Home | About | Journals | Submit | Contact Us | Français |

**|**Springer Open Choice**|**PMC5591609

Formats

Article sections

- Abstract
- Introduction
- The device
- It can’t be built!
- Oh, but it can be built!
- Why?
- Under the mathematical microscope
- Conclusion?
- References

Authors

Related links

Quantum Information Processing

Quantum Inf Process. 2016; 15(3): 1043–1056.

Published online 2015 December 28. doi: 10.1007/s11128-015-1206-7

PMCID: PMC5591609

University of Maryland Baltimore County (UMBC), Baltimore, MD 21250 USA

Samuel J. Lomonaco, Email: ude.cbmu@ocanomoL, http://www.csee.umbc.edu/~lomonaco.

Received 2015 October 27; Accepted 2015 November 27.

Copyright © The Author(s) 2015

In this paper, we show how the GHZ paradox can be used to design a computing device that cannot be physically implemented within the context of classical physics, but nonetheless can be within quantum physics, i.e., in a quantum physics laboratory. This example gives an illustration of the many subtleties involved in the quantum control of distributed quantum systems. We also show how the second elementary symmetric Boolean function can be interpreted as a quantification of the nonlocality and indeterminism involved in the GHZ paradox.

This paper began with an invitation to give the Annual George Washington University Mathematics Department April Fools Day Lecture in April of 2014. After some thought, I decided what better topic to choose for the talk than how quantum mechanics makes fools of us all. For that reason, I chose to speak on the GHZ paradox, as embodied in Mermin’s machine [5].

In this paper, we show how the Greenberger–Horne–Zeilinger (GHZ) paradox can be used to design a computing device that cannot be physically implemented within the context of classical physics, but nonetheless can be within quantum physics, i.e., in a quantum physics laboratory. This example gives an illustration of the many subtleties involved in the quantum control of distributed quantum systems [8].

Corollary 1 in Sect. 6 can be
interpreted as showing that the second elementary symmetric Boolean function
*σ*_{2} explicitly quantifies the nonlocality and
indeterminism involved in the GHZ paradox.

A blueprint describing Mermin’s machine [5, 6] is shown below in Fig. 1:

As illustrated, the device consists of two different types of components, i.e., a
**source**
** S**, and three identical

The source, as illustrated below in Fig. 2, is a device that contains three objects, called
**particles**, labeled ** A**,

Each detector, upon encountering an incoming particle, flashes either red
** R** or green

As stated below in the design specifications, the only switch settings of interest are those for which an odd number of the three switches is set to 1, i.e.,

No other switch settings are important, i.e., of interest.

The design specifications are as follows:

*Spec 1*- After all particles are detected, for switch settings
001, 010, and 100,
**only an odd number**of the detectors flash red.*R* *Spec 2*- After all particles are detected, for switch setting
111,
**only an even number**of detectors flash red.*R*

These design specifications are subject to the following three constraints:

*Constraint 1*- The detectors cannot communicate with one another. (They are separated by a spacelike distance.)
*Constraint 2*- After being ejected from the source
*S*, the particles can no longer communicate with one another. *Constraint 3*- The particles only communicate with the detector upon impact.

Because of the above constraints, each particle must locally carry instructions
telling its respective detector whether to flash red ** R** or green

For example, particle ** A** must carry a

where *c*_{A0} = ** R** or

Let us rename the colors ** R** and

It is now immediate that Specs 1 and 2 are equivalent to the following linear system of equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{f}_{A}(0)+{f}_{B}(0)+{f}_{C}(1)=1\left(\text{mod}2\right)\hfill \\ {f}_{A}(0)+{f}_{B}(1)+{f}_{C}(0)=1\left(\text{mod}2\right)\hfill \\ {f}_{A}(1)+{f}_{B}(0)+{f}_{C}(0)=1\left(\text{mod}2\right)\hfill \\ {f}_{A}(1)+{f}_{B}(1)+{f}_{C}(1)=0\left(\text{mod}2\right)\hfill \end{array}\right.\end{array}$$

which is obviously inconsistent.

In other words, the device cannot be built! It’s simply impossible. □

However, within the context of quantum physics, it can actually be built, i.e., can be physically implemented.

But before we can show how this device can actually be built, we need a few definitions.

We define a **Boolean unitary transformation** as a map from
{0,1}^{k} into a group of unitary
transformations. In like manner, a **Boolean Hermitian operator**
is defined as a map from {0,1}^{k} into an algebra of observables. If
*b* ( = 0 or 1) and if *U* is a
unitary transformation, then *U*^{b} will denote the Boolean unitary
transformation

where *I* denotes the identity operator.
In like manner, if *Ω* is an observable, then *b**Ω* will denote the Boolean
observable

where *O* denotes the zero operator.

In other words, Boolean unitary and Boolean Hermitian operators are unitary and Hermitian transformations controlled by classical bits.

There is much more that could be said in regard to Boolean unitary and
Hermitian operators. But that would take us too far afield of the intended
objectives of this paper. So the following will have to suffice: Let
𝔹 be a Boolean algebra or Boolean ring.
Let 𝕌 be a unitary group, and let
**u** denote its Lie algebra. The set
𝕌^{𝔹} = *m**a**p*(𝔹, 𝕌) of Boolean unitary operators forms a
Lie group containing the group 𝕌 as a sub-Lie group. Moreover, the set
**u**^{𝔹} = *m**a**p*(𝔹, **u**) of Boolean Skew Hermitian operators is
the Lie algebra of 𝕌^{𝔹} and contains **u** as a sub-Lie algebra.

Let *X*, *Y*, *Z*, respectively,
denote the Pauli spin operators

$$\begin{array}{c}\hfill X=\left(\begin{array}{cc}\hfill 0& \hfill 1\\ \hfill 1& \hfill 0\end{array}\right)\text{,}\phantom{\rule{4pt}{0ex}}Y=\left(\begin{array}{cc}\hfill 0& \hfill -i\\ \hfill i& \hfill 0\end{array}\right)\text{,}\phantom{\rule{4pt}{0ex}}Z=\left(\begin{array}{cc}\hfill 1& \hfill 0\\ \hfill 0& \hfill -1\end{array}\right).\end{array}$$

Moreover, let *H* denote
the Hadamard gate

$$\begin{array}{c}\hfill H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 1& \hfill -1\end{array}\right)\phantom{\rule{0.333333em}{0ex}}\text{,}\end{array}$$

and let *U* be the
single-qubit gate

$$\begin{array}{c}\hfill U={e}^{\left[\frac{i\mathit{\pi}}{3}\left(\frac{X+Y+Z}{\sqrt{3}}\right)\right]}=\frac{1+i}{2}\left(\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill i& \hfill -i\end{array}\right)\phantom{\rule{0.333333em}{0ex}}\text{.}\end{array}$$

A wiring diagram summarizing a physical implementation of Mermin’s machine is shown in Fig. 4.

In this diagram, a single line indicates a wire carrying a qubit and a double line indicates a wire carrying a classical bit. The graphics

denote, respectively, a
**Controlled-Not** and a **measurement in the standard
basis.** Finally the graphic

denotes the Boolean gate ${H}^{{s}_{j}^{\ast}}$, controlled by the classical bit
${s}_{j}^{\ast}$, where ${s}_{j}^{\ast}$ denotes the **complement** of
the *j*-th switch setting *s*_{j}. In other words,

Please note that *H**Z**H* = *X*. Hence, if |*φ*〉 is a single-qubit state, then
measurement of ${H}^{{s}_{j}^{\ast}}\left|\mathit{\phi}\right.\u232a$ with respect to the observable
*Z* is equivalent to measurement of |*φ*〉 with respect to the Boolean observable
${H}^{{s}_{j}^{\ast}}Z{H}^{{s}_{j}^{\ast}}={s}_{j}^{\ast}X+{s}_{j}Z$. So, each detector portion of the
wiring diagram can be simplified to a local measurement with respect to the
Boolean observable ${s}_{j}^{\ast}X+{s}_{j}Z$, for *j* = 1, 2, 3 (please refer to Fig. 5). In fact, each detector portion of
the diagram can be even further simplified to local measurement of the GHZ
state with respect to Boolean observable ${U}^{\u2020}\left({s}_{1}^{\ast}X+{s}_{1}Z\right)U={s}_{j}^{\ast}Y+{s}_{j}X$, for *j* = 1, 2, 3.

An equivalent wiring diagram for Mermin’s machine, where
Υ(*s*_{j}) is the Boolean observable
$\mathrm{{\rm Y}}\left({s}_{j}\right)={s}_{j}^{\ast}X+{s}_{j}Z$

The three leftmost gates provide a preparation of the GHZ state

$$\begin{array}{c}\hfill \frac{1}{\sqrt{2}}\left(\left|000\right.\u232a+\left|111\right.\u232a\right)\phantom{\rule{0.333333em}{0ex}}\text{.}\end{array}$$

The local unitary^{1} transformation *U*^{⊗3} = *U* ⊗ *U* ⊗ *U* transforms the GHZ state into the
entangled state

$$\begin{array}{c}\hfill \left|\mathit{\psi}\right.\u232a=\frac{1}{2}\left(\left|000\right.\u232a-\left|011\right.\u232a-\left|101\right.\u232a-\left|110\right.\u232a\right)\phantom{\rule{0.333333em}{0ex}}\text{,}\end{array}$$

which will be used to control the flashing
light patterns of the three detectors.^{2}

Let ℋ_{even} and ℋ_{odd} denote the Hilbert subspaces of the
underlying three-qubit Hilbert space ℋ spanned, respectively, by the standard
basis elements labeled by bit strings of even and odd Hamming weight. It now
follows from the following table:

that

$$\begin{array}{c}\hfill \left({H}^{{s}_{1}^{\ast}}\otimes {H}^{{s}_{2}^{\ast}}\otimes {H}^{{s}_{3}^{\ast}}\right)\left|\mathit{\psi}\right.\u232a\in \left\{\begin{array}{cc}{\mathcal{H}}_{even}\hfill & \text{if}\phantom{\rule{0.166667em}{0ex}}s=111\hfill \\ \hfill & \hfill \\ {\mathcal{H}}_{odd}\hfill & \text{if}\phantom{\rule{0.166667em}{0ex}}s=001,010,100\hfill \end{array}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\right.\end{array}$$

Thus, if the switch setting is
*s* = 111, application of each and all local
detector measurements with respect to the standard basis (no matter in which
temporal order) will project the state $\left({H}^{{s}_{1}^{\ast}}\otimes {H}^{{s}_{2}^{\ast}}\otimes {H}^{{s}_{3}^{\ast}}\right)\left|\mathit{\psi}\right.\u232a$ into ℋ_{even}, necessarily resulting in a standard
basis state |*c*_{1}*c*_{2}*c*_{3}〉 of even Hamming weight, and
corresponding eigenvalues (-1)^{c1}, (-1)^{c2}, (-1)^{c3} with *c*_{1} + *c*_{2} + *c*_{3} = 0(mod2). Using the same argument for the switch
settings *s* = 001, 010, 001, the three local detector measurements
of $\left({H}^{{s}_{1}^{\ast}}\otimes {H}^{{s}_{2}^{\ast}}\otimes {H}^{{s}_{3}^{\ast}}\right)\left|\mathit{\psi}\right.\u232a$ will result in a standard basis element
|*c*_{1}*c*_{2}*c*_{3}〉 of odd Hamming weight with
corresponding eigenvalues (-1)^{c1}, (-1)^{c2}, (-1)^{c3} with *c*_{1} + *c*_{2} + *c*_{3} = 1(mod2).

Thus, using *c*_{j} = 0 as the control bit instruction to flash
Green G and *c*_{j} = 1 as the control bit instruction to flash
Red, we have shown that the device defined by the wiring diagram satisfies
all the required specs and constraints.

So the device can be built after all!

So where has the impossibility argument given in Sect. 3 of this paper gone awry?

Certainly the proof in Sect. 3 of this paper of the following proposition, on which the proof of impossibility is based, is beyond reproach:

There exist no Boolean functions

such that

$$\begin{array}{c}\hfill {f}_{A}\left({s}_{1}\right)+{f}_{B}\left({s}_{2}\right)+{f}_{C}\left({s}_{3}\right)\equiv \left\{\begin{array}{cc}1\phantom{\rule{4pt}{0ex}}\left(\text{mod}2\right)\hfill & \text{if}\phantom{\rule{0.333333em}{0ex}}s=\left({s}_{1},{s}_{2},{s}_{3}\right)=001\text{,}\phantom{\rule{0.333333em}{0ex}}010\text{, or}\phantom{\rule{0.333333em}{0ex}}100\text{.}\hfill \\ 0\phantom{\rule{4pt}{0ex}}\left(\text{mod}2\right)\hfill & \text{if}\phantom{\rule{0.333333em}{0ex}}s=\left({s}_{1},{s}_{2},{s}_{3}\right)=111\hfill \end{array}\right.\end{array}$$

The logic is flawless.^{3} But the crux of
the matter is that the argument of impossibility found in
Sect. 3 is only as sound
as the assumptions upon which it is based.

More specifically, the argument of impossibility fails because at least one of the following two tacitly assumed premises is false:

Premise 1. **Reality Principle: **
*What is measured is completely determined before it is
measured* (for a more refined definition of this principle and
the concept of an element of reality, please refer to [1] and [7]).

Premise 2. **Principle of Locality:**
*Spacelike separated regions of spacetime are physically
independent*.

It is not clear that these are fully independent principles. For how can that which is not fully determined already be localized? Moreover, can that which is not localized already be fully determined?

The above two premises lead to the following unfounded conclusions:

Unfounded Conclusion 1.
*Based on Premise 1 (The Reality Principle), the detector lamp
instructions *
*f*_{A}, *f*_{B}, *f*_{C}
* must already be predetermined well-defined total functions*
^{4} at the time of particle ejection.

Unfounded Conclusion 2.
*Based on Premise 2 (The Principle of Locality), the detector lamp
instructions *
*f*_{A}
*, *
*f*_{B}
*, *
*f*_{C}
* must be local. Hence, *
*f*_{j}
* is a function only of the *
*j*
*th switch setting *
*s*_{j}
* and independent of the two other switch settings.*

We will show in the next section that the detector lamp instructions
*f*_{A}, *f*_{B}, *f*_{C} are neither predetermined well-defined
functions before ejection, nor local independent functions.

It is instructive to take a closer look at Mermin’s machine.

We will now explicitly compute the random functions *f*_{A}, *f*_{B}, *f*_{C}. In so doing, we will find, contrary to the
unfounded conclusions given in the previous section, that these functions
are:

- Random partial functions,
- Global interdependent functions of the switch settings, and
- Not fully defined until measured by the detectors.

For reasons of transparency, it will prove more convenient to work with the equivalent wiring diagram shown in Fig. 5, where

denotes the *s*_{j}-controlled gate for the Boolean
observable

That this wiring diagram is equivalent to the one found in
Fig. 4 follows from the fact
that *H**Z**H* = *X*. Hence, measurement of ${H}^{{s}_{j}^{\ast}}\left|\mathit{\psi}\right.\u232a$ with respect to *Z* is
equivalent to measurement of |*ψ*〉 with respect to ${H}^{{s}_{j}^{\ast}}Z{H}^{{s}_{j}^{\ast}}={s}_{j}^{\ast}X+{s}_{j}Z$.

We will also need to use the **quantum measurement function**
*Q*, which takes as input a pair consisting of an existing
quantum state and a quantum observable and then upon evaluation produces as
output a pair consisting of a resulting eigenstate and the corresponding
eigenvalue. For example, if *ρ* is a density operator representing the
state of a quantum system and if *Ω* an observable with spectral
decomposition

$$\begin{array}{c}\hfill \mathit{\Omega}={\displaystyle \sum _{j}^{n}}{\mathit{\lambda}}_{j}{P}_{j}\text{,}\end{array}$$

then on evaluation *Q*(*ρ*, *Ω*) produces

$$\begin{array}{c}\hfill Q\left(\mathit{\rho},\mathit{\Omega}\right)=\left(\frac{{P}_{j}\mathit{\rho}{P}_{j}}{Tr\left({P}_{j}\mathit{\rho}\right)},{\mathit{\lambda}}_{j}\right)\phantom{\rule{0.333333em}{0ex}}\text{,}\phantom{\rule{0.333333em}{0ex}}\end{array}$$

where *P*_{j} is the projection operator for the
eigenspace corresponding to the eigenvalue *λ*_{j}.

Please note that the function *Q* is a random output function,
very much like the random number generator found on most classical computers,
except that its output is not pseudorandom, but actually truly random. A
pseudorandom number generator is a **predeterministic** function, i.e.,
a function fully predefined before evaluation, which upon evaluation
deterministically produces an output. On the other hand, the function
*Q* is **indeterministic**,^{5} i.e., it is a function that is not fully defined (and not
fully determined) as a function until it is evaluated.

We finally are ready to take a closer look at the implementation of Mermin’s machine, as described by the wiring diagram found in Fig. 5.

After the state preparation of the entangled state |*ψ*〉 and before ejection of the particles, the
detector lamp instructions *f*_{A}, *f*_{B}, *f*_{C} are indeterministic, i.e., only partially
defined (and only partially localized) by the entangled state |*ψ*〉. This is a result of the state of each
individual qubit of |*ψ*〉 being indeterministic, i.e., not yet fully
defined, and not yet fully localized.

In Sect. 4, it was pointed out
that the property that the final resulting light pattern always satisfies the
machine specifications and constraints is independent of the temporal order of
the detector measurements. For this reason, we focus only on the case for which
the detector measurements occur in the temporal order *t*_{A} < *t*_{B} < *t*_{C}, where *t*_{A}, *t*_{B}, *t*_{C} denote the measurement times for detectors
*A*, *B*, *C*,
respectively.

The topic of the temporal order of measurements is remarkably subtle. To say that the detector light pattern is independent of the order of the measurements is counterfactual and hence physically meaningless. However, it is meaningful (not counterfactual) to say that the state specifications and constraints are met, independent of the order of measurements. On the other hand, because of relativity, there can be, for each possible temporal order, a different observer that observes the measurements in that order. The fact that each of three different observers sees the measurements in a different temporal order is not counterfactual because all observers are viewing the same measurements.

We recall that the spectral decompositions of the Pauli spin operators
*X* and *Z* are,
respectively,

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{P}_{+}=\left|+\right.\u232a\left.\u2329+\right|\\ \\ {P}_{-}=\left|-\right.\u232a\left.\u2329-\right|\end{array}\right.\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\left\{\begin{array}{c}{P}_{0}=\left|0\right.\u232a\left.\u23290\right|\\ \\ {P}_{1}=\left|1\right.\u232a\left.\u23291\right|\end{array}\right.\phantom{\rule{0.333333em}{0ex}}\text{,}\end{array}$$

and where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left|+\right.\u232a=\frac{\left|0\right.\u232a+\left|1\right.\u232a}{\sqrt{2}}\\ \\ \left|-\right.\u232a=\frac{\left|0\right.\u232a-\left|1\right.\u232a}{\sqrt{2}}\end{array}\right.\phantom{\rule{0.333333em}{0ex}}\text{.}\end{array}$$

In the calculations to follow, we use the following notational convention:

At the time *t*_{A}, the function *f*_{A}(*s*_{1}) is evaluated as follows:

where *j*_{1} = 0 or 1, and where *T**r*_{23}(|*ψ*〉〈*ψ*|) is the partial trace of |*ψ*〉〈*ψ*| over qubits 2 and 3. The resulting
state of the three qubits is

$$\begin{array}{c}\hfill \left|{\mathit{\psi}}^{\prime}\right.\u232a=\frac{\left({P}_{{j}_{1}^{{s}_{1}^{\ast}}}\otimes 1\otimes 1\right)\left|\mathit{\psi}\right.\u232a}{\sqrt{\u2329\mathit{\psi}\left|{P}_{{j}_{1}^{{s}_{1}^{\ast}}}\otimes 1\otimes 1\right|\mathit{\psi}\u232a}}\phantom{\rule{0.333333em}{0ex}}\text{.}\end{array}$$

At the time *t*_{B}, the function *f*_{B}(*s*_{2}) is evaluated as follows:

where *j*_{2} = 0 or 1, and where *T**r*_{13}(|*ψ*^{′}〉〈*ψ*^{′}|) is the partial trace of |*ψ*^{′}〉〈*ψ*^{′}| over qubits 1 and 3. The resulting
state of the three qubits is

$$\begin{array}{c}\hfill \left|{\mathit{\psi}}^{\u2033}\right.\u232a=\frac{\left(1\otimes {P}_{{j}_{2}^{{s}_{2}^{\ast}}}\otimes 1\right)\left|{\mathit{\psi}}^{\prime}\right.\u232a}{\sqrt{\left.\u2329\mathit{\psi}\right|\left(1\otimes {P}_{{j}_{2}^{{s}_{2}^{\ast}}}\otimes 1\right)\left|{\mathit{\psi}}^{\prime}\right.\u232a}}\phantom{\rule{0.333333em}{0ex}}\text{.}\end{array}$$

At the time *t*_{C}, the function *f*_{C}(*s*_{3}) is evaluated as follows:

where *j*_{3} = 0 or 1, and where *T**r*_{12}(|*ψ*^{″}〉〈*ψ*^{″}|) is the partial trace of |*ψ*^{″}〉〈*ψ*^{″}| over qubits 1 and 2 The resulting state
of the three qubits is

$$\begin{array}{c}\hfill \left|{\mathit{\psi}}^{\u2033\prime}\right.\u232a=\frac{\left(1\otimes 1\otimes {P}_{{j}_{3}^{{s}_{3}^{\ast}}}\right)\left|{\mathit{\psi}}^{\u2033}\right.\u232a}{\sqrt{\left.\u2329{\mathit{\psi}}^{\u2033}\right|\left(1\otimes 1\otimes {P}_{{j}_{3}^{{s}_{3}^{\ast}}}\right)\left|{\mathit{\psi}}^{\u2033}\right.\u232a}}\phantom{\rule{0.333333em}{0ex}}\text{.}\end{array}$$

Please note that each of the instructions *f*_{A}(*s*), *f*_{B}(*s*), *f*_{C}(*s*) can only be a nonlocal function of
*s* = (*s*_{1}, *s*_{2}, *s*_{3}). For from relativity, there can be
three different observers Alice, Bob, and Charlie each observing the same
measurements, but each observing the same measurements in the three
different temporal orders *t*_{A} < *t*_{B} < *t*_{C}, *t*_{B} < *t*_{C} < *t*_{A}, *t*_{C} < *t*_{A} < *t*_{B}, respectively. If Alice observes
*f*_{A} as only a function of *s*_{1}, so would Bob and Charlie.

We are now in a position to explicitly quantify the interdependence of the
random Boolean partial functions *f*_{A}, *f*_{B}, *f*_{C}. To do so, we will make use of the
following well-known combinatorial formula [3]:

Let *b* = (*b*_{1}, *b*_{2}, *b*_{3}, …, *b*_{n}) be a binary string of length
*n* > 0. The binary expansion of the Hamming
weight *W**t*(*b*) of *b* is given by the
following formula:

$$\begin{array}{c}\hfill Wt\left(b\right)={\displaystyle \sum _{k=0}^{O\left(logn\right)}}{\mathit{\sigma}}_{{2}^{k}}\left(b\right)\xb7{2}^{k}\phantom{\rule{0.333333em}{0ex}}\text{,}\end{array}$$

where *σ*_{2k}(*b*) denotes the 2^{k}-th elementary symmetric function modulo
2, i.e.,

$$\begin{array}{c}\hfill {\mathit{\sigma}}_{{2}^{k}}\left(b\right)={\displaystyle \sum _{1\le {\ell}_{1}<{\ell}_{2}<\dots <{\ell}_{n}\le {2}^{k}}}{b}_{{\ell}_{1}}{b}_{{\ell}_{2}}{b}_{{\ell}_{3}}\cdots {b}_{{\ell}_{n}}\phantom{\rule{1em}{0ex}}\left(\text{mod}2\right)\phantom{\rule{0.333333em}{0ex}}\text{.}\end{array}$$

In light of the above theorem, an immediate consequence of the above measurement calculations is the following lemma and corollary:

If the switch setting *s* = (*s*_{1}, *s*_{2}, *s*_{3}) is of odd Hamming weight,
then

$$\begin{array}{c}\hfill \left({P}_{{j}_{1}^{{s}_{1}^{\ast}}}\otimes {P}_{{j}_{2}^{{s}_{2}^{\ast}}}\otimes 1\right)\left|\mathit{\psi}\right.\u232a\phantom{\rule{0.333333em}{0ex}}\text{lies in}\phantom{\rule{0.333333em}{0ex}}\left(1\otimes 1\otimes {P}_{{j}_{3}^{{s}_{3}^{\ast}}}\right)\mathcal{H}\phantom{\rule{0.333333em}{0ex}}\text{,}\end{array}$$

where

and where *σ*_{2}(*s*) denotes the second elementary symmetric
function

Thus,

|*ψ*^{″′}〉 = |*ψ*^{″}〉 .

For a switch setting *s* = (*s*_{1}, *s*_{2}, *s*_{3}) of odd Hamming weight, the detector
lamp instructions *f*_{A}, *f*_{B}, *f*_{C} are the random partial functions given
by:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{f}_{A}\left(s\right)={j}_{1}\hfill \\ \hfill \\ {f}_{B}\left(s\right)={j}_{2}\hfill \\ \hfill \\ {f}_{C}\left(s\right)={j}_{3}\hfill \end{array}\right.\phantom{\rule{0.333333em}{0ex}}\text{,}\end{array}$$

with the Boolean algebraic dependence

where *σ*_{2} denotes the second elementary symmetric
function

Hence, the random Boolean instruction functions *f*_{A}, *f*_{B}, *f*_{C} are global and interdependent partial
functions, thereby refuting Unfounded Conclusions 1 and 2, found in
Sect. 5 of this
paper.

It is interesting to note that the Boolean function *σ*_{2}(*s*_{1}, *s*_{2}, *s*_{3}), involved in the above algebraic
interdependence, in some way fully encapsulates the entire paradox. In other
words, this second elementary symmetric Boolean function somehow quantifies
the nonlocality and the indeterminism involved in the GHZ paradox.

We conclude with no conclusion, but with a question:

Is quantum mechanics trying to tell us that the very fabric of reality is indeterminate, i.e., not fully defined until it is observed?

I would like to thank Jozef Przytycki and Valentina Harizanov for their gracious invitation to give the George Washington University April Fools Day Lecture in April 2014. I would like to thank John Meyers for a number of helpful discussions and Lou Kauffman for his encouragement to write up my lecture. I would also like to thank Alain Connes and Rad Balu for their helpful conversations. Finally, I would like to thank my good friend Howard Brandt for our many challenging discussions on quantum physics. This work was partially supported by NASA Grant No. NNX15AK58, by Mathematisches Forschungsinstitut Oberwolfach, by the 2015 Army Research Laboratory Faculty Research Program, and by the L-O-O-P Fund.

^{1}It is important to note that *U*^{⊗3} is a local unitary transformation that
does not change entanglement type. For a better understanding of the
significance of this fact, please refer to [4].

^{2}For a more in-depth explanation of the use of entanglement as a distributed
control mechanism, please refer to [8].

^{3}Actually, as we will see, the above proposition is a proof of the
counterfactuality of the lamp instructions being total functions, and not a
proof that the device cannot be built.

^{4}A **total function** is a function that is defined for all possible
values of its arguments. A **partial function** is a function
defined for some of its argument values, but not necessarily all. For more
information, please refer to any text on recursive function theory.

^{5}Please note that we have avoided use of the term
“nondeterministic” because this term has an entirely
different meaning in the theory of computation.

1. Bub J. Interpreting the Quantum World. Cambridge: Cambridge University Press; 1997.

2. Greenberger DM, Horne M, Zeillenger A. Going beyond Bell’s theorem. In: Kafatos M, editor. Bell’s Theorem, Quantum Theory, and Conceptions of the
Universe. Dordrecht: Kluwer Academic; 1989.

3. Knuth D. The Art of Computer Programming. 2. Boston: Addison-Wesley; 1981.

4. Lomonaco, S. J., Jr.: An Entangled Tale
of Quantum Entanglement, AMS PSAPM, vol. 58, pp. 305–349 (2002).
http://arxiv.org/pdf/quant-ph/0101120.pdf

5. Mermin ND. Quantum mysteries revisited. Am. J. Phys. 1990;58(750):731–734. doi: 10.1119/1.16503. [Cross Ref]

6. Mermin ND. Quantum Computer Science: An Introduction. Cambridge: Cambridge University Press; 2007.

7. Redhead M. Incompleteness, Nonlocality, and Realism: A Prolegomenon to the
Philosophy of Quantum Mechanics. Oxford: Oxford University Press; 2002.

8. Yimsiriwattana, A., Lomonaco, S.J.,
Jr.: Generalized GHZ States and Distributed Quantum Computing, AMS CONM/381,
pp. 131–147 (2005) . http://xxx.lanl.gov/abs/quant-ph/0402148

Articles from Springer Open Choice are provided here courtesy of **Springer**

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |