Quantum Science Philippines
Quantum Science Philippines

On the EPR paper “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?”

LUTCHIE DYAN S. MENDOZA

In the May 15, 1935 issue of Physical Review , Albert Einstein co-authored a paper with his two postdoctoral research associates at the Institute for Advanced Study, Boris Podolsky and Nathan Rosen. The paper, known as EPR, became a centerpiece in debates, challenging the validity of Quantum Theory.

The paper features a striking case where two quantum systems interact in such a way as to link both their spatial coordinates in a certain direction and also their linear momenta (in the same direction). As a result of this “entanglement”, determining either position or momentum for one system would fix (respectively) the position or the momentum of the other.  In quantum mechanics, in the case of two physical quantities described by non- commuting operators, the knowledge of one prevents the knowledge of the other (Heisenberg Uncertainty Principle). Thus, the paper asserts that, either (1) the quantum- mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities, like position and momentum, do not commute the two quantities can not have simultaneous reality. The authors affirm that one or another of these assertions must hold, giving rise to these two premises: (1) if quantum mechanics were complete (first option failed) then the second option would hold, that is, incompatible quantities cannot have real values simultaneously but (2) that if quantum mechanics were complete, then incompatible quantities (in particular position and momentum) could indeed have simultaneous, real values. They conclude that quantum mechanics is incomplete. The conclusion certainly follows since otherwise if the theory were complete one would have a contradiction. To establish the premises, the authors discuss the idea of a complete theory, offering only a necessary condition. In order for a theory to be complete, every element of the physical reality must have a counterpart in the physical theory, further requiring the criterion: If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity. This is the famous EPR Criterion of Reality.

To realize their assertions further, the authors provided a thought experiment wherein the two quantum systems (one system is labeled Albert while the other system is named Neils) interact in such a way that  conservation of relative position and momentum hold following their interaction. The paper constructs an explicit wave function for the combined system that satisfies both conservation principles. The critical point in the paper centers on the two assumptions made by the authors, namely, separability and locality.  The first assumption states that at the time when measurements will be performed on Albert’s system there is some reality that pertains to Niels’ system alone. In effect, they assume that Niels’ system maintains its separate identity even though it is correlated with Albert’s. The second assumption supposes that no real change can take place in Niels’ system as a consequence of a measurement made on Albert’s system. Locality implies that the prediction of the position of Niels’ system does not involve any change in the reality of Niels’ system. Since the prediction does not disturb Neils’ system, all the pieces are in place to apply the Criterion of Reality. Hence, the authors concluded that Niels’ system can have real values or elements of reality for both position and momentum simultaneously. The negation of the first premise leads to the negation of the only alternative.

Following the result of the thought experiment, separability, locality as well as the application of the Criterion of Reality, EPR concludes that quantum- mechanical description of a physical reality given by the wave functions is not complete.

Reference:

A. EINSTEIN, N. ROSEN and B. PODOLSKY, Phys. Rev. 47, 777 (1935).

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Commutation of Spin, Angular and Spin-Orbital Momentum

Marichu T. Miscala

In quantum mechanics, the presence of spin-orbit coupling gives rise to the Hamiltonian that will no longer commute with [eq]\vec{L}[/eq], and [eq]\vec{S}[/eq], so the spin and orbital momenta are not separately conserved.

In order to understand this concept better, a commutation problem for orbital angular momentum [eq]\vec{L}[/eq], spin [eq]\vec{S}[/eq], and spin-orbital momentum [eq]\vec{J}[/eq] is presented here.

  • Consider the fundamental commutation relations for angular momentum. The individual components of the spin [eq]\vec{S}[/eq] do not commute with each other. That is,

[eq][S_x,S_y] = i\hbar S_z [/eq];         [eq][S_y,S_z] = i\hbar S_x [/eq];            [eq][S_z,S_x] = i\hbar S_y [/eq];

  • The commutation relation for the ‘instrinsic’ angular momentum [eq]\vec{S}[/eq] is much like a ‘carbon copy’ to that of the ‘extrinsic’ angular momentum [eq]\vec{L}[/eq]

[eq][L_x,L_y] = i\hbar L_z [/eq];         [eq][L_y,L_z] = i\hbar L_x [/eq];            [eq][L_z,L_x] = i\hbar L_y [/eq];

But the spin-obit Hamiltonian does commute with [eq]L^2[/eq], [eq]S^2[/eq] and the total angular momentum [eq]\vec{J}[/eq] which is

[eq]\vec{J} = \vec{L} + \vec{S}[/eq]

From the given fundamental commutation relations above, we then now seek the commutators of the following commutation relations:

(a.) [eq][\vec{L}.\vec{S},\vec{L}][/eq]          (b.) [eq][\vec{L}.\vec{S},\vec{S}][/eq]          (c.) [eq][\vec{L}.\vec{S},J][/eq]

(d.) [eq][\vec{L}.\vec{S},L^2][/eq]          (e.) [eq][\vec{L}.\vec{S},S^2][/eq]            (f.)  [eq][\vec{L}.\vec{S},J^2][/eq]

As a hint here, [eq]\vec{L}[/eq] and [eq]\vec{S}[/eq] satisfy the fundamental commutation relations for angular momentum, but [eq]\vec{L}[/eq] and [eq]\vec{S}[/eq] commute with each other.

(a.) [eq][\vec{L} . \vec{S} , L_x] = [L_x S_x + L_y S_y + L_z S_z , L_x] [/eq]

[eq] = S_x [L_x, L_x] + S_y [L_y, L_x] + S_z [L_z, L_x][/eq]
[eq] = S_x (0) + S_y (-i \hbar L_z) + S_z (i \hbar L_y)[/eq]
[eq] = i \hbar (L_y S_z – L_z S_y) = i \hbar (\vec{L} \times \vec{S})_x[/eq]

same goes for the other two-components, so

[eq][\vec{L}.\vec{S}, \vec{L}] = i \hbar (\vec{L} \times \vec {S})[/eq]

(b.) [eq][\vec{L}.\vec{S},\vec{S}][/eq] is identical only with [eq] \vec{L} \Longleftrightarrow \vec{S}[/eq]

[eq][\vec{L}.\vec{S},\vec{S}] = i \hbar (\vec{S} \times \vec{L})[/eq]

(c.) [eq][\vec{L}.\vec{S},\vec{J}] = [\vec{L}.\vec{S},\vec{L}] + [\vec{L}.\vec{S},\vec{S}] = i \hbar (\vec{L} \times \vec{S} + \vec{S} \times \vec{L}) = 0[/eq]

(d.) [eq]L^2[/eq] commutes with all the components of [eq]\vec{L}[/eq] (and [eq]\vec{S}[/eq]), so

[eq][\vec{L} . \vec{S} , L^2] = 0 [/eq]

(e.) Likewise,

[eq][\vec{L} . \vec{S} , S^2] = 0 [/eq]

(f.) [eq][\vec{L} . \vec{S} , J^2] = [\vec{L} . \vec{S} , L^2] + [\vec{L} . \vec{S} , S^2] + 2[\vec{L} . \vec{S} , \vec{L} . \vec{S}][/eq]

where [eq]J^2 = (\vec{L} + \vec{S}) . (\vec{L} + \vec{S}) = L^2 + S^2 + 2 \vec{L} . \vec{S} [/eq]

The first, second and third terms vanish from the results of (d.) and (e.). Thus,

[eq][\vec{L} . \vec{S} , J^2] = 0 [/eq]

This means that the quantities [eq]L^2[/eq], [eq]S^2[/eq] and [eq]\vec{J}[/eq] are conserved. That is, the eigenstates of [eq]L_z[/eq] and [eq]S_z[/eq] are not “good” states to use in perturbation theory, but the eigenstates of [eq]L^2[/eq], [eq]S^2[/eq] and [eq]J_z[/eq] are.

The problem presented here is based on the problem 6.16 from D.J. Griffiths “Introduction to Quantum Mechanics”.

About the author: Marichu T. Miscala is currently taking up her masters in Mindanao State University – Iligan Institute of Technology. She is most interested in pursuing a career in advanced research.

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Quantum Mechanics Violating Bell’s Inequality

Lotis R. Racines


Einstein never liked quantum mechanics. He didn’t like the idea that the momentum of a particle, if it’s position was known, was completely unknowable: that is, random [1]. He even said that: “God does not play dice with the universe.” which was referred to Copenhagen Interpretation of quantum mechanics that there exists no objective physical reality other than that which is revealed through measurement and observation.

Einstein wasn’t the only one who didn’t like Quantum Mechanics. In 1935 he got two other physicists, Boris Podolsky and Nathan Rosen, and wrote a famous paper entitled Can Quantum-Mechanical Description of Physical Reality be Considered Complete? We now refer to it as simply the EPR Paradox. EPR Paradox occurs when phenomenon appears disobeying local realism, EPR’s assumption. With this, EPR concluded that since there were “real” properties of the world not even definable in quantum theory, then quantum theory is “incomplete”.

Aside of Einstein, Podolsky and Rosen, there were already several physicists trying to convey quantum mechanics. One of these physicists was David Joseph Bohm, a British quantum physicist who made significant contributions in the fields of theoretical physics, philosophy, neuropsychology and to the Manhattan Project. He published his first book, Quantum Theory, on 1951 but was not satisfied with some he had written in that  book. So he began to develop his own approach, Bohm interpretation, a non-local hidden variable deterministic theory whose predictions agree perfectly with the nondeterministic quantum theory. In 1959, Bohm and his student, Yakir Aharonov, discovered the Aharonov-Bohm effect, which states that a quantum mechanical phenomenon in which an electrically charged particle shows a measurable interaction with an electromagnetic field despite being confined to a region in which both the magnetic field B and electric field E are zero. (In fact this effect was predicted first by Werner Ehrenberg and Raymond Siday in 1949.) He then claimed that either we abandon the locality principle or we are forced to accept the realization that the electromagnetic potential offers a more complete description of electromagnetism than the electric and magnetic fields can [2]. His work and the EPR argument became the major motivation of John Bell deriving the Bell’s theorem.

In 1964, after a year’s leave from CERN that he spent at Stanford University, the University of Wisconsin-Madison and Brandeis University, he wrote a paper entitled “On the Einstein-Podolsky-Rosen Paradox“. Applying the EPR’s assumption, a mathematical relation that was expressed by an inequality shown below, was derived concerning outcome of some measurements of microscopic particles.

[eq] 1 + P(\vec{b},\vec{c}) \ge | P(\vec{a},\vec{b}) – P(\vec{a},\vec{c})| [/eq]

With this, he showed that carrying forward EPR’s analysis permits one to derive the Bell’s inequality. This inequality conflicts with the predictions of quantum theory. That is, Bell’s inequality shows that there are limits that apply to local hidden variable modes of quantum systems, and that quantum mechanics predicts that these limits will be exceeded by measurements performed on entangled pairs of particles.

Now, there are several yet different responses [3] regarding violation of Bell’s inequality.

  1. to  simply assume that quantum mechanics is wrong. However, experiments undergone have supported quantum mechanics!
  2. to abandon the notion of hidden variables and to argue that the wave function does not contain any information about the outcome of the measurement of the values in the particles which leads to the Copenhagen interpretation of quantum mechanics.
  3. to give up locality in favor of the non-locality principle which leads to Bohm’s interpretation of quantum mechanics. However, this type of interpretation is regarded as inelegant, since it requires all particles in the universe to be able to instantaneously exchange information with all other particles in the universe!
  4. to assume counterfactual definiteness. However, in the Everett many-worlds interpretation, the assumption of counterfactual definiteness is abandoned because this interpretation assumes that the universe branches into many different observers each which measures a different observation!

After Bell’s article appear many experiments from all over the world to test Bell’s theorem. But the most interesting experiment was carried out by a physicist at the University of Geneva, Switzerland, Nicolas Gisin in 1997. He was able to measure entangled particles (emanating from a single particle to two particles) about 10 kilometers apart through a detector. This could mean that particles no matter how distant they maybe, could somehow communicate with each other and somehow affect each others’ measurement.

Up to these days, Bell’s inequality with EPR’s assumed principles is still being debated. However, most of the experiments we’re to violate Bell’s Inequality which implies that locality should not be generally assumed, since there are instances that an action of  a particle could affect the behavior of the other particle though remote; hence, pushes quantum theory  as a better candidate to theoretical and applied physics.

This was based from John Bell’s article in 1964 entitled, “On the Einstein-Podolsky-Rosen Paradox” [4] which basically shows that the principle (“local realism”) assumed by Einstein and colleagues contradicts to the statistical predictions of quantum mechanics.

Reference:

[1] http://library.thinkquest.org/C008537/cool/bellsinequality/bellsinequality.html

[2] http://en.wikipedia.org/wiki/David_Bohm

[3] http://www.starrepublic.org/encyclopedia/wikipedia/b/be/bell_s_theorem.html

[4] J. S. Bell, “On the Einstein Podolsky Rosen Paradox”, Physics Vol. 1, No. 3, 195-200, 1964

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If there’s no separation of reality from the observer, could it be that the universe only exists because we are conscious of it? Or perhaps we only exist because someone or something is conscious of us? This is where physics wonderfully melds with philosophy and religion.

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A little about the Author:

Lotis R. Racines is an MS Physics Student of Mindanao State University – Iligan Institute of Technology, Iligan City. Hopefully, she’ll take up other fields of Science after her Master’s Degree in MSU-IIT.

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Three Dimensional Virial Theorem for the Hydrogen Atom

Catherine Therese J. Quiñones

The virial theorem is a general theorem relating the potential energy (V) and the kinetic energy (T) in a bound system.  A simple physical example is a small object orbiting around another object  bound by a force as in the case of a hydrogen atom. The average kinetic energy and potential energies of a system of particle that interact by Coulomb forces are related by

[eq] \langle T \rangle = -\frac{1}{2} \langle V \rangle [/eq]                                                                                                                  (1)

Since, the Hamiltonian H of the given system is

[eq] \langle H \rangle = \langle T \rangle + \langle V \rangle = E_n [/eq]                                                                                                   (2)

Thus substituting Eqn (1) to Eqn (2) yields,

[eq] -\frac{1}{2}\langle V \rangle + \langle V \rangle = E_n [/eq]                                                                                                       (3)

[eq] \frac{\langle V \rangle}{2} = E_n [/eq]                                                                                                                        (4)

Now, we will derive the expectation value of  [eq]\frac{1}{r}[/eq] in the unperturbed state of the of a hydrogen atom. We can use the virial theorem to easily solve the expectation value since the system can be considered a bound system with the electron orbiting around the proton which is bound by the Coulombic force.

For a hydrogen atom, the potential energy is expressed as

[eq] V = -\frac{e^2}{4\pi\epsilon_0} \frac{1}{r} [/eq]                                                                                                                  (5)

where [eq]e[/eq] is the charge of the electron and the proton, [eq]r[/eq] represents the separation distance between the two charges and [eq]\epsilon_0[/eq] is the permittivity of free space . The negative sign indicates that the force is attractive.

The allowed energies [eq]E_n[/eq] is given by

[eq]E_n = – \Bigg[\frac{m}{2\hbar^2}\Bigg(\frac{e^2}{4\pi\epsilon_0}\Bigg)^2\Bigg]\Bigg{\frac{1}{n^2}\Bigg} [/eq]                                                                                     (6)

where [eq]m[/eq] is the mass of the particle, [eq]\hbar[/eq] is Planck’s constant over [eq]2\pi[/eq] and [eq]n= 0,1,2,3,..[/eq] which indicates the quantization of the energy level. The solution is very straight forward. All we need is to plug in eqn (5) and (6) to eqn (4).  Hence,

[eq]-\frac{e^2}{4\pi\epsilon_0}\Big\langle\frac{1}{r}\Big\rangle = -2\Bigg[ \frac{m}{2\hbar^2}\Bigg(\frac{e^2}{4\pi\epsilon_0}\Bigg)^2\Bigg] \Bigg{\frac{1}{n^2}\Bigg} [/eq]                                                                                                                                           (7)

[eq]\Big\langle\frac{1}{r}\Big\rangle=\Bigg(\frac{me^2}{4\pi\epsilon_0\hbar^2}\Bigg)\frac{1}{n^2}[/eq]                                                                                                        (8)

Note that the term inside the parenthesis is just [eq]\frac{1}{a_0}[/eq], where [eq]a_0[/eq] is the Bohr radius . Hence we can write the expectation value of [eq]\frac{1}{r}[/eq] as,

[eq]\Big\langle\frac{1}{r}\Big\rangle = \frac{1}{a_0n^2}[/eq]                                                                                                                      (9)

Thus we have derived the expectation value, [eq]\Big\langle\frac{1}{r}\Big\rangle[/eq], of the hydrogen atom in the unperturbed state using the virial theorem.

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BATTLING DECOHERENCE: THE FAULT-TOLERANT QUANTUM COMPUTER

EDWIN B. FABILLAR

Introduction

Information carried by a quantum system has notoriously weird properties. Physicists and engineers are now learning how to put that weirdness to work. Quantum computers, which manipulate quantum states rather than classical bits, may someday be able to perform tasks that would be inconceivable with conventional digital technology. A particularly daunting difficulty is that quantum computers are highly susceptible to making errors. The magical power of the quantum computer comes from its ability to process coherent quantum states; but such states are very easily damaged by uncontrolled interactions with the environment—a process called decoherence. In response to the challenge posed by decoherence, the new discipline of quantum error correction has arisen at the interface of physics and computer science. We have learned that quantum states can be cleverly encoded so that the debilitating effects of decoherence, if not too severe, can be resisted.

The power of the quantum computer

A classical computer executes a series of simple operations (often called “gates”), each of which acts on a single bit or pair of bits. By executing many gates in succession, the computer can evaluate any Boolean function of a set of input bits. Also a quantum computer executes a series of elementary quantum gates, each of which is a unitary transformation that acts on a single qubit or pair of qubits. By executing many such gates in succession, the quantum computer can apply a complicated unitary transformation to a particular initial state of a set of qubits.  A classical computer can faithfully simulate a quantum  computer, so that anything the quantum computer could do, the classical computer could also do. Still, there is a sense in which the quantum computer appears to be a more powerful device: In more physical terms, running a classical simulation of a quantum computer is hard because (as exemplified by John Bell’s famous inequalities) correlations among quantum bits are qualitatively different from correlations among classical bits. The exponential explosion in the size of Hilbert space as we increase the number of qubits arises because the correlations among qubits are too weird to be expressed easily in classical language.

The challenge of error correction

Are there any obstacles that might be fundamental matters of principle, that would prevent us from ever constructing a quantum computer? In fact, there is a problem of principle that is potentially very serious: decoherence. Unavoidable interactions with the environment will cause the quantum information stored in a quantum computer to decay, thus inducing errors in the computation. If quantum computers are ever to be capable of solving hard problems, a means must be found to control decoherence and other potential sources of error.

Quantum error-correcting codes

In 1995, Shor and Andrew Steane discovered that the obstacles were illusory— that quantum error correction really is possible. To appreciate the insights of Shor and Steane, lets first consider how to defend quantum information against a dragon who performs only bit flips . We are to protect the state

(1)               [eq] a|0> + b|1> [/eq],

a coherent superposition of the red (|0>) and green (|1>) states of a single qubit, where the complex coefficients a and b are unknown. Were the dragon to attack, the bit flip would transform the state to

(2)               [eq] a|1> + b|0>[/eq],

and damage would be inflicted unless a =±b. The beaver’s assignment is to diagnose and reverse bit flips, but without disturbing the delicate superposition state, that is without modifying a and b, the beaver applies the principle of redundant storage by encoding the qubit in a state of three qubits.

The red state is encoded as three red qubits, and the green state as three green qubits; that is,

(3.a)            [eq] |0> \rightarrow |0> = |000>[/eq],

(3.b)            [eq] |1> \rightarrow |1> = |111>[/eq],

Thus the unknown superposition state becomes

(4)                [eq] a|0> + b|1> \rightarrow a|0> + b|1> = a|000> + b|111> [/eq],

This redundant state is not the same as three identical copies of the original unknown state, which would be

(5)               [eq](a|0> + b|1>) (a|0> + b|1>) (a|0> + b|1>)[/eq],

Although it is impossible to copy unknown quantum information, nothing prevents us from building a (unitary) machine that will execute the encoding transformation given as equation 4.

Now suppose that the dragon flips one of the three qubits, let’s say the first one, so that the state becomes

(6)                  [eq] a|100> + b|011>[/eq],

and the beaver is to detect and reverse the damage. His first impulse would be to open the boxes and look to see if one ball was a different color from the others, just as he would to diagnose errors in classical information, but he must resist that temptation. If he were to open door 1 of all three boxes, he would find either I100>  (with probability |a|2), or I011 >  (with probability IbI2); either way, the coherent quantum information (the values of a and b) would be irrevocably lost. But he is a clever beaver who knows he need not restrict his attention to single-qubit measurements. Instead, he performs collective measurements on two qubits at once. The beaver won that round, but now the dragon tries a more subtle approach. Rather than flipping the first qubit, he rotates it only slightly, so that the three-qubit state becomes

(7)                    [eq] a|000> + b|111> \rightarrow a|000> + b|111> [/eq]

[eq] + e(a|100> + b|011>) + o (e2)[/eq],                    

where |e| >> 1. What should the beaver do now? In fact, he can do the same thing as before. If he performs a collective measurement on the first two qubits, then most of the time

(with probability [eq] 1 – |e|^2 [/eq]), the measurement well project the damaged state (equation 7) back to the completely undamaged state (equation 4). Only much more rarely (with probability [eq] |e|^2 [/eq]) will the measurement project onto the state given as equation 6 with a bit-flip error. But then the measurement outcome tells the beaver what action to take to repair the damage, just as in theprevious case.

Collective measurement and fault tolerance

Topological ideas arise naturally in the theory of fault tolerance. The topological properties of an object remain invariant when we smoothly deform the object. Similarly, how a fault-tolerant gate acts on encoded information should remain unchanged when we deform the gate by introducing a small amount of noise. In seeking fault-tolerant implementations of quantum logic, we are led to contemplate physical interactions with a topological character. What comes quickly to mind is the Aharonov-Bohm effect. When an electron is transported around a magnetic flux tube, its wave function acquires a phase that depends only on the winding number of the electron about the solenoid; it is unmodified if the electron’s trajectory is slightly deformed. If we could perform quantum logic by means of topological interactions, then we would be able to give the beaver a rest! We could protect encoded information not by vigilantly checking for errors and reversing them, but rather by weaving fault tolerance into the design of our hardware.

Outlook

For a quantum computer to compete with a state-of-the-art classical computer, we will need machines with hundreds or thousands of qubits capable of performing millions or billions of operations. The technology clearly has far to go before quantum computers can assume their rightful place as the world’s fastest machines. But now that we know how to protect quantum information from errors, there are no known insurmountable obstacles blocking the path. Quantum computers of the 21st century may well unleash the vast computational power woven into the fundamental laws of physics. Apart from enabling a new technology, the discovery of fault-tolerant methods for quantum error correction and quantum computation may have deep implications for the future of physics. What else might coherent quantum systems be capable of? In what ways will they surprise, baffle, and delight us? Armed with new tools for maintaining and controlling intricate quantum states, physicists of the next century will seek the answers.

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