blocklengths, it turns out that a "strong" analysis of the polarization of the underlying martingale would lead to such constructions. Indeed for the martingale associated with $G_2$ such a strong polarization was shown in two independent works ([Guruswami and Xia, IEEE IT '15] and [Hassani et al., IEEE IT '14]), resolving a major theoretical challenge of the efficient attainment of Shannon capacity. In this work we extend the result above to cover martingales associated with all matrices that satisfy the necessary condition for (weak) polarization. In addition to being vastly more general, our proofs of strong polarization are also simpler and modular. Specifically, our result shows strong polarization over all prime fields and leads to efficient capacity-achieving codes for arbitrary symmetric memoryless channels. We show how to use our analyses to achieve exponentially small error probabilities at lengths inverse polynomial in the gap to capacity. Indeed we show that we can essentially match any error probability with lengths that are only inverse polynomial in the gap to capacity.Comment: 73 pages, 2 figures. The final version appeared in JACM. This paper combines results presented in preliminary form at STOC 2018 and RANDOM 201

Polar Codes with exponentially small error at finite block length

We show that the entire class of polar codes (up to a natural necessary condition) converge to capacity at block lengths polynomial in the gap to capacity, while simultaneously achieving failure probabilities that are exponentially small in the block length (i.e., decoding fails with probability $\exp(-N^{\Omega(1)})$ for codes of length $N$). Previously this combination was known only for one specific family within the class of polar codes, whereas we establish this whenever the polar code exhibits a condition necessary for any polarization. Our results adapt and strengthen a local analysis of polar codes due to the authors with Nakkiran and Rudra [Proc. STOC 2018]. Their analysis related the time-local behavior of a martingale to its global convergence, and this allowed them to prove that the broad class of polar codes converge to capacity at polynomial block lengths. Their analysis easily adapts to show exponentially small failure probabilities, provided the associated martingale, the ``Arikan martingale'', exhibits a corresponding strong local effect. The main contribution of this work is a much stronger local analysis of the Arikan martingale. This leads to the general result claimed above. In addition to our general result, we also show, for the first time, polar codes that achieve failure probability $\exp(-N^{\beta})$ for any $\beta < 1$ while converging to capacity at block length polynomial in the gap to capacity. Finally we also show that the ``local'' approach can be combined with any analysis of failure probability of an arbitrary polar code to get essentially the same failure probability while achieving block length polynomial in the gap to capacity.Comment: 17 pages, Appeared in RANDOM'1

Universal Polarization

A method to polarize channels universally is introduced. The method is based on combining two distinct channels in each polarization step, as opposed to Arikan's original method of combining identical channels. This creates an equal number of only two types of channels, one of which becomes progressively better as the other becomes worse. The locations of the good polarized channels are independent of the underlying channel, guaranteeing universality. Polarizing the good channels further with Arikan's method results in universal polar codes of rate 1/2. The method is generalized to construct codes of arbitrary rates. It is also shown that the less noisy ordering of channels is preserved under polarization, and thus a good polar code for a given channel will perform well over a less noisy one.Comment: Submitted to the IEEE Transactions on Information Theor

Fast Polarization for Processes with Memory

Fast polarization is crucial for the performance guarantees of polar codes. In the memoryless setting, the rate of polarization is known to be exponential in the square root of the block length. A complete characterization of the rate of polarization for models with memory has been missing. Namely, previous works have not addressed fast polarization of the high entropy set under memory. We consider polar codes for processes with memory that are characterized by an underlying ergodic finite-state Markov chain. We show that the rate of polarization for these processes is the same as in the memoryless setting, both for the high and for the low entropy sets.Comment: 17 pages, 3 figures. Submitted to IEEE Transactions on Information Theor

Hamilton's turns as visual tool-kit for designing of single-qubit unitary gates

Unitary evolutions of a qubit are traditionally represented geometrically as rotations of the Bloch sphere, but the composition of such evolutions is handled algebraically through matrix multiplication [of SU(2) or SO(3) matrices]. Hamilton's construct, called turns, provides for handling the latter pictorially through the as addition of directed great circle arcs on the unit sphere S$^2 \subset \mathbb{R}^3$, resulting in a non-Abelian version of the parallelogram law of vector addition of the Euclidean translation group. This construct is developed into a visual tool-kit for handling the design of single-qubit unitary gates. As an application, it is shown, in the concrete case wherein the qubit is realized as polarization states of light, that all unitary gates can be realized conveniently through a universal gadget consisting of just two quarter-wave plates (QWP) and one half-wave plate (HWP). The analysis and results easily transcribe to other realizations of the qubit: The case of NMR is obtained by simply substituting $\pi/2$ and $\pi$ pulses respectively for QWPs and HWPs, the phases of the pulses playing the role of the orientation of fast axes of these plates.Comment: 16 Pages, 14 Figures, Published versio

A compact and robust method for full Stokes spectropolarimetry

We present an approach to spectropolarimetry which requires neither moving parts nor time dependent modulation, and which offers the prospect of achieving high sensitivity. The technique applies equally well, in principle, in the optical, UV or IR. The concept, which is one of those generically known as channeled polarimetry, is to encode the polarization information at each wavelength along the spatial dimension of a 2D data array using static, robust optical components. A single two-dimensional data frame contains the full polarization information and can be configured to measure either two or all of the Stokes polarization parameters. By acquiring full polarimetric information in a single observation, we simplify polarimetry of transient sources and in situations where the instrument and target are in relative motion. The robustness and simplicity of the approach, coupled to its potential for high sensitivity, and applicability over a wide wavelength range, is likely to prove useful for applications in challenging environments such as space.Comment: 36 pages, 11 figures, 3 tables; accepted for publication in Applied Optic