{"id":1785,"date":"2018-03-29T15:47:23","date_gmt":"2018-03-29T14:47:23","guid":{"rendered":"https:\/\/www.quantum-bits.org\/?p=1785"},"modified":"2022-08-12T17:17:49","modified_gmt":"2022-08-12T16:17:49","slug":"having-fun-with-bell-states-and-q","status":"publish","type":"post","link":"https:\/\/www.quantum-bits.org\/?p=1785","title":{"rendered":"Having fun with Bell states and Q#"},"content":{"rendered":"

In a previous post<\/a>, we introduced the basic concepts of quantum computing, during which quantum entanglement, Bell states and usual quantum gates such as Hadamard’s or Pauli’s were quickly addressed.<\/p>\n

Today, we’re going to explore some of these concepts and start having fun with quantum programming<\/strong> to illustrate these ideas.<\/p>\n

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Quantum entanglement<\/strong><\/p>\n

Quantum entanglement is a phenomenon which occurs when pairs (or groups) of particles are generated (or interact) in such a way that the quantum state of each particle cannot be described independently of the state of the other(s), even when the particles are separated by a large distance:  the quantum state must be described for the system as a whole.<\/p>\n

Measurements of physical properties (position, momentum, spin, polarization …) performed on entangled particles are found to be correlated. Let’s imagine a pair of particles generated in such a way that their total spin is known to be zero.<\/p>\n

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Now, let’s imagine that one particle is found to have clockwise spin on a certain axis. Because they are entangled, the spin of the other particle, measured on the same axis, will be found to be counterclockwise !<\/p>\n

This behavior gives rise to paradoxical effects: any measurement of a property of a particle can be seen as acting on that particle. In the case of entangled particles, such a measurement will be on the entangled system as a whole.<\/p>\n

It seems to appear that one particle of an entangled pair “knows” what measurement has been performed on the other, and with what outcome, even though there is no known means for such information to be communicated between the particles, which at the time of measurement may be separated by any arbitrarily large distances (Einstein referring to it as “spooky action at a distance”).<\/p>\n

EPR paradox<\/strong><\/p>\n

Such phenomena were the subject of a 1935 paper by Albert Einstein<\/a>, Boris Podolsky<\/a>, and Nathan Rosen<\/a> (and several papers by Erwin Schr\u00f6dinger<\/a>) describing what came to be known as the EPR paradox<\/a>.<\/p>\n

This thought experiment<\/a> aims at the Copenhagen interpretation<\/a> (more precisely, at the wave function collapse<\/a>). Resolutions of the paradox have important implications for the interpretation of quantum mechanics<\/a>.<\/p>\n

In quantum mechanics, a particle is described by a quantum state. This quantum state can be represented as a superposition (i.e. a linear combination) of basis states. In principle one is free to choose the set of basis states, as long as they span the space. If one chooses the eigenfunctions<\/a> of the position operator<\/a> q as a set of basis functions, one speaks of a state as a wave function<\/a> position space.<\/p>\n

For a quick recap on bra-ket notations, inner products, operators, observables and all that jazz, please, have a look at this (old) post<\/a>.<\/p>\n

Let’s define the ket |\\psi\\rangle, representing a quantum state. The wave function (position space) \\psi(q) can be defined as \\psi(q) = \\langle q | \\psi\\rangle .<\/p>\n

This wave function is interpreted as a (complex-valued) probability amplitude. The probabilities \\mathcal{P}(q) for the possible results of measurements made on the system are :<\/p>\n

\\mathcal{P}(q) = |\\psi(q)|^2 = \\psi(q)^*\\psi(q)=|\\langle q | \\psi\\rangle|^2<\/p>\n

Quantum frameworks do not provide single measurement outcomes in a deterministic way. According to the understanding of quantum mechanics known as the Copenhagen interpretation, measurement causes an instantaneous collapse of the wave function describing the quantum system into an eigenstate of the observable that was measured.<\/p>\n

Einstein was one of the most prominent opponent of the Copenhagen interpretation. In his view, quantum mechanics was incomplete. Commenting on this, other writers (such as John von Neumann<\/a> and David Bohm<\/a><\/sup>) hypothesized that consequently there would have to be ‘hidden’ variables<\/a> responsible for random measurement results.<\/p>\n

The article that first brought forth these matters, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” was published in 1935 by Einstein, Podolsky, and Rosen (EPR). The paper prompted a response by Niels Bohr<\/a>, which he published in the same journal, in the same year, using … the same title \ud83d\ude42<\/p>\n

Einstein had been skeptical of the Heisenberg uncertainty principle<\/a> and the role of chance in quantum theory.<\/p>\n

But the crux of this debate was not about chance, but something deeper:<\/p>\n