## The More You Know: Bell States/EPR Pairs

Bell States (named after John S. Bell) or EPR pairs (after Einsetin, Podolski and Rosen Paradox) are the maximally entangled quantum states of a two qubit system (i.e. a quantum mechanical system composed of two interacting two-level subsystems).

In the computational basis $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ the four maximally entangled Bell states are expressed as

$|\Phi^\pm\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}}$,

$|\Psi^\pm\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}}$.

To understand the correlations present in these states consider the Bell state $|\Phi^+\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}$. If a measurement is performed on the first qubit the result may be 0 or 1, after which the two qubit state collapses to $|00\rangle$ or $|11\rangle$ respectively. Then, a measurement on the second qubit must result in 0 or 1, yielding the same result as the measurement on the first one. That is, the outcomes of the measurements are correlated.

John Bell proved that the correlations present in EPR pairs cannot be accounted for by any classical theory. See this article for more.

## The More You Know: Bloch Sphere

The Bloch Sphere is a useful geometrical representation of the state of a Qubit or any other two level quantum system. As discussed in this article an arbitrary pure state $|\psi\rangle$ of a Qubit can be expressed in the computational basis as

$|\psi\rangle=\alpha|0\rangle + \beta|1\rangle$,

where $\alpha$ and $\beta$ are complex number such that $|\alpha|^2+|\beta|^2=1$. Since an overall phase has no observable effects, $\alpha$ or $\beta$ can be chosen  as real numbers and $|\psi\rangle$ can written as

$|\psi\rangle=\cos\frac{\theta}{2}|0\rangle+\exp^{i\phi}\sin\frac{\theta}{2}|1\rangle$,

where $\theta$ and $\phi$ $\in \mathbb{R}$. The parameters $0\leq\theta\leq\pi$ and $0\leq\phi\leq2\pi$ define a point on the $3D$ sphere called the Bloch Sphere (Fig. 1).

Fig. 1: Bloch Sphere representation of the State of a Qubit.

The points in the surface of the sphere represent pure states of the qubit, with $|0\rangle$ as the North pole and $|1\rangle$ as the South pole, and the equator states of the form $|\psi\rangle=|0\rangle+\exp^{i\phi}|1\rangle$. Additionally any mixed state of a qubit can be represented as a point inside the sphere, with maximally mixed states at the center of the Bloch Sphere.

This representation is useful in Quantum Computation to visualize the effect of single qubit gates.

## The More You Know: Qubit

A Qubit, or quantum bit is the fundamental unit of quantum information, it is a mathematical concept satisfying particular properties. The Qubit is the quantum counterpart of the classical bit.

A classical bit can be in one of two different states: 0 or 1, whereas a Qubit can be in the quantum states $|1\rangle$$|0\rangle$ but also in any superposition of these two. That is, the state $|\psi\rangle$  of a Qubit will be of the form

$|\psi\rangle=\alpha|0\rangle + \beta|1\rangle$,

where $\alpha$ and $\beta$ are complex number satisfying

$|\alpha|^2+|\beta|^2=1$.

The states $|1\rangle$ and $|0\rangle$ form the so-called computational basis $\{|1\rangle,|0\rangle\}$. Then, as a mathematical concept, a Qubit is a vector normalized to length 1 in a $2D$ complex vector field.

Just as it’s classical analogue, a Qubit can have many physical realizations: The polarization of a photon, the spin of an electron or a nucleus, the localization of a charge, among other two-state quantum systems.

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## Announcement: Introducing Entangled Physics New Logo

I am very proud to present Entangled Physics new identity.

This logo is composed by two entangled atoms bound by an infinite. It’s design is clean, modern and friendly, and the text “Entangled Physics” is written in an easily readable font. Not only the colors radiate energy, warmth, strength and inspiration, but their intensity brings the image to the foreground.

The logo was created by amazing graphic designer Saida de la Roca.

All text and Images copyright © Marco Vinicio Sebastian Cerezo de la Roca.

## The More You Know: Loss-DiVicenzo Quantum Computer

The Loss-DiVicenzo Quantum Computer is a hardware proposal for the implementation of a scalable spin-based quantum computer in a coupled quantum dot system. The qubits of their quantum computer are realized as the two spin states of a confined electron.

In this proposal, initialization is achieved by applying a strong magnetic field and letting the electrons reach their thermodynamic ground state. Single-qubit operations are performed by varying the effective Zeeman interaction on each dot individually.

Two-qubit quantum gate operations are performed by controlling the electrostatic barrier between neighbouring quantum dots. Thus, if the barrier potential is high, the electrons cannot tunnel from one quantum dot to another and the spins are decoupled. When the barrier is pulsed to a low voltage, the two electron wave functions overlap and the spins will be subject to a non-zero Heisenberg exchange coupling:

$H_{s}(t)=J(t)\vec{S}_1\cdot\vec{S}_2$.

Finally, readout could be performed using spin-to-charge conversion (high-fidelity, single-shot electron charge detection).

References:

[1] “Quantum Computation with Quantum Dots”, D. Lossa, and D. P. DiVincenzo. arXiv:cond-mat/9701055

## The More You Know: EPR Paradox

The EPR Paradox is a thought experiment published in a paper in May 1935 by Albert Einstein and two of his postdoctoral research associates Boris Podolsky and Nathan Rosen that was meant to prove that Quantum Mechanics showed internal contradictions in it’s formulation. The authors claimed that if the description of physical reality given by the wave function is complete, then two quantities described by non-commuting operators could have simultaneous real values. We could for instance measure the position and the momentum of two entangled particles with more accuracy than that allowed by Heisenberg’s Uncertainty Principle. Therefore concluding that quantum-mechanical’s wave function description of reality is incomplete.

This thought experiment is considered to be one of the first papers to put the spotlight on quantum entanglement.

For a more extensive explanation you can visit this post.

This apparent paradox was finally solved by an experiment proposed by physicist J. S. Bell (see Bell’s Inequalities).

References:

[1] “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?“, A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).

## The More You Know: DiVicenzo Criteria

The DiVicenzo Criteria for implementing a quantum computer are a set of requirements that any candidate (circuit model) quantum computer must satisfy. These five requirements, plus two relating to the communication of quantum information were formulated by David P. DiVincenzo in 2008 and are stated as follows:

1. A scalable physical system with well characterized qubits.
2. The ability to initialize the state of the qubits to a simple fiducial state, such as $|000...\rangle$.
3. Long relevant decoherence times, much longer than the gate operation time.
4. A “universal” set of quantum gates.
5. A qubit-specific measurement capability.
6. The ability to interconvert stationary and flying qubits $^1$.
7. The ability faithfully to transmit flying qubits between specified locations.

$^1$ Flying qubit:  qubits that are readily transmitted from place to place.

References:

[1] “The Physical Implementation of Quantum Computation”, D. P. DiVincenzo. arXiv:quant-ph/0002077

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## The More You Know: Boson Sampling

Boson Sampling is a model for non-universal quantum computation which performs specific tasks that are thought to be hard to efficiently perform on a classical computer. By exploiting the processing power of quantum mechanics, the output photon number distribution of a linear optical interferometer with single photon inputs can be sampled. Unlike other models for quantum computation boson sampling does not require any special quantum gate, and this is one of the main reasons for which boson sampling has attracted so much attention among physicists.

## New Article Series: “The More You Know”

In the spirit of writing a “faq / Dictionary of Quantum Information and Quantum Computation” for the blog I’m starting a new article series called: The More You Know.

In each article I’m gonna give a brief definition of terms which are of interest to those beginning to study (or currently studying) Quantum Information and Quantum Computation.

Additionally, I’m also gonna add a new page to the blog called “Dictionary of Quantum Information and Quantum Computation” where all the articles will be listed for easy access.

Hope you like it!

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## Luis Masperi’s Price Special Mention

Every year  during the annual Argentinean Physicists Meeting (Reunión anual de la AFA), all students that got their physics degree can participate in the Luis Masperi Price by presenting the work they did for their thesis. The presentations are graded by a panel of  three judges and results are announced by the end of the meeting.

Last month, during the 100th Annual Meeting I participated for this award, presenting my work on “Non-transverse factorizing fields and entanglement in finite spin systems” (in collaboration with my director Dr. R. Rossignoli and co-director Dr. N. Canosa) and got the second place (special mention)!

You can check my paper based on my thesis here.