# Vector Representations and Why Would They Deviate From Projective Geometry? Note Quote. There is, of course, a definite reason why von Neumann used the mathematical structure of a complex Hilbert space for the formalization of quantum mechanics, but this reason is much less profound than it is for Riemann geometry and general relativity. The reason is that Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics turned out to be equivalent, the first being a formalization of the new mechanics making use of l2, the set of all square summable complex sequences, and the second making use of L2(R3), the set of all square integrable complex functions of three real variables. The two spaces l2 and L2(R3) are canonical examples of a complex Hilbert space. This means that Heisenberg and Schrödinger were working already in a complex Hilbert space, when they formulated matrix mechanics and wave mechanics, without being aware of it. This made it a straightforward choice for von Neumann to propose a formulation of quantum mechanics in an abstract complex Hilbert space, reducing matrix mechanics and wave mechanics to two possible specific representations.

One problem with the Hilbert space representation was known from the start. A (pure) state of a quantum entity is represented by a unit vector or ray of the complex Hilbert space, and not by a vector. Indeed vectors contained in the same ray represent the same state or one has to renormalize the vector that represents the state after it has been changed in one way or another. It is well known that if rays of a vector space are called points and two dimensional subspaces of this vector space are called lines, the set of points and lines corresponding in this way to a vector space, form a projective geometry. What we just remarked about the unit vector or ray representing the state of the quantum entity means that in some way the projective geometry corresponding to the complex Hilbert space represents more intrinsically the physics of the quantum world as does the Hilbert space itself. This state of affairs is revealed explicitly in the dynamics of quantum entities, that is built by using group representations, and one has to consider projective representations, which are representations in the corresponding projective geometry, and not vector representations.

# Philosophy of Quantum Entanglement and Topology Many-body entanglement is essential for the existence of topological order in condensed matter systems and understanding many-body entanglement provides a promising approach to understand in general what topological orders exist. It also leads to tensor network descriptions of many-body wave functions potentializing the classification of phases of quantum matter. The generic many-body entanglement is reduced to specifically 2-body systems for choice of entanglement. Consider the equation,

S(A) ≡ −tr(ρA log2A)) —– (1)

where, ρA ≡ trBAB ⟩⟨ΨAB | is the density matrix for part A, and where we assumed that the whole system is in a pure state AB.

Specializing AB⟩ to a ground state in a local Hamiltonian in D dimensions spatially, the central observation being that the entanglement between of a region A of size LD and the (much larger) rest B of the lattice is then often proportional to the size |σ(A)| of the boundary σ(A) of region A,

S(A) ≈ |σ(A)| ≈ LD−1  —– (2)

where, the correction -1 is due to the topological order of the topic code, thus signifying adherence to Boundary Law observed in the ground state of gapped local Hamiltonian in arbitrary dimension D, as well as in some gapless systems in D > 1 dimensions. Instead, in gapless systems in D = 1 dimensions, as well as in certain gapless systems in D > 1 dimensions (namely systems with a Fermi surface of dimension D − 1), ground state entanglement displays a logarithmic correction to the boundary law,

S(A) ≈ |σ(A)| log2 (|σ(A)|) ≈ LD−1 log2(L) —– (3)

At an intuitive level, the boundary law of (2) is understood as resulting from entanglement that involves degrees of freedom located near the boundary between regions A and B. Also intuitively, the logarithmic correction of (3) is argued to have its origin in contributions to entanglement from degrees of freedom that are further away from the boundary between A and B. Given the entanglement between A and B, introducing an entanglement contour sA that assigns a real number sA(i) ≥ 0 to each lattice site i contained in region A such that the sum of sA(i) over all the sites i ∈ A is equal to the entanglement entropy S (A),

S(A) = Σi∈A sA(i) —– (4)

and that aims to quantifying how much the degrees of freedom in site i participate in/contribute to the entanglement between A and B. And as Chen and Vidal put it, the entanglement contour sA(i) is not equivalent to the von Neumann entropy S(i) ≡ −tr ρ(i) log2 ρ(i) of the reduced density matrix ρ(i) at site i. Notice that, indeed, the von Neumann en- tropy of individual sites in region A is not additive in the presence of correlations between the sites, and therefore generically

S(A) ≠ Σi∈A S(i)

whereas the entanglement contour sA(i) is required to fulfil (4). Relatedly, when site i is only entangled with neighboring sites contained within region A, and it is thus uncorrelated with region B, the entanglement contour sA(i) will be required to vanish, whereas the one-site von Neumann entropy S(i) still takes a non-zero value due to the presence of local entanglement within region A.

As an aside, in the traditional approach to quantum mechanics, a physical system is described in a Hilbert space: Observables correspond to self-adjoint operators and statistical operators are associated with the states. In fact, a statistical operator describes a mixture of pure states. Pure states are the really physical states and they are given by rank one statistical operators, or equivalently by rays of the Hilbert space. Von Neumann associated an entropy quantity to a statistical operator and his argument was a gedanken experiment on the ground of phenomenological thermodynamics. Let us consider a gas of N(≫ 1) molecules in a rectangular box K. Suppose that the gas behaves like a quantum system and is described by a statistical operator D, which is a mixture λ|φ1⟩⟨φ1| + (1 − λ)|φ1⟩⟨φ2|, |φi⟩ ≡ φ is a state vector (i = 1, 2). We may take λN molecules in the pure state φ1 and (1−λ)N molecules in the pure state φ2. On the basis of phenomenological thermodynamics, we assume that if φ1 and φ2 are orthogonal, then there is a wall that is completely permeable for the φ1-molecules and isolating for the φ2-molecules. We add an equally large empty rectangular box K′ to the left of the box K and we replace the common wall with two new walls. Wall (a), the one to the left is impenetrable, whereas the one to the right, wall (b), lets through the φ1-molecules but keeps back the φ2-molecules. We add a third wall (c) opposite to (b) which is semipermeable, transparent for the φ2-molecules and impenetrable for the φ1-ones. Then we push slowly (a) and (c) to the left, maintaining their distance. During this process the φ1-molecules are pressed through (b) into K′ and the φ2-molecules diffuse through wall (c) and remain in K. No work is done against the gas pressure, no heat is developed. Replacing the walls (b) and (c) with a rigid absolutely impenetrable wall and removing (a) we restore the boxes K and K′ and succeed in the separation of the φ1-molecules from the φ2-ones without any work being done, without any temperature change and without evolution of heat. The entropy of the original D-gas ( with density N/V ) must be the sum of the entropies of the φ1- and φ2-gases ( with densities λ N/V and (1 − λ)N/V , respectively). If we compress the gases in K and K′ to the volumes λV and (1 − λ)V , respectively, keeping the temperature T constant by means of a heat reservoir, the entropy change amounts to κλN log λ and κ(1 − λ)N log(1 − λ), respectively. Indeed, we have to add heat in the amount of λiNκT logλi (< 0) when the φi-gas is compressed, and dividing by the temperature T we get the change of entropy. Finally, mixing the φ1- and φ2-gases of identical density we obtain a D-gas of N molecules in a volume V at the original temperature. If S0(ψ,N) denotes the entropy of a ψ-gas of N molecules (in a volume V and at the given temperature), we conclude that

S0(φ1,λN)+S0(φ2,(1−λ)N) = S0(D, N) + κλN log λ + κ(1 − λ)N log(1 − λ) —– (5)

must hold, where κ is Boltzmann’s constant. Assuming that S0(ψ,N) is proportional to N and dividing by N we have

λS(φ1) + (1 − λ)S(φ2) = S(D) + κλ log λ + κ(1 − λ) log(1 − λ) —– (6)

where S is certain thermodynamical entropy quantity ( relative to the fixed temperature and molecule density ). We arrived at the mixing property of entropy, but we should not forget about the initial assumption: φ1 and φ2 are supposed to be orthogonal. Instead of a two-component mixture, von Neumann operated by an infinite mixture, which does not make a big difference, and he concluded that

S (Σiλi|φi⟩⟨φi|) = ΣiλiS(|φi⟩⟨φi|) − κ Σiλi log λi —– (7)

Von Neumann’s argument does not require that the statistical operator D is a mixture of pure states. What we really needed is the property D = λD1 + (1 − λ)D2 in such a way that the possible mixed states D1 and D2 are disjoint. D1 and D2 are disjoint in the thermodynamical sense, when there is a wall which is completely permeable for the molecules of a D1gas and isolating for the molecules of a D2-gas. In other words, if the mixed states D1 and D2 are disjoint, then this should be demonstrated by a certain filter. Mathematically, the disjointness of D1 and D2 is expressed in the orthogonality of the eigenvectors corresponding to nonzero eigenvalues of the two density matrices. The essential point is in the remark that (6) must hold also in a more general situation when possibly the states do not correspond to density matrices, but orthogonality of the states makes sense:

λS(D1) + (1 − λ)S(D2) = S(D) + κλ log λ + κ(1 − λ) log(1 − λ) —– (8)

(7) reduces the determination of the (thermodynamical) entropy of a mixed state to that of pure states. The so-called Schatten decomposition Σi λi|φi⟩⟨φi| of a statistical operator is not unique even if ⟨φi , φj ⟩ = 0 is assumed for i ≠ j . When λi is an eigenvalue with multiplicity, then the corresponding eigenvectors can be chosen in many ways. If we expect the entropy S(D) to be independent of the Schatten decomposition, then we are led to the conclusion that S(|φ⟩⟨φ|) must be independent of the state vector |φ⟩. This argument assumes that there are no superselection sectors, that is, any vector of the Hilbert space can be a state vector. On the other hand, von Neumann wanted to avoid degeneracy of the spectrum of a statistical operator. Von Neumann’s proof of the property that S(|φ⟩⟨φ|) is independent of the state vector |φ⟩ was different. He did not want to refer to a unitary time development sending one state vector to another, because that argument requires great freedom in choosing the energy operator H. Namely, for any |φ1⟩ and |φ2⟩ we would need an energy operator H such that

eitH|φ1⟩ = |φ2⟩

This process would be reversible. Anyways, that was quite a digression.

Entanglement between A and B is naturally described by the coefficients {pα} appearing in the Schmidt decomposition of the state |ΨAB⟩,

AB⟩ = Σα √pαAα ⟩ ⊗ |ΨBα ⟩ —– (9)

These coefficients {pα} correspond to the eigenvalues of the reduced density matrix ρA, whose spectral decomposition reads

ρA = ΣαpAα⟩⟨ΨAα—– (10)

defining a probability distribution, pα ≥ 0, Σα pα = 1, in terms of which the von Neumann entropy S(A) is

S(A) = − Σαpα log2(pα—– (11)

On the other hand, the Hilbert space VA of region A factorizes as the tensor product

VA = ⊗ i∈A V(i) —– (12)

where V(i) describes the local Hilbert space of site i. The reduced density matrix ρA in (10) and the factorization of (12) define two inequivalent structures within the vector space VA of region A. The entanglement contours A is a function from the set of sites i∈A to the real numbers,

sA : A → ℜ —– (13)

that attempts to relate these two structures, by distributing the von-Neumann entropy S(A) of (11) among the sites i ∈ A. According to Chen and Vidal, there are five conditions/requirements on entanglement contours that need satiation.

a. Positivity: sA(i) ≥ 0

b. Normalization: Σi∈AsA(i) = S(A)

These constraints amount to defining a probability distribution pi ≡ sA(i)/S(A) over the sites i ∈ A, with pi ≥ 0 and i Σipi = 1, such that sA(i) = piS(A), however, do not requiring sA to inform us about the spatial structure of entanglement in A, but only relating to the density matrix ρA through its total von Neumann entropy S(A).

c. Symmetry: if T is a symmetry of ρA, that is AT = ρA, and T exchanges site i with site j, then sA(i) = sA(j).

This condition ensures that the entanglement contour is the same on two sites i and j of region A that, as far as entanglement is concerned, play an equivalent role in region A. It uses the (possible) presence of a spatial symmetry, such as invariance under space reflection, or under discrete translations/rotations, to define an equivalence relation in the set of sites of region A, and requires that the entanglement contour be constant within each resulting equivalence class. Notice, however, that this condition does not tell us whether the entanglement contour should be large or small on a given site (or equivalence class of site). In particular, the three conditions above are satisfied by a canonical choice sA(i) = S (A)/|A|, that is a flat entanglement contour over the |A| sites contained in region A, which once more does not tell us anything about the spatial structure of the von Neumann entropy in ρA.

The remaining conditions refer to subregions within region A, instead of referring to single sites. It is therefore convenient to (trivially) extend the definition of entanglement contour to a set X of sites in region A, X ⊆ A, with vector space

VX = ⊗i∈X V(i) —– (14)

as the sum of the contour over the sites in X,

sA(X) ≡  Σi∈XsA(i) —– (15)

It follows from this extension that for any two disjoint subsets X1, X2 ⊆ A, with X1 ∩ X2 = ∅, the contour is additive,

sA(X1 ∪ X2) = sA(X1) + sA(X2—– (16)

In particular, condition 2 can be now recast as sA(A) =S(A). Similarly, if X, X ⊆ A, are such that all the sites of X1 are also contained in X2, X1X2 ,then the contour must be larger on X2 than on X1 (monotonicity of sA(X)),

sA(X1) ≤ sA(X2) if X1 ⊆ X2 —– (17)

d. Invariance under local unitary transformations: if the state |Ψ′AB is obtained from the state AB by means of a unitary transformation UX that acts on a subset X ⊆ A of sites of region A, that is |Ψ′AB⟩ ≡ UXAB, then the entanglement contour sA(X) must be the same for state AB and for state |Ψ′AB.

That is, the contribution of region X to the entanglement between A and B is not affected by a redefinition of the sites or change of basis within region X. Notice that it follows that  Ucan also not change sA(X’), where X’ ≡ A − X is the complement of X in A.

To motivate our last condition, let us consider a state AB that factorizes as the product

AB⟩ = |ΨXXB⟩ ⊗ |ΨX’X’B—– (18)

where X ⊆ A and XB ⊆ B are subsets of sites in regions A and B, respectively, and X’ ⊆ A and X’B ⊆ B are their complements within A and B, so that

VA = VX ⊗ VX’, —– (19)

VB = VXB ⊗ VX’B —– (20)

in this case the reduced density matrix ρA factorizes as ρA = ρX ⊗ ρX’ and the entanglement entropy is additive,

S(A) = S(X) + S(X’) —– (21)

Since the entanglement entropy S(X) of subregion X is well-defined, let the entanglement profile over X be equal to it,

sA(X) = S(X) —– (22)

The last condition refers to a more general situation where, instead of obeying (18), the state AB factorizes as the product

AB⟩ = |ΨΩAΩB⟩ ⊗ |ΨΩ’AΩ’B, —– (23)

with respect to some decomposition of VA and VB as

tensor products of factor spaces,

VA = VΩA ⊗ VΩ’A, —– (24)

VB = VΩB ⊗ VΩ’B —– (25)

Let S(ΩA) denote the entanglement entropy supported on the first factor space VΩA of  VA, that is

S(ΩA) = −tr(ρΩA log2ΩA)) —– (26)

ρΩA ≡ trΩB |Ψ ΩA ΩB⟩⟨Ψ ΩA ΩB| —– (27)

and let X ⊆ A be a subset of sites whose vector space VX is completely contained in VΩA , meaning that VΩA can be further decomposed as

VΩA  ≈ VX VX’ —– (28)

e. Upper bound: if a subregion X ⊆ A is contained in a factor space ΩA (24 and 28) then the entanglement contour of subregion X cannot be larger than the entanglement entropy S(ΩA) (26)

sA(X) S(ΩA) —– (29)

This condition says that whenever we can ascribe a concrete value S(ΩA) of the entanglement entropy to a factor space ΩA within region A (that is, whenever the state AB factorizes as in (24) then the entanglement contour has to be consistent with this fact, meaning that the contour S(X) in any subregion X contained in the factor space ΩA is upper bounded by S(ΩA).

Let us consider a particular case of condition e. When a region X ∈ A is not at all correlated with B, that is ρXBX ⊗ ρB,then it can be seen that X is contained in some factor space ΩA such that the state |Ψ ΩA ΩB itself further factorizes as |Ψ ΩA⟩ ⊗ |ΨΩB, so that (23) becomes

AB⟩ = |Ψ ΩA⟩ ⊗ |ΨΩB ⊗ |ΨΩ’AΩ’B ⟩, —– (30)

and S(ΩA) = 0. Condition e then requires that sA(X) = 0, that is

ρXBX ⊗ ρB sA(X) = 0, —– (31)

reflecting the fact that a region X ⊆ A that is not correlated with B does not contribute at all to the entanglement between A and B. Finally, the upper bound in e can be alternatively announced as a lower bound. Let Y ⊆ A be a subset of sites whose vector space VY completely contains VΩA in (24), meaning that VY can be further decomposed as

VY VΩA ⊗ VΩ’A —– (32)

e’. Lower bound: The entanglement contour of subregion Y is at least equal to the entanglement entropy S(ΩA) in (26),

sA(Y) ≥ S(ΩA) —– (33)

Conditions a-e (e’) are not expected to completely determine the entanglement contour. In other words, there probably are inequivalent functions sA : A → ℜ that conform to all the conditions above. So, where do we get philosophical from here? It is through the entanglement contour through selected states that a time evolution ensuing a global or a local quantum quench characterizing entanglement between regions rather than within regions, revealing a a detailed real-space structure of the entanglement of a region A and its dynamics, well beyond what is accessible from the entanglement entropy alone. But, that isn’t all. Questions of how to quantify entanglement and non-locality, and the need to clarify the relationship between them are important not only conceptually, but also practically, insofar as entanglement and non-locality seem to be different resources for the performance of quantum information processing tasks. Whether in a given quantum information protocol (cryptography, teleportation, and algorithm . . .) it is better to look for the largest amount of entanglement or the largest amount of non-locality becomes decisive. The ever-evolving field of quantum information theory is devoted to using the principles and laws of quantum mechanics to aid in the acquisition, transmission, and processing of information. In particular, it seeks to harness the peculiarly quantum phenomena of entanglement, superposition, and non-locality to perform all sorts of novel tasks, such as enabling computations that operate exponentially faster or more efficiently than their classical counterparts (via quantum computers) and providing unconditionally secure cryptographic systems for the transfer of secret messages over public channels (via quantum key distribution). By contrast, classical information theory is concerned with the storage and transfer of information in classical systems. It uses the “bit” as the fundamental unit of information, where the system capable of representing a bit can take on one of two values (typically 0 or 1). Classical information theory is based largely on the concept of information formalized by Claude Shannon in the late 1940s. Quantum information theory, which was later developed in analogy with classical information theory, is concerned with the storage and processing of information in quantum systems, such as the photon, electron, quantum dot, or atom. Instead of using the bit, however, it defines the fundamental unit of quantum information as the “qubit.” What makes the qubit different from a classical bit is that the smallest system capable of storing a qubit, the two-level quantum system, not only can take on the two distinct values |0 and |1 , but can also be in a state of superposition of these two states: |ψ = α0 |0 + α1 |1.

Quantum information theory has opened up a whole new range of philosophical and foundational questions in quantum cryptography or quantum key distribution, which involves using the principles of quantum mechanics to ensure secure communication. Some quantum cryptographic protocols make use of entanglement to establish correlations between systems that would be lost upon eavesdropping. Moreover, a quantum principle known as the no-cloning theorem prohibits making identical copies of an unknown quantum state. In the context of a C∗-algebraic formulation,  quantum theory can be characterized in terms of three information-theoretic constraints: (1) no superluminal signaling via measurement, (2) no cloning (for pure states) or no broadcasting (mixed states), and (3) no unconditionally secure bit commitment.

Entanglement does not refute the principle of locality. A sketch of the sort of experiment commonly said to refute locality runs as follows. Suppose that you have two electrons with entangled spin. For each electron you can measure the spin along the X, Y or Z direction. If you measure X on both electrons, then you get opposite values, likewise for measuring Y or Z on both electrons. If you measure X on one electron and Y or Z on the other, then you have a 50% probability of a match. And if you measure Y on one and Z on the other, the probability of a match is 50%. The crucial issue is that whether you find a correlation when you do the comparison depends on whether you measure the same quantity on each electron. Bell’s theorem just explains that the extent of this correlation is greater than a local theory would allow if the measured quantities were represented by stochastic variables (i.e. – numbers picked out of a hat). This fact is often misrepresented as implying that quantum mechanics is non-local. But in quantum mechanics, systems are not characterised by stochastic variables, but, rather, by Hermitian operators. There is an entirely local explanation of how the correlations arise in terms of properties of systems represented by such operators. But, another answer to such violations of the principle of locality could also be “Yes, unless you get really obsessive about it.” It has been formally proven that one can have determinacy in a model of quantum dynamics, or one can have locality, but cannot have both. If one gives up the determinacy of the theory in various ways, one can imagine all kinds of ‘planned flukes’ like the notion that the experiments that demonstrate entanglement leak information and pre-determine the environment to make the coordinated behavior seem real. Since this kind of information shaping through distributed uncertainty remains a possibility, folks can cling to locality until someone actually manages something like what those authors are attempting, or we find it impossible. If one gives up locality instead, entanglement does not present a problem, the theory of relativity does. Because the notion of a frame of reference is local. Experiments on quantum tunneling that violate the constraints of the speed of light have been explained with the idea that probabilistic partial information can ‘lead’ real information faster than light by pushing at the vacuum underneath via the ‘Casimir Effect’. If both of these make sense, then the information carried by the entanglement when it is broken would be limited as the particles get farther apart — entanglements would have to spontaneously break down over time or distance of separation so that the probabilities line up. This bodes ill for our ability to find entangled particles from the Big Bang, which seems to be the only prospect in progress to debunk the excessively locality-focussed.

But, much of the work remains undone and this is to be continued…..