At Los Alamos National Laboratory, we’re pioneering the next era of quantum information science. Our efforts span foundational research, experimental systems, and real-world applications that align with national security priorities. Learn how LANL is shaping the quantum future.
About Quantum at LANL
Leading the Future of Quantum at LANL
Where theory meets mission-critical innovation
Intro to Quantum
Quantum information is an exciting new paradigm for information science which makes use of the counterintuitive concept of quantum superpositions of information. This new concept raises the possibility of capabilities for information transmission, storage, and manipulation that are simply impossible with conventional information technologies. In the past few years there have been advances in the experimental study of the foundations of quantum mechanics, photonics, and atomic physics that have made accessible these novel uses of quantum states.
Revolution in Computer Science
Moreover, important applications of quantum mechanical concepts to information security and information assurance have been identified, and so this field has recently undergone a dramatic revolution from an essentially academic subject to one with an enormous potential to revolutionize computer science. The realization of these new information science concepts requires the ability to "engineer" quantum mechanical (coherent) states of several particles which have hitherto only been used in quite limited forms for testing the foundations of quantum mechanics.
Foundations of Information Science
The foundations of information science were laid out during and shortly after World War II and tacitly assumed that information, whether in the form of ink on paper or voltages in a microprocessor, would be represented by processes obeying classical physics. However, in the early 1980s Richard Feynman and Charles Bennett (among others) began to investigate the generalization to information represented by quantum physical processes. That is, they considered the representation of binary numbers by orthogonal quantum states (|0> or |1>) of some suitable two-level quantum system. (The representation of a single bit of information in this form has come to be known as a "qubit.")
Examples of Qubit Representation in Systems
Examples of physical systems that permit such a qubit representation are ubiquitous: vertical and horizontal photon polarization states ; single-photon interference states in which a photon can emerge from one or the other exit ports of a Mach-Zehnder interferometer; and the electronic ground and first (metastable) excited state of a (trapped) ion, to list only three.
New Methods for Information Storage, Transmission, and Manipulation
From this pioneering work it has been shown that quantum mechanics opens up powerful new methods for information storage, transmission and manipulation because of the superposition principle, the indivisibility of quanta and the peculiarities of measurement in quantum mechanics.
Qubit="quantum"+"bit"
A qubit is exactly what it sounds like: a bit of information represented by a quantum object, such as a single atom, ion, or photon. Just as a bit is the basic unit of information in a classical information system, a qubit is the basic unit of information of a quantum information system. The power of a qubit is that it is not limited to a value of 0 or 1. It can actually be in a "superposition" state of any combination of 0 or 1. This is the difference between a classical and a quantum computer, which theoretically allows quantum computers to be much more efficient at solving certain types of problems.
Light Polarization
In classical physics, light of a single color is described by an electromagnetic field in which electric and magnetic fields oscillate at a frequency, that is related to the wavelength, by the relation c = , where c is the velocity of light. For example, visible light has wavelengths in the range from 400-750 nm, while longer wavelength radiation (invisible to the eye) is known as infra- red.
Polarized Optical Waves
An important property of optical waves is their polarization: we will define a vertically polarized ("V") wave as one for which the electric field is restricted to lie along the z-axis for a wave propagating along the x-axis, and similarly a horizontally polarized ("H") wave will be defined as one in which the electric field lies along the y-axis. It is a remarkable property of light that any other polarization state of light propagating along the x-axis can be resolved into a linear superposition of vertically polarized and horizontally polarized waves with a particular relative phase. In the case of linearly polarized light the amplitude of the two components is determined by the projections of the polarization direction along the V or H polarization axes. For instance, light linearly polarized along the +45† direction in the y-z plane is an equal amplitude, in-phase superposition of V and H, while light polarized along the -45° direction in the y-z plane is an equal amplitude, opposite-phase superposition. For obvious reasons V and H polarization states (or +45° and -45° polarizations) will be referred to as orthogonal polarizations, while two polarizations (such as V and +45°) that have a non-zero projection will be called non-orthogonal .
Crossed Polarizers
Light of a particular linear polarization can be produced by sending unpolarized light through a polarizing medium (Polaroid) whose polarizing axis is oriented along the direction of the desired linear polarization. When this light is passed through a second polarizer, only the component polarized parallel to the polarizing axis emerges, while the orthogonal component is absorbed. For example if V light impinges on a polarizer oriented at +45° the emerging light is reduced in amplitude by a factor of 2^-1/2 (one over the square root of two), has the +45° polarization and an intensity (proportional to the square of the amplitude) which is 50% of the incident intensity. Likewise, if V light impinges on an H polarizer, no light emerges, and we refer to this configuration as having "crossed polarizers."
Foundations of Information Science
These are the essential features of classical polarized light, but in quantum cryptography we deal with very low intensity light where quantum mechanics must be used. Specifically, during propagation such light has wave-like properties, but on detection exhibits particle-like behavior: the optical energy is quantized into indivisible units of size h, called photons (the elementary particles of electromagnetic radiation) where h is a fundamental constant of Nature known as Planck's constant. The indivisibility of photons raises the interesting question of how a photon of polarized light behaves when it encounters a polarizer. Clearly, there is no difficulty of interpretation if we are only concerned with orthogonal polarizations: a V-photon would pass a V-polarizer with certainty, but be absorbed by an H-polarizer with certainty, for instance. However, once non-orthogonal polarizations are introduced the peculiarities of quantum mechanics become evident.
Consider a +45°-polarized photon impinging on a V-polarizer: there is no such object as a "half photon" by analogy with the classical 50% transmission intensity.
Collapsing the Photon's Wave Function
Instead, quantum mechanics predicts that there is a 50% probability that the photon will be absorbed, and a 50% probability that it will be transmitted, with V-polarization, in any given trial (experiment). Of particular relevance for quantum cryptography is that beyond these probabilities we cannot predict how a particular +45°-photon will behave at a V-polarizer. (Of course the result of a given experiment is either a definite absorption or transmission, but with many repetitions of the experiment we would build up a set of results reflecting the 50-50 absorption-transmission probabilities.) A further relevant quantum peculiarity is that if a +45°-photon passes the V- polarizer it loses all of its "+45°-ness." Specifically, if the emerging photon is made to impinge on a second polarizer oriented at +45° it will be absorbed or transmitted with 50% probability in each case, even though it was originally a +45°-photon. This randomization of properties by non- orthogonal measurements is a crucial element in the detectability of eavesdropping in quantum cryptography. In the terminology of quantum mechanics one says that the V-polarizer has "collapsed the photon's wavefunction."
Orthogonal States
The examples of photons with vertical ("V") or horizontal ("H") polarization introduce the concept of orthogonal quantum states. A "V" photon will never pass a test for "H" polarization (and vice versa), and so using the language of vectors, we say that "V" and "H" are two orthogonal quantum states of a photon. It is a remarkable property of photons that any other single-photon polarization state can be formed from a suitable linear combination of "V" and "H" states, possibly with complex coefficients. We say that single-photon polarization is a two-state (or two-level) quantum system, and that "V" and "H" form a basis for the space of polarizations (an example of a Hilbert space).
Quantum Notation
The allusion to vectors representing quantum states can be made precise. Using a notation introduced by Dirac, it is common to represent quantum states as abstract vectors such as |V> (vertical photon polarization) or |H> (horizontal photon polarization) in Hilbert space. These objects are known as "kets," (the right portion of the word "bracket"). For our purposes we need not be concerned with the formal properties of this vector space, and will merely concern ourselves with some of the basic results.
One of the important properties of vectors is their length, or more generally the length of the component of one vector along another, which is known as the scalar product of the two vectors. In quantum notation, it is useful to introduce the complex conjugate vectors, <V| and <H|, known as "bras" (from the left hand portion of the word bracket) to define the scalar product using the relations:
< V|V> = <H|H> = 1, <V|H> = <H|V> = 0
The length of a quantum state vector is related to the probability of an experimental outcome or measurement, which mathematically is represented by the idea of projection: the component of a vector along another vector. For example, a measurement of "V" polarization would be represented by the projection operator, P(V) = |V><V|. Thus, a measurement for "V" polarization on a |V> photon state produces the state |V>, corresponding to the 100% probability that this state has the "V" polarization on measurement. Conversely, there is no component of an |H> state along the vector |V> which corresponds to the experimental result that an "H" photon will never pass a test for "V"-ness.
Why is Integer Factorization Important?
The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length.... Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.
—Karl Friedrich Gauss, Disquisitiones Arithmeticae (1801) (translation: A. A. Clarke)
Much of modern encryption is accomplished by a system known as public-key cryptography. In this system, a person has a public key and a private key. The public key is used for encryption and may be used by anyone: thus it can, even should, be made public. The private key is used for decryption and therefore must be kept a secret by the person who holds it. Public-key cryptography is designed so that private keys are "computationally infeasible" to obtain from the corresponding public keys. Many public-key cryptosystems are based on the fact that it is very difficult for classical computers to factor large integers: a problem which is theoretically quite simple for a quantum computer. The world of cryptology is very concerned with the feasibility of quantum computers for obvious reasons. Interestingly, while quantum mechanics poses such a threat to cryptography, it also provides a solution in the form of quantum cryptography.
