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As theoretical physics delves deeper into the fundamental nature of reality, we’re left to grapple with the questions it leaves us. For example, some physicists claim that our universe is merely an illusion, a product of quantum machinations happening in a lower-dimensional setting—in other words, a hologram. But do these latest theoretical insights offer revelations into reality, itself, or merely serve as mathematical tools to help us solve thorny problems?…….Continue reading….
Source: Popular Mechanics
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The history of quantum information theory began at the turn of the 20th century when classical physics was revolutionized into quantum physics. The theories of classical physics were predicting absurdities such as the ultraviolet catastrophe, or electrons spiraling into the nucleus. At first these problems were brushed aside by adding ad hoc hypotheses to classical physics.
Soon, it became apparent that a new theory must be created in order to make sense of these absurdities, and the theory of quantum mechanics was born.Quantum mechanics was formulated by Erwin Schrödinger using wave mechanics and Werner Heisenberg using matrix mechanics. The equivalence of these methods was proven later.
Their formulations described the dynamics of microscopic systems but had several unsatisfactory aspects in describing measurement processes. Von Neumann formulated quantum theory using operator algebra in a way that it described measurement as well as dynamics.
These studies emphasized the philosophical aspects of measurement rather than a quantitative approach to extracting information via measurements. With the advent of Alan Turing’s revolutionary ideas of a programmable computer, or Turing machine, he showed that any real-world computation can be translated into an equivalent computation involving a Turing machine. This is known as the Church–Turing thesis.
Soon enough, the first computers were made, and computer hardware grew at such a fast pace that the growth, through experience in production, was codified into an empirical relationship called Moore’s law. This ‘law’ is a projective trend that states that the number of transistors in an integrated circuit doubles every two years. As transistors began to become smaller and smaller in order to pack more power per surface area, quantum effects started to show up in the electronics resulting in inadvertent interference.
This led to the advent of quantum computing, which uses quantum mechanics to design algorithms. At this point, quantum computers showed promise of being much faster than classical computers for certain specific problems. One such example problem was developed by David Deutsch and Richard Jozsa, known as the Deutsch–Jozsa algorithm. This problem however held little to no practical applications.
Peter Shor in 1994 came up with a very important and practical problem, one of finding the prime factors of an integer. The discrete logarithm problem as it was called, could theoretically be solved efficiently on a quantum computer but not on a classical computer hence showing that quantum computers should be more powerful than Turing machines.
Quantum information differs strongly from classical information, epitomized by the bit, in many striking and unfamiliar ways. While the fundamental unit of classical information is the bit, the most basic unit of quantum information is the qubit. Classical information is measured using Shannon entropy, while the quantum mechanical analogue is Von Neumann entropy. Given a statistical ensemble of quantum mechanical systems with the density matrix.
Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy and the conditional quantum entropy. Unlike classical digital states (which are discrete), a qubit is continuous-valued, describable by a direction on the Bloch sphere. Despite being continuously valued in this way, a qubit is the smallest possible unit of quantum information, and despite the qubit state being continuous-valued, it is impossible to measure the value precisely.
Five famous theorems describe the limits on manipulation of quantum information. No-teleportation theorem, which states that a qubit cannot be (wholly) converted into classical bits; that is, it cannot be fully “read”. No-cloning theorem, which prevents an arbitrary qubit from being copied. No-deleting theorem, which prevents an arbitrary qubit from being deleted.
No-broadcast theorem, which prevents an arbitrary qubit from being delivered to multiple recipients, although it can be transported from place to place (e.g. via quantum teleportation). No-hiding theorem, which demonstrates the conservation of quantum information. These theorems are proven from unitarity, which according to Leonard Susskind is the technical term for the statement that quantum information within the universe is conserved.
The five theorems open possibilities in quantum information processing.Quantum mechanics is the study of how microscopic physical systems change dynamically in nature. In the field of quantum information theory, the quantum systems studied are abstracted away from any real world counterpart. A qubit might for instance physically be a photon in a linear optical quantum computer, an ion in a trapped ion quantum computer, or it might be a large collection of atoms as in a superconducting quantum computer.
Regardless of the physical implementation, the limits and features of qubits implied by quantum information theory hold as all these systems are mathematically described by the same apparatus of density matrices over the complex numbers. Another important difference with quantum mechanics is that while quantum mechanics often studies infinite-dimensional systems such as a harmonic oscillator, quantum information theory is concerned with both continuous-variable systems and finite-dimensional systems.
Quantum communication is one of the applications of quantum physics and quantum information. There are some famous theorems such as the no-cloning theorem that illustrate some important properties in quantum communication. Dense coding and quantum teleportation are also applications of quantum communication. They are two opposite ways to communicate using qubits.
While teleportation transfers one qubit from Alice and Bob by communicating two classical bits under the assumption that Alice and Bob have a pre-shared Bell state, dense coding transfers two classical bits from Alice to Bob by using one qubit, again under the same assumption, that Alice and Bob have a pre-shared Bell state.
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