Quantum Entanglement: “Spooky action at a distance”, or an inflection towards the extension of classical set-theoretic description?

Our understanding of numbers and their relations has developed from counting using fingers, through glyphs, say on the Ishango bone, to our modern-day quaternions: of a truth, we have come a long way. One defining feature of the development of our number systems and the mathematical possibilities that eventually erupts from each stage of development is the extension of the meaning and implications of each new number systems that is discovered/invented.

If you missed the first part of this newsletter, please read for continuity: https://www.dhirubhai.net/pulse/quantum-entanglement-spooky-action-distance-towards-felix-wejeyan-d1c0c

The most basic number system is the natural numbers 1, 2, 3, and so forth. They were apparently designed to count and label. Afterwards, zero was discovered and used as a place holder, before actually being assigned as a number of its own. My four years old daughter always says “zero means nothing”, so how does nothing have anything to do with counting? Then the negative numbers were discovered, and this extended the natural number system into the integers, and the number zero eventually found a rational structural position as the boundary between the positive and negative numbers.

“Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty’s sake and pulls it down to earth.” – Morse Maston

Later on, the concept of division gave birth to the rational and irrational numbers, and Pythagoras theorem and the nature of the completeness of the number line led us to the concept of the real numbers. The square root of minus one lead us to the complex numbers, and at this point we once again needed to redesign/extend our description of the number line to accommodate this new concept, as it was obvious that the complex number has no place within the number line. In 1806 Jean-Robert Argand described the complex numbers as lying one unit above the origin of the real number line, and this enlarged schema now accommodates this new disturbing number without any problem or inconsistency with the initial structure.

Then eventually the quaternions were invented, and these seems to be the elder brother(s) of the complex number(s), but couldn’t satisfy the commutativity of multiplication; using the concept of a division ring, it was grouped as a hypercomplex number. The important point to note here is that through every step of the way, we encounter phenomena that each new number system can describe that the previous system couldn’t, and this expands our schema and the subsequent ability to discover and understand more concepts and structures.

“Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.” – Joseph Fourier

We have grouped all of our discovered numbers into a set-theoretical structure that is supposedly consistent, and applied it in the description of every natural phenomenon we have encountered in classical physics with an outstanding pace of success and technological advancement that we have tagged them as “axiomatic” – fundamentally complete! Then, Kurt Godel showed that such type of system cannot prove its own consistency, and like icing on this cake of logical conundrums, quantum phenomena has opened new spectrum of experiences that our consistent set-theoretic structure now struggles with and seems to need substantial restructuring, or better still an expansion – like the Argand diagram did for complex numbers.

The first of these phenomena was quantum superposition, which implies that numbers (a quantum system) can be in more than one state at the same time – 0 and 1 – until certain criteria make it collapse and reveals itself when measured. Next, quantum entanglement tells us that natural phenomena and processes can be connected in such a way as to defy the classical effects of causality through space-time. These phenomena have no classical counterparts, and as such expresses the truth of Godel’s incompleteness theorem that maybe the classical set-theoretic description isn’t as complete as we thought it was. In an attempt to resolve these issues, there has been a plethora of amendment, restructuring and redesigning of our most powerful and axiomatic set-theoretical structure to find a consistent description for these phenomena; and, in order to get a better understanding of these contemporary struggles it is imperative to theoretically analyze these phenomena into its most basic parts.

“Entanglement can best be understood as this: When subatomic matter is in a process together, subsequently the subatomic particles go apart from each other and go across the universe. When they do this, they will remain entangled. That means if you do something to one, the other one responds immediately, instantaneously.” – Edgar Mitchel

It was explained in the previous part of this publication that if the quantum state with which we use to describe a pair of quantum entities A and B is entangled, then when measurement is made on one of such pair, say A, it is discovered that such measurement is strongly correlated with the result of measurement upon the other pair B. The entangled system of A and B is called a bipartite composite system – this simply implies a composite system composed of two (“bi”) parts.

For such an entangled pair the basis states of the composite system are inseparable; this means that it has become impossible to attribute a pure state to either system A or system B. This is a quintessence of what is called a maximally entangled pure state, as the composite Hilbert space of A and B can no longer be separated into pure Hilbert state of each quantum entities. Notice that in a maximally entangled state between two quantum entities, causality seems to be preserved between the measurement of each state regardless of spatial separation in such a way that violates Bell’s inequality.

The amount of information one has about a quantum entangled system can help us decide whether the system is in a pure ensembled state or in a mixed ensembled state. In the former, the system is regarded as a rank 1 matrix using the spectral theorem, and with the latter the quantum system is represented using density matrices. In a mixed ensemble the number of populations is usually composed of a large number of different states, and possibly different spatial orientations.

The important question to ask at this point is whether this mixed state is separable into pure ensembles or not. When we can write the density matrix of this system as the sum of pure ensembles and expanding, then we can say that they are themselves pure ensembles, otherwise, the state of the entire system is said to be entangled. This becomes a sort of multipartite composite system, and Herman Weyl (1931) observed that such kind of system exhibits a sort of “Gestalt” behavior. Generally, it is difficult and usually considered to be a NP-hard problem to decide whether a mixed state is entangled or not.

“Of course, minute as its impact may be in our physical universe, the fact of quantum entanglement is this: If one logically inexplicable thing is known to exist, then this permits the existence of all logically inexplicable things. A thing may be of deeper impossibility than another, in the sense that you can be more deeply underwater--but whether you are five feet or five fathoms from the surface you are still all wet.” – Brian McGreevy

Regardless of the difficulties in deciding whether a composite system is entangled or not, we have come to use the phenomena of entanglement as a resource in quantum information theory to implement valuable transformations. One of such transformation is quantum teleportation where, for instance, a bipartite composite system destroys its quantum state and information at one location due to measurement and the corresponding wave function collapse, and transmits its information to its entangled pair in a spatially different location. Another resource phenomenon is entanglement swapping, and this was described in the previous publication. The important point to note here is that quantum entanglement can be used as a resource to enable quantum interaction through a sort of quantum channels such as used in Quantum Key Distribution (QKD), but these entangled states are consumed in the process.

The entanglement relationship between two maximally entangled bipartite composite system A and B is usually monogamous. This means that if A and B is maximally entangled, then they can not be entangled to any other system. This can be described as an equivalent relation as used in the set-theoretical description of Zermelo-Fraenkel (ZFC) – a “strictly” bipartite system. A and B can be described as belonging to the set that forms the bipartite entanglement, and the relation between each can be succinctly described with an ordered pair. In other words, a change in A leads to a corresponding change in B, and each cannot be described without the other: this simple relationship can be mapped out with a function. This is a quintessential characteristic of an entangled bipartite composite system, and this fits neatly into the standard set-theoretic structure.

“Logic issues in tautologies, mathematics in identities, philosophy in definitions; all trivial, but all part of the vital work of clarifying and organizing our thought.” – Frank Plumpton Ramsey

Following this train of thought, there are no set-theoretic analogy for multipartite composite systems. The Greenberger-Horne-Zeilinger (GHZ) state was used to describe a tripartite entangled composite system: they struggled to carve the mathematical structure to fit into the classical set-theoretic formulation and won a Nobel price for their accomplishment. Their struggle indicates a need for an expansion of our current schema to accommodate a new phenomenon that seems to defy all classical interpretations.

For an entangled tripartite composite system A, B and C, the relation between each system does not fit naturally into the ZFC structure, because a change in A leads to a corresponding change in B and C, and this occurs all the way round each system; this relationship cannot be described using an ordered pair, and the function designed by GHZ has no classical counterpart. This is a clear indication for the need for an extension of our current set-theoretical description, and possibly a redesign of the foundations of mathematics. How can we extend our current set-theoretic schema to accommodate for this? What then are the possible mathematical structures with which we can use to describe a multipartite composite system without any struggle?

To be continued…

Please share your thoughts on the ontological commitments and sacrifice/compromise contemporary science research needs to make in order to adapt to these new phenomena.?

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