Tensor-Centric Warfare VI : A Global Warfare Model

We propose a global warfare model that integrates the models of the whole tensor-centric warfare series, represented as a high-dimensional entangled warfare category. Its underpinning metaphysics is “entangled fusion”: this is the macroscopic entanglement concept inspired by high-dimensional (HD) quantum computation (the “quantum brain”), in which any number of entangled wave-functions can be highly correlated, with neuron-like signaling among them. From this entangled perspective, war and battle is seen essentially as a holistic phenomenon: if any one of a set of mutually entangled warring parties is removed from the equation, then the war as it is instantly stops, possibly to be replaced by a new conflict between the remaining parties but distinct from that which it supplants. The formal global warfare framework developed in this paper expresses this fundamental idea of arbitrary many interrelated/entangled conflicts, each of them defined by its own battle-manifold (with warfighting tensor fields acting on it) and occurring (more-or-less) simultaneously on the planet; we call this entangled category Warfare .

The basic concept of Global Warfare introduced in this paper concerns an attempt to realize a holistic approach to modeling war and battle.Until recently, the vast majority of works in this topic have focused on representing dynamics V. Ivancevic  of individual battles or missions.Yet, this approach fails to address the complexity of modern warfare, and particularly the nonlinear and non-local nature of interactions and consequences, thus compelling us to examining wars not just as series of battles, and battles as not just as series of missions, in temporal sequence.
Rather, we seek here to model war as arbitrary collections of overlapping battles and battles as collections of missions, which can unfold simultaneously and can interact with one another.
This increasing complexity and interdependence means that to better design appropriate future force capability and operating methods, strategies and tactics, a more holistic approach to modeling is required.As a consequence of this need for more holistic methods, we need to be able to capture all the interdependencies between various overlapping battles and simultaneous operations in different theaters, and among events during any given war, battle or mission.The intent in this paper is to create a formal mathematical framework through the utility of Tensors and the concept of entanglement to capture this global view of war and battle, which constitutes then a national or multi-national perspective.
It has been long recognized that warfare is a highly complex phenomenon that contains both elements of order and of chaos, usually with both co-occurring.It can be said that, all other things being equal, the victor will be the side that makes the least errors; or, to put it another way, the side that is able to cope with the chaos the most effectively.We recognize that complex systems lead to emergent patterns and behaviors which can only be understood through modeling and simulations if the model or simulation has captured adequately the large-scale interrelationships that connect all the activities and events that collectively comprise a global picture of war and battle from a national perspective.
The central idea of the present paper, the underlying thought-current reflecting our holistic view of warfare, is the concept of a large-scale warfighting fusion, or complex, consisting of many interrelated battlefields, that is, an entangled fusion 1 of warfighting.In other words, we see a modern warfare as a compound whole, an inseparable fusion, almost having a mind-like character, comprising a number of non-local yet interacting battlefields-rather than a reducible, disjoint sum of passive mechanistic components.
The entangled fusion is a macroscopic concept inspired by high-dimensional (HD) quantum computation, called the "quantum brain", in which any number of entangled wave-functions can be highly-correlated, inseparable and with neuron-like signaling among them.The entanglement is visible in any macroscopic 1 The general concept of entanglement subsumes three fundamental scientific concepts: correlation, causality and function between two systems or processes (see [6]).Its micro + macro nature is arguably best defined by the popular "second Einstein equation" (coined by L. Susskind from Stanford and J. Maldacena from Princeton): or ,

ER EPR
Wormhole Entanglement =⇔ = which relates two Einstein's papers from 1935: ER, referring to the Einstein-Rosen bridge [7] (or, a cosmological wormhole that connects two distant parts of the universe); and EPR, referring to the Einstein, Podolsky and Rosen paradox [8] in which the entanglement was born as a quantum non-locality (to be three decades later governed by Bell's theorem; see [9] and the references therein).Intelligent Control and Automation system or process which cannot be completely reduced to its components, but that manifests behavior only as a whole compound/fused system.Biological systems like various swarming behaviors of bees, birds and bats show this characteristic.By analogy, in this paper we claim that the concept of a global warfare is essentially an entangled fusion of two (or possibly more) parties at least some of whom are in a state of conflict; the war cannot exist without conflicting parties though not all parties need to simultaneously be in conflict at any one time.Any attempt to reduce such a phenomenon into independent passive components is necessarily limited at best and generally unrealistic and hence misleading.Furthermore, if any one of mutually entangled parties is removed from the equation then the state of warfare as it has existed until that point instantly stops, possibly to be replaced with another.In other words, war is essentially a holistic phenomenon, both in terms of human populations involved and in terms of (futuristic) autonomous cyber-physical-cognitive systems (CPC-systems, introduced in [6]).
Seen in this way, major historical conflicts such as World War I and II comprise not distinct events but more like pronounced clusters of overlapping component conflicts that evolved over time, as parties shifted alliances and entered and exited different levels of states of hostility and cooperation with each other.The change in stance in World War 2 of the USA from supporting Britain with materiel to entering fully into the conflict, the Japanese attack on Pearl Harbor, the breakdown of the German-Soviet non-aggression pact with Germany's attack on the Soviet Union, the Winter War between Russia and the USSR, and the capitulation of Italy all represent shifting elements of the overall conflict with changing allegiances and levels of conflict.Furthermore, the Second World War did not occur in isolation of other conflicts, but rather was preceded and influenced by a host of disparate conflicts, including the Spanish Civil War, Italy's invasion of Ethiopia, the Japanese invasion of China, and the ongoing sociopolitical consequences of the First World War, and was then immediately supplanted by the Cold War.
A formal framework developed in this paper to represent national and international viewpoint of war and battles and to express this fundamental observation that conflicts are not isolated but, at least from a national perspective, comprise many interrelated/entangled conflicts at various degrees of involvement, occurring (more-or-less) simultaneously on the planet, is called the entangled warfare category.

Entangled Tensor Categories
The basic algebraic operator which we use to represent entangled fusion is the ordinary tensor product and its categorical abstraction.Specifically, the Kronecker tensor product, usually denoted by " ⊗ " and based on the Cartesian product 2  In the same way, both linear and quadratic Lanchester-type TCW-terms (see [1]): . . .
In addition, the Red + Blue battle Hamiltonian, , with its Ising-type connection tensor, ), is explicitly written as: . In all these TCW examples, the tensor product ⊗ is used as a "glue" to stick together various tensor fields into coherent/inseparable terms, representing coherent/inseparable warfighting actions.
Similarly, in quantum computation, a compound system functor" because it applies suitably both to the sets and to the functions between them.
3 For example, in a simple quantum computation involving only two qubits, the Bell states represent four maximally-entangled quantum states of the qubits.Generalization to any number of multicubits (or, n-cubits) is straightforward, and the degree of entanglement is measured by the von Neumann entropy: ( ) , where ρ is the density matrix that describes a quantum system in a mixed state (a statistical ensemble of several states).4 The fundamental phenomenon of entanglement arises in compound quantum systems defined by tensor products of their component systems.The general form of a vector , , This two-fold collection of objects/systems and morphisms/processes between them is governed by the functional composition, denoted by "  ", a temporal/sequential composition in which the output of the first process : f A B → becomes the input to the second process : g B C → etc., formally defined as: The extension to a chain of many sequential processes is straightforward.
Secondly, to be able to include compound/fused CPC-systems of arbitrary nature, where two subsystems act as an entangled/coherent whole, we need to extend the standard definition of a category, which is based on the serial/temporal composition "  ", with a new notion of parallel/spatial composition (i.e., entangled fusion) 6 .Such an enriching of a standard category theory can be done using the special functor ⊗ , called the symmetric monoidal tensor product 7 (or, tensoring functor ⊗ , for short)-a categorical generalization of the Kronecker tensor product (used above to glue together tensorial expressions), which acts both on objects and on morphisms of a category, along with its compound operations: A Bf gC D ⊗ ⊗ ⊗ , inherited from the operations on the individual subsystems.
Briefly, we define a symmetric monoidal tensor (SMT) category  8 , using the tensoring functor All quantum-computational processes Q (including: quantum information-flow, teleportation, entanglement swapping and communication protocols, as well as quantum cryptography, games and gambling (see [10] [11] and the references therein)-are naturally occurring between two quantum state-agents, traditionally called Alice (A) and Bob (B).In this paper, we generalize this Alice--Bob metaphor to arbitrary interacting CPC-systems. 6 The simplest way of thinking of the parallel/spatial composition (entangled fusion) is juxtaposition: putting two (or, several) systems side-by-side to make them act as one (a coherent pair or team).However, from the "second Einstein's equation" it follows that the entangled systems can be non-local, i.e., physically separated by large distances, e.g.situated on different continents, like a mother and daughter, or twin siblings and yet fully entangled and functioning as a coherent one system.

7
A monoid is a category with one object, which is a group without inverses (a group is a category with one object in which all morphisms are isomorphisms).
The main characteristic of the tensoring functor : ⊗ × →    is the so-called commutative bifunctoriality 9 : the order in which two operations, f (applied to one subsystem) and g (applied to another subsystem), does not matter (for the proof of the commutative bifunctoriality see [12] as well as [13], [14]).Briefly, for any quadruple ( 1 2 1 2 , , , A A B B ) of CPC-systems with the corresponding quadruple of processes ( 1 2 1 2 , , , f f g g ) between them, we have the following bi-compositional rule: This rule is the "signature" of any SMT category  .It states that any combinations of serial and parallel compositions of CPC-systems are commutative-their order doesn't matter, due to the symmetry of the tensoring functor ⊗ (in spite of the time-asymmetry of the serial composition "  ").
In this way defined SMT category  represents a collection of objects/systems and morphisms/processes, combined in two ways: using both sequential composing and parallel tensoring operations-in order to formulate a rigorous compositional theory (see [15]).This powerful architecture is capable of assembling large and open 10 This implies that the SMT category  is not an ordinary category, but rather Bénabou's bicategory [16] [17], which also includes an identity object 1 and some natural isomorphisms obeying MacLane's coherence conditions [12], including the fancy "pentagon" diagram.In the bicategory  , the usual categorical composition  is naturally used to represent physical processes combined in series, while the tensoring ⊗ represents physical processes combined in parallel.A bifunctor is a generic picture projecting (all objects and morphisms of) a source bicategory into a target bicategory.Our tensoring functor : ⊗ × →    is technically a bifunctor.For a "readable" introduction to HD categories (or, n-categories), see [18] and the references therein.Intelligent Control and Automation where the tensor product of Hilbert spaces: ( ) , , weighted by for all , Generalization of the 2D quantum-computational SMT category 2 Hilb to a large-scale SMT category N Hilb , in which the quantum-computational fusion is called the quantum neural network (QNN, which can be promoted into a "quantum brain" if its dimension N is very large; see [23]), is defined by the HD tensoring functor, : Hilb , given by: with objects given by the following ( N N × )-matrix of Hilbert spaces: ( ) .
Every element in the matrix (1) represents the Hilbert state-space of a single multiqubit circuit with its own quantum state: .

The Entangled Warfare Category
Now, inspired by the QNN model ( 1)-( 2), we can introduce our main result, the 11 Technically speaking, a class of quantum circuits that is closed under both ⊗ and  is the class of Clifford stabilizer circuits (see, e.g.[19] [20] [21] [22]).Intelligent Control and Automation HD warfare-category Warfare -as a tensor entanglement of all existing Red and Blue battle-manifolds 12 : Red Blue .
NN NN M M = ⊗ Warfare Specifically, the objects and morphisms of the Warfare -category are defined by the action of the HD warfare tensoring functor, : ⊗ × → Warfare Warfare Warfare , given by: (with all indices going from 1, , N  ).
The extended warfare (bi)category 12 Warfare shows that whenever there are two apparently unrelated warfighting conflicts in different parts of the world (e.g., on different continents)-ultimately they would be entangled into the same global warfare.In other words, even though we have two geographically-separated conflicts, the proposed framework suggests that they can be linked through 13 As noted before, the Warfare -category is the flexible architecture which is not restricted to Red vs. Blue warfare, but can include any number p of warfighting parties, in such a way that all tensor products in Warfare would have p entangled components.Intelligent Control and Automation entanglement fusion.
The concept of Global Warfare was illustrated in this section and we have shown a practical approach to the holistic modeling of warfare.The framework illustrated here has clearly allowed for the capture of the complexity of the Global Warfare through the use of the concept of entanglement.
We are confident that this framework is appropriate for the evaluation of future force structures, capability, strategies and tactics.As a result our approach may be thought of as a tool to aid the future planning and design process, being incorporated in such work-flow by the military.We have been successful in creating a formal mathematical framework through the utility of Tensors and the concept of entanglement to capture this global view of warfare.

Conclusions
In this paper we have proposed a global warfare model, represented by the high-dimensional entangled category Warfare .The metaphysics of this global warfare model is "entangled fusion", the macroscopic entanglement concept, inspired by high-dimensional (HD) quantum computation (the "quantum brain") in which any number of entangled wave-functions can be highly-correlated, with neuron-like signaling among them.In this "entangled view", the warfare is essentially a holistic phenomenon, thus if any one of mutually entangled warfighting parties is removed from the equation, the conflict, in its present form, instantly stops, though possibly being instantly replaced by another.
This delineation of conflicts within our global entangled model thus ascribes a different identity to distinct phases or threads within a larger surrounding web of conflict, as parties enter or exit different states of conflict or cooperation with other parties.We believe this more accurately reflects the way in which past and present geopolitical conflict manifests; in particular, our model here is intended to represent war and battle more from a national perspective, rather than from the point of view of isolated individual battles or missions temporally linearly arranged.
As an integration of the whole tensor centric warfare series, the formal global warfare framework developed in this paper expresses this fundamental idea of many simultaneously interrelated/entangled battlefields, each of them defined by its own battle-manifold (with warfighting tensor fields actin on it) and occurring (more-or-less) simultaneously on the planet which is called the entangled category Warfare .The action of the Warfare -category is formally defined using the warfare tensoring (bi) functor : ⊗ × → Warfare Warfare Warfare .
We are confident that the framework presented here is appropriate for the evaluation of future force structures, capability, strategies and tactics, and can be used as a tool to aid the future planning and design process, incorporating a holistic work-flow for the military.We have been successful in creating a formal mathematical framework through the utility of Tensors and the concept of entanglement to capture this global view of warfare.

2 ψ and 2  4 . 2 The
, which can be physically far-apart but still have their maximally-entangled Bell-states 3 : Therefore, in quantum computation, the Cartesian product, X Y × , of two sets, X and Y, consists of all ordered pairs ( ) , x y of the elements x X ∈ and y Y ∈ .It can be generalized (lifted/abstracted) to the notion of a "tensoring so that all quantum states form the corresponding complex-valued ( N N × )-matrix, in which every tensor product the following (N, N)-matrices of Red and Blue battle-manifolds, respectively: ), the Red and Blue vectorfields, governed by the HD generalization of the standard TCW-combat equations: the Warfare -category, i.e. the number of entangled battle-manifolds Red Blue NN NN M M ⊗, for simplicity equal to N for both Red and Blue warfighting parties); delta strikes (for technical details on included tensor terms, see[3] [4]).