2-2

ENTANGLED?

Notes for a Reed College Physics Seminar presented 11 October 2006

REMARK ADDED AFTER COMPLETION OF THE NOTEBOOK: In the seminar I allowed myself, in the interest of expository simplicity, to assume that all vector components are real.  In an addendum—written partly in response to a remark by Tom Wieting—I develop the quite independent separation criteria that arise when complex phase factors are assigned to the components of composite state vectors.

NOTE: This notebook was originally written in Mathematica 5.2, and converted to v7.0 on 11 February 2013. It is based upon material I wrote in an effort to be of assistance to Julia Keller in her thesis project ("Quantifying Entanglement" (2007)), at a time when I was largely ignorant of the literature and standard methods that bear on the problem addressed. The work seems in retrospect to be of interest in only one respect: the use made of the Moore-Penrose pseudo-inverse.

Introduction: correlations latent in the states of composite quantum systems

Alice is in her laboratory, studying quantum system (Subscript[𝒮, Alice]). Bob is in his lab, studying his quite independent system (Subscript[𝒮, Bob]).

For expository convenience we assume both systems to be 2-state systems (which, experimentally, happens actually to be the most commonly encountered situation). Alice writes

(Subscript[ψ, Alice])=(({{(Subscript[α, 1])}, {(Subscript[α, 2])}}));

to describe the (in this instance "pure") quantum state of (Subscript[𝒮, Alice]) , and

A=(({{(Subscript[a, 11]), (Subscript[a, 12])}, {(Subscript[a, 21]), (Subscript[a, 22])}}));

to describe her Hamiltonian, meters, etc. Bob writes

(Subscript[ψ, Bob])=(({{(Subscript[β, 1])}, {(Subscript[β, 2])}}));

B=(({{(Subscript[b, 11]), (Subscript[b, 12])}, {(Subscript[b, 21]), (Subscript[b, 22])}}));

Constructions like this

A.(Subscript[ψ, Alice])//MatrixForm
B.(Subscript[ψ, Bob])//MatrixForm

assume importance when they sit down to do standard quantum calculations.

God, looking down from on high, gives equal attention simultaneously to both Alice and Bob. How does He do it?

Introduction of the KRONECKER PRODUCT:

Let P and Q be matrices of arbitrary—and typically dissimilar—dimension:

P=(({{(Subscript[p, 11]), (Subscript[p, 12])}, {(Subscript[p, 21]), (Subscript[p, 22])}, {(Subscript[p, 31]), (Subscript[p, 32 ])}}));
Q=(({{(Subscript[q, 11]), (Subscript[q, 12])}, {(Subscript[q, 21]), (Subscript[q, 22])}}));

The Kronecker product has this meaning

P⊗Q≡(({{(Subscript[p, 11])Q, (Subscript[p, 12])Q}, {(Subscript[p, 21])Q, (Subscript[p, 22])Q}, {(Subscript[p, 31])Q, (Subscript[p, 32 ])Q}}))

and is produced by the following command:

P_⊗Q_:=Flatten[Table[Flatten[Table[Part[Outer[Times,P,Q],i,j,k],{j,Dimensions[P][[2]]}]],{i,Dimensions[P][[1]]},{k,Dimensions[Q][[1]]}],1]

P⊗Q//MatrixForm

(({{(Subscript[p, 11]) (Subscript[q, 11]), (Subscript[p, 11]) (Subscript[q, 12]), (Subscript[p, 12]) (Subscript[q, 11]), (Subscript[p, 12]) (Subscript[q, 12])}, {(Subscript[p, 11]) (Subscript[q, 21]), (Subscript[p, 11]) (Subscript[q, 22]), (Subscript[p, 12]) (Subscript[q, 21]), (Subscript[p, 12]) (Subscript[q, 22])}, {(Subscript[p, 21]) (Subscript[q, 11]), (Subscript[p, 21]) (Subscript[q, 12]), (Subscript[p, 22]) (Subscript[q, 11]), (Subscript[p, 22]) (Subscript[q, 12])}, {(Subscript[p, 21]) (Subscript[q, 21]), (Subscript[p, 21]) (Subscript[q, 22]), (Subscript[p, 22]) (Subscript[q, 21]), (Subscript[p, 22]) (Subscript[q, 22])}, {(Subscript[p, 31]) (Subscript[q, 11]), (Subscript[p, 31]) (Subscript[q, 12]), (Subscript[p, 32]) (Subscript[q, 11]), (Subscript[p, 32]) (Subscript[q, 12])}, {(Subscript[p, 31]) (Subscript[q, 21]), (Subscript[p, 31]) (Subscript[q, 22]), (Subscript[p, 32]) (Subscript[q, 21]), (Subscript[p, 32]) (Subscript[q, 22])}}))

The Kronecker product provides a natural mechanism for developing multiple copies of linear algebra simultaneously , and at the same time for establishing cross-talk among them.

To described the state of the composite Alice/Bob system, God writes

Ψ=(Subscript[ψ, Alice])⊗(Subscript[ψ, Bob]);

Ψ//MatrixForm

and to describe their respective Hamiltonians/meters/etc. He writes

(Subscript[A, God])=A⊗(({{1, 0}, {0, 1}}));
(Subscript[B, God])=(({{1, 0}, {0, 1}}))⊗B;

(Subscript[A, God])//MatrixForm
(Subscript[B, God])//MatrixForm

He is thus enabled to describe Alice/Bob's activity in a unified way:

((Subscript[A, God])).Ψ==(A.(Subscript[ψ, Alice]))⊗(Subscript[ψ, Bob])//Simplify
((Subscript[B, God])).Ψ==(Subscript[ψ, Alice])⊗(B.(Subscript[ψ, Bob]))//Simplify

TYPICAL CONCRETE CASE (of, as it happens, very great special interest):

Alice selects the measurement device (or "meter") described by the hermitian matrix

A=(({{1, 0}, {0, -1}}));

From

Eigenvalues[A]

we see that her meter always announces either +1 or -1 (up/down, true/false, yes/no). God writes

(Subscript[A, God])=A⊗(({{1, 0}, {0, 1}}));

(Subscript[A, God])//MatrixForm

to describe Alice's meter. The "spectral resolution" of (Subscript[A, God]) is immediate:

(Subscript[A, God])==(+1)(({{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}))+(-1)(({{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}))

where the matrices are obviously projective (project onto orthogonal 2-spaces.

We will suppose Bob to have selected a similar meter

B=(({{1, 0}, {0, -1}}));

which leads God to write

(Subscript[B, God])=(({{1, 0}, {0, 1}}))⊗B;

(Subscript[B, God])//MatrixForm

Immediately

(Subscript[B, God])==(+1)(({{1, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 0}}))+(-1)(({{0, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 1}}))

Suppose God—trickster that He is—presents Alice serially with many copies of the state

Ψ=(((({{1}, {0}}))⊗(({{0}, {1}}))+(({{0}, {1}}))⊗(({{1}, {0}})))/(Sqrt[2]));

Ψ//MatrixForm

Alice's meter announces +1 with probability

Transpose[Ψ].(({{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}})).Ψ

and on such occasions prepares the state (here the (Sqrt[2]) is  normalization factor)

AliceOutUp=(Sqrt[2])(({{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}})).Ψ;

AliceOutUp//MatrixForm

Alternatively, her meter announces -1 with probabilty

Transpose[Ψ].(({{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}})).Ψ

and prepares the state

AliceOutDn=(Sqrt[2])(({{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}})).Ψ;

AliceOutDn//MatrixForm

Because Bob—gentleman that he is—always lets Alice go first, he with his B-meter examines states that have been altered by Alice's prior measurement activity. When he inherits an AliceOutUp state his meter reads +1 with probability

Transpose[AliceOutUp].(({{1, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 0}})).AliceOutUp

—which is to say: never—and reads -1 with probability

Transpose[AliceOutUp].(({{0, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 1}})).AliceOutUp

—which is to say: with certainty. On  the other hand, when he inherits an AliceOutDn state his meter reads +1 with probability

Transpose[AliceOutDn].(({{1, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 0}})).AliceOutDn

—which is to say: with certainty—and never reads -1:

Transpose[AliceOutDn].(({{0, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 1}})).AliceOutDn

Those statistical results are summarized in the following table, where the red entries are JOINT probabilities, while the blue entries are MARGINAL probabilities, got by adding all joint probabilities in a  row/column.

({{\[Placeholder], BobUp, BobDn, Alice Marginal}, {AliceUp, 0, (1/2), (1/2)}, {AliceDn, (1/2), 0, (1/2)}, {Bob Marginal, (1/2), (1/2), \[Placeholder]}})

Alice's meter reads ± with equal frequency, and so does Bob's. But when they come together (Alice works at Stanford, Bob at Harvard) they find that their respective meter readings are perfectly anticorrelated!

What we have encountered here is David Bohm's sharp illustration (1951) of the essential upshot of the famous EPR paradox (1935). It was in this setting that John Bell (1964)—after supplying Alice and Bob with more general devices

A=(({{(Subscript[a, 3]), (Subscript[a, 1])-I (Subscript[a, 2])}, {(Subscript[a, 1])+I (Subscript[a, 2]), -(Subscript[a, 3])}}))
B=(({{(Subscript[b, 3]), (Subscript[b, 1])-I (Subscript[b, 2])}, {(Subscript[b, 1])+I (Subscript[b, 2]), -(Subscript[b, 3])}}))

—was led to the Bell inequality that sparked the on-going  experimental inquiry into the foundations of quantum mechanics, one early result of which was to rule out the possibility that behind quantum theory lurks a deterministic local hidden variable theory.

To lend quantitative precision to our intuitive allusion to "correlation", write

Ψ=(({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}, {(Subscript[ψ, 3])}, {(Subscript[ψ, 4])}}))

and define

correlation≡(averaged product)-(product of averages)

Are led by just such argumentation—analysis of Alice/Bob statistics—to define

correlation[w_,x_,y_,z_]:=(w^2)-(x^2)-(y^2)+(z^2)-((w^2)+(x^2)-(y^2)-(z^2)) ((w^2)-(x^2)+(y^2)-(z^2))

which when applied to Ψ  gives

correlation[(Subscript[ψ, 1]),(Subscript[ψ, 2]),(Subscript[ψ, 3]),(Subscript[ψ, 4])]

and in the example just studied

Ψ=(({{0}, {(1/(Sqrt[2]))}, {(1/(Sqrt[2]))}, {0}}))

supplies

correlation[0,(1/(Sqrt[2])),(1/(Sqrt[2])),0]

If, on the other hand, God had respected the "physical independence" of Alice/Bob's systems we would have had

Ψ=(({{(Subscript[α, 1]) (Subscript[β, 1])}, {(Subscript[α, 1]) (Subscript[β, 2])}, {(Subscript[α, 2]) (Subscript[β, 1])}, {(Subscript[α, 2]) (Subscript[β, 2])}}))

giving the intuitively more comprehensible result

FullSimplify[correlation[(Subscript[α, 1]) (Subscript[β, 1]),(Subscript[α, 1]) (Subscript[β, 2]),(Subscript[α, 2]) (Subscript[β, 1]),(Subscript[α, 2]) (Subscript[β, 2])]]

%/.{((Subscript[α, 1]^2)+(Subscript[α, 2]^2))->1,((Subscript[β, 1]^2)+(Subscript[β, 2]^2))->1}

Entangled states of composite quantum systems

Correlation effects have been shown to be absent when the composite state vector Ψ possesses the Kronecker factored form α⊗β. In all other ("non-separable") cases they are present, and Ψ is said to describe an entangled composite state.  One can then not speak separately of 
	• the quantum physics of (Subscript[𝒮, Alice])
	• the quantum physics of (Subscript[𝒮, Bob])
but can speak only of 
	• the quantum physics of (Subscript[𝒮, composite])≡ (Subscript[𝒮, Alice])⊗(Subscript[𝒮, Bob]).
It is the entanglement phenomenon that lies at the base of
	• quantum computation
	• quantum encryption
	• teleportation, etc.
It was Schrödinger—writing in response to publication of the EPR paper—who coined the term "entanglement," and who wrote
"When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled.
…Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separate and therefore virtually capable of being ‘best possibly known,’ i.e., of possessing, each of them, a representative of its own. The lack of knowledge is by no means due to the interaction being insufficiently known — at least not in the way that it could possibly be known more completely — it is due to the interaction itself."

DECISION PROBLEM and the related QUANTIFICATION PROBLEM

I quote from "The decision problem for entanglement," by Wayne C. Myrvold (1997):

In March 1995, Abner Shimony [professor of philosophy at Boston University, and an influential theorist in this area] attended a conference held in honor of hte 60th anniversary of the famous EPR paper at Technion University in Haifa, Israel. Among the lecturers at the conference was Alain Aspect [a leading experimentalist]. In Paris, immediately after the conference, Abner had a dream in which Aspect…posed the problem: is it algorithmically decidable whether a given quantum mechanical state is decidable or not?  Upon returning to the United States, Abner posed the question to me [Myrvold, then a recent PhD in philosophy, who specialized in "computability theory"].
[Within a couple of weeks, Myrvold had established that] the question is almost decidable—there is an algorithmic procedure that answers correctly in all but the borderline cases, and for these cases fails to terminate…  The function that gives the minimum distance from Ψ  to a product state is a computable function, even though there is no effective procedure that always produces the product state [in question]."

Beyond answering "yes/no" to the question "Is Ψ separable?" one would like, in non-separable cases, to be in position to respond to the question "How nearly separable is it; how entangled is the state Ψ?"

The literature provides many attempted answers to these questions. 
	• Many are elegantly abstruse;
	• Some claim to be "stronger" than others;
	• Most are computationally awkward.

Elementary solution of the decision problem

QUESTION: Do there exist normalized vectors (({{(Subscript[a, 1])}, {(Subscript[a, 2])}}))and (({{(Subscript[b, 1])}, {(Subscript[b, 2])}}))that satisfy the non-linear coupled system of equations

(({{(Subscript[a, 1])(Subscript[b, 1])}, {(Subscript[a, 1])(Subscript[b, 2])}, {(Subscript[a, 2])(Subscript[b, 1])}, {(Subscript[a, 2])(Subscript[b, 2])}}))=(({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}, {(Subscript[ψ, 3])}, {(Subscript[ψ, 4])}}))

where the ψs are the specified components of a normalized 4-vector?

REMARK: From

(Subscript[ψ, 1])(({{1}, {0}}))⊗(({{1}, {0}}))+(Subscript[ψ, 2])(({{1}, {0}}))⊗(({{0}, {1}}))+(Subscript[ψ, 3])(({{0}, {1}}))⊗(({{1}, {0}}))+(Subscript[ψ, 4])(({{0}, {1}}))⊗(({{0}, {1}}))//MatrixForm

we see that it is always possible to display Ψ as a linear combination of not more than four separated terms. And the elegant Schmidt decomposition theorem—which is based upon the SVD of the associated density matrix

ρ≡Ψ.Transpose[Ψ]

—asserts that it is always possible to write

Ψ=(Sqrt[(Subscript[μ, 1])])(Subscript[u, 1])⊗(Subscript[v, 1])+(Sqrt[(Subscript[μ, 2])])(Subscript[u, 2])⊗(Subscript[v, 2])

where (Subscript[u, 1]) and (Subscript[u, 2]) (ditto (Subscript[v, 1]) and (Subscript[v, 2])) are orthogonal 2-vectors. This method is restricted, however, to bipartite systems, cannot be extended to (for example) tripartite systems, where in separable/disentangled  cases one has

Ψ=((({{(Subscript[a, 1])}, {(Subscript[a, 2])}}))⊗(({{(Subscript[b, 1])}, {(Subscript[b, 2])}})))⊗(({{(Subscript[c, 1])}, {(Subscript[c, 2])}}))//MatrixForm

Constructive solution of the decision problem

Step One:

Take logs, get

(({{Log[(Subscript[a, 1])]+Log[(Subscript[b, 1])]}, {Log[(Subscript[a, 1])]+Log[(Subscript[b, 2])]}, {Log[(Subscript[a, 2])]+Log[(Subscript[b, 1])]}, {Log[(Subscript[a, 2])]+Log[(Subscript[b, 2])]}}))=(({{Log[(Subscript[ψ, 1])]}, {Log[(Subscript[ψ, 2])]}, {Log[(Subscript[ψ, 3])]}, {Log[(Subscript[ψ, 4])]}}))

Step Two:

Write

(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))(({{Log[(Subscript[a, 1])]}, {Log[(Subscript[a, 2])]}, {Log[(Subscript[b, 1])]}, {Log[(Subscript[b, 1])]}}))=(({{Log[(Subscript[ψ, 1])]}, {Log[(Subscript[ψ, 2])]}, {Log[(Subscript[ψ, 3])]}, {Log[(Subscript[ψ, 4])]}}))

Straightforward inversion is not possible, because

Det[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))]

But the singularity (non-invertibility) of the matrix signals not that the equations cannot be solved, but only that the solutions are non-unique: from

Transpose[NullSpace[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))]]//MatrixForm

we see that to every purported solution vector

(({{Log[(Subscript[a, 1])]}, {Log[(Subscript[a, 2])]}, {Log[(Subscript[b, 1])]}, {Log[(Subscript[b, 1])]}}))

can be added any vector of the form

(({{-λ}, {-λ}, {λ}, {λ}}))

since

(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}})).(({{-λ}, {-λ}, {λ}, {λ}}))==(({{0}, {0}, {0}, {0}}))

We are led thus to write

(({{Log[(Subscript[a, 1])]}, {Log[(Subscript[a, 2])]}, {Log[(Subscript[b, 1])]}, {Log[(Subscript[b, 1])]}}))=PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))].(({{Log[(Subscript[ψ, 1])]}, {Log[(Subscript[ψ, 2])]}, {Log[(Subscript[ψ, 3])]}, {Log[(Subscript[ψ, 4])]}}))+(({{-λ}, {-λ}, {λ}, {λ}}))

That the Moore-Penrose "pseudo-inverse"—which can be considered to be a child of the SVD—is in this intance a funny-looking thing

PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))]//MatrixForm

need not concern us. If we set

λ=Log[k]

then the preceding equation becomes

(({{Log[k (Subscript[a, 1])]}, {Log[k (Subscript[a, 2])]}, {Log[(Subscript[b, 1])/k]}, {Log[(Subscript[b, 1])/k]}}))=PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))].(({{Log[(Subscript[ψ, 1])]}, {Log[(Subscript[ψ, 2])]}, {Log[(Subscript[ψ, 3])]}, {Log[(Subscript[ψ, 4])]}}))

DIGRESSION: It helps to notice that matrices are lists of lists, and that functions like Log and Exp are, in Mathematica's opinion, "listable" in the very useful sense that I now demonstrate:

Log[(({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}, {(Subscript[ψ, 3])}, {(Subscript[ψ, 4])}}))]//MatrixForm

Exp[%]//MatrixForm

End of digression.

We are placed thus in position to set

Exp[(({{Log[k (Subscript[a, 1])]}, {Log[k (Subscript[a, 2])]}, {Log[(Subscript[b, 1])/k]}, {Log[(Subscript[b, 1])/k]}}))]//MatrixForm

equal to

Exp[PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))].Log[(({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}, {(Subscript[ψ, 3])}, {(Subscript[ψ, 4])}}))]]

Or, still more efficiently, to write

(({{(Subscript[a, 1])}, {(Subscript[a, 2])}, {(Subscript[b, 1])}, {(Subscript[b, 2])}}))=(({{1/(Subscript[k, a]), 0, 0, 0}, {0, 1/(Subscript[k, a]), 0, 0}, {0, 0, (Subscript[k, b]), 0}, {0, 0, 0, (Subscript[k, b])}})).Exp[PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))].Log[(({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}, {(Subscript[ψ, 3])}, {(Subscript[ψ, 4])}}))]]

MatrixForm[%]

The multipliers (Subscript[k, a]) and (Subscript[k, b]) acquire values from the requirements that α and β be unit vectors:

Solve[((((((Subscript[ψ, 1]^(3/8)) (Subscript[ψ, 2]^(3/8)))/((Subscript[k, a]) (Subscript[ψ, 3]^(1/8)) (Subscript[ψ, 4]^(1/8))))))^2)+((((((Subscript[ψ, 3]^(3/8)) (Subscript[ψ, 4]^(3/8)))/((Subscript[k, a]) (Subscript[ψ, 1]^(1/8)) (Subscript[ψ, 2]^(1/8))))))^2)==1,(Subscript[k, a])]
Solve[((((((Subscript[k, b]) (Subscript[ψ, 1]^(3/8)) (Subscript[ψ, 3]^(3/8)))/((Subscript[ψ, 2]^(1/8)) (Subscript[ψ, 4]^(1/8))))))^2)+((((((Subscript[k, b]) (Subscript[ψ, 2]^(3/8)) (Subscript[ψ, 4]^(3/8)))/((Subscript[ψ, 1]^(1/8)) (Subscript[ψ, 3]^(1/8))))))^2)==1,(Subscript[k, b])]

The composite state vector Ψ will be separable if and only if

(Subscript[k, a])=(Subscript[k, b])

FIRST EXAMPLE:

Look to the case

Ψ=(({{(Sqrt[3/8])}, {(Sqrt[3/8])}, {(Sqrt[1/8])}, {(Sqrt[1/8])}}));

We in this case have

(({{1/(Subscript[k, a]), 0, 0, 0}, {0, 1/(Subscript[k, a]), 0, 0}, {0, 0, (Subscript[k, b]), 0}, {0, 0, 0, (Subscript[k, b])}})).Exp[PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))].Log[Ψ]]

Solve[(((((3^(3/8))/((2^(3/4)) (Subscript[k, a])))))^2)+((((1/((2^(3/4)) (3^(1/8)) (Subscript[k, a])))))^2)==1,(Subscript[k, a])]//Simplify
Solve[((((((3^(1/8)) (Subscript[k, b]))/(2^(3/4)))))^2)+((((((3^(1/8)) (Subscript[k, b]))/(2^(3/4)))))^2)==1,(Subscript[k, b])]//Simplify

Here  (Subscript[k, a ])= (Subscript[k, b])  so the composite state Ψ is disentangled/separable. Our calculation has supplied

α={{((3^(3/8))/((2^(3/4)) (Subscript[k, a])))},{(1/((2^(3/4)) (3^(1/8)) (Subscript[k, a])))}}/.(Subscript[k, a])->((2^(1/4))/(3^(1/8)))
β={{(((3^(1/8)) (Subscript[k, b]))/(2^(3/4)))},{(((3^(1/8)) (Subscript[k, b]))/(2^(3/4)))}}/.(Subscript[k, b])->((2^(1/4))/(3^(1/8)))

and it is indeed the case that

α⊗β==Ψ

SECOND EXAMPLE:

Ψ=(1/(Sqrt[(8^2)+(10^2)+(12^2)+(15^2)]))(({{8}, {10}, {12}, {15}}));

(({{1/(Subscript[k, a]), 0, 0, 0}, {0, 1/(Subscript[k, a]), 0, 0}, {0, 0, (Subscript[k, b]), 0}, {0, 0, 0, (Subscript[k, b])}})).Exp[PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))].Log[Ψ]]

Solve[(((((2 ((((10/1599)))^(1/4)))/(Subscript[k, a]))))^2)+(((((((((10/533)))^(1/4)) (3^(3/4)))/(Subscript[k, a]))))^2)==1,(Subscript[k, a])]//N
Solve[(((2 ((((3/2665)))^(1/4)) (2^(3/4)) (Subscript[k, b])))^2)+(((((((3/1066)))^(1/4)) (5^(3/4)) (Subscript[k, b])))^2)==1,(Subscript[k, b])]//N

We have come again upon an disentangled case, and proceed now to check out that claim:

α={{((2 ((((10/1599)))^(1/4)))/(Subscript[k, a]))},{((((((10/533)))^(1/4)) (3^(3/4)))/(Subscript[k, a]))}}/.(Subscript[k, a])->1.0139337054294644`
β={{2 ((((3/2665)))^(1/4)) (2^(3/4)) (Subscript[k, b])},{((((3/1066)))^(1/4)) (5^(3/4)) (Subscript[k, b])}}/.(Subscript[k, b])->1.0139337054294646`

α⊗β-Ψ//Chop//MatrixForm

THIRD EXAMPLE:

Ψ=(1/(Sqrt[(23^2)+(12^2)+(42^2)+(37^2)]))(({{23}, {12}, {42}, {37}}));

(({{1/(Subscript[k, a]), 0, 0, 0}, {0, 1/(Subscript[k, a]), 0, 0}, {0, 0, (Subscript[k, b]), 0}, {0, 0, 0, (Subscript[k, b])}})).Exp[PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))].Log[Ψ]]//N

Solve[(((((0.41811171195346475`)/(Subscript[k, a]))))^2)+(((((0.9921179316209712`)/(Subscript[k, a]))))^2)==1,(Subscript[k, a])]//N
Solve[(((0.7822153877636896` (Subscript[k, b])))^2)+(((0.5303093410572127` (Subscript[k, b])))^2)==1,(Subscript[k, b])]//N

Here (Subscript[k, a])  and  (Subscript[k, b])  are distinct, so Ψ is NOT separable, describes an entangled composite state. If we nevertheless construct

α={{((0.41811171195346475`)/(Subscript[k, a]))},{((0.9921179316209712`)/(Subscript[k, a]))}}/.(Subscript[k, a])->1.0766222150394869`
β={{0.7822153877636896` (Subscript[k, b])},{0.5303093410572127` (Subscript[k, b])}}/.(Subscript[k, b])->1.0581631947198626`

then

α⊗β//MatrixForm

describes a disentangled "neighbor" of the entangled state

Ψ//N//MatrixForm

FOURTH EXAMPLE:

Look finally to the state

Ψ=(({{0}, {(1/(Sqrt[2]))}, {(1/(Sqrt[2]))}, {0}}));

that a playful God inflicted upon Alice. Mathematica, when asked to execute our algorithm, complains

(({{1/(Subscript[k, a]), 0, 0, 0}, {0, 1/(Subscript[k, a]), 0, 0}, {0, 0, (Subscript[k, b]), 0}, {0, 0, 0, (Subscript[k, b])}})).Exp[PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))].Log[Ψ]]

because it has encountered terms of the indeterminate form Log[0]. These we eliminate by setting

(Subscript[Ψ, God])=(({{ϵ}, {(1/(Sqrt[2]))}, {(1/(Sqrt[2]))}, {ϵ}}));

with the intention of letting ϵ ↓ 0 at the end of the day. We get

(({{1/(Subscript[k, a]), 0, 0, 0}, {0, 1/(Subscript[k, a]), 0, 0}, {0, 0, (Subscript[k, b]), 0}, {0, 0, 0, (Subscript[k, b])}})).Exp[PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))].Log[(Subscript[Ψ, God])]]

Proceeding in the standard way

Solve[(((((ϵ^(1/4))/((2^(1/8)) (Subscript[k, a])))))^2)+(((((ϵ^(1/4))/((2^(1/8)) (Subscript[k, a])))))^2)==1,(Subscript[k, a])]
Solve[((((((ϵ^(1/4)) (Subscript[k, b]))/(2^(1/8)))))^2)+((((((ϵ^(1/4)) (Subscript[k, b]))/(2^(1/8)))))^2)==1,(Subscript[k, b])]

we find that  (Subscript[k, a])  and  (Subscript[k, b])  are certainly unequal (the state Ψ is certainly entangled) and that in the limit

(Subscript[k, a])↓0
(Subscript[k, b])↑∞

(Subscript[k, a])  and  (Subscript[k, b])  could not be more different!

The state (Subscript[Ψ, God]) was actually drawn from the population of so-called "Bell states," which are commonly held to be composite states of maximal entanglement.

Concluding comments

1. Quantification

One cannot speak of "maximal entanglement" except in reference to some well-defined measure of entanglement (though one can speak unambituously of the absence of entanglement). Of those, easily half a dozen have been proposed, of which the most widely accepted is the entropy of entanglement, the definition of which may be taken up in Julia Keller's thesis seminar, and is patterned upon von Neumann's definition of the thermodynamic

entropy of mixture =  -Tr[ρ Log[ρ]]

where ρ is the density matrix. I am prepared, by the pattern of the preceding argument, to suggest this

Clear[ka,kb]

NewMeasureOfEntanglement[ka_, kb_] := 1-(E^(-(((kb-ka))^2)))

For unentangled states we would then have

NewMeasureOfEntanglement[(Subscript[k, a]),(Subscript[k, a])]

while for God's Bell state

NewMeasureOfEntanglement[0,∞]

2. Source of the k-ambiguity

Clear[α,β]
(Subscript[a, 1])=.
(Subscript[a, 2])=.
(Subscript[b, 1])=.
(Subscript[b, 2])=.

The k-ambiguity has a very simple and very general origin, which I illustrate in the simplest instance. Suppose once again that

α=(({{(Subscript[a, 1])}, {(Subscript[a, 2])}}));
β=(({{(Subscript[b, 1])}, {(Subscript[b, 2])}}));

Then

α⊗β//MatrixForm

renders transparent the fact that

(k α)⊗(β/k)==α⊗β

3. It takes two to entangle, but…

The realization that entanglement makes a difference, joined to the observation that every "isolated" system a detached fragment of what was once a larger system, points up the fact that in quantum mechanics the "isolated system" concept is an idealization. But so also, for that same reason, is the bipartite "Alice/Bob system." One is driven to the view that in quantum physics there is really only one system, only one state function, and that refers to the whole universe! (But what, by any semi-standard interpretation of quantum mechanics, can one mean by (Subscript[Ψ, universe])?)
The question therefore arises: Why does the "isolated system" concept enjoy such success in work-a-day quantum mechanics?
That entanglement does not make a difference in classical physics—that one classically can contemplate systems in isolation (I set aside "Mach's Principle" and all similarly vague suggestions)—emerges therefore as "the most characteristic feature of physics in the limit ℏ → 0." And a major mystery, since ℏ does, in point of physical fact, not → 0: all "classical" systems are ultimately quantum mechanical.

4. Generalization to the case of tripartite 2-component composite systems

Introduce

γ=(({{(Subscript[c, 1])}, {(Subscript[c, 2])}}));

construct

(α⊗β)⊗γ//MatrixForm

and attempt to solve

(({{(Subscript[a, 1]) (Subscript[b, 1]) (Subscript[c, 1])}, {(Subscript[a, 1]) (Subscript[b, 1]) (Subscript[c, 2])}, {(Subscript[a, 1]) (Subscript[b, 2]) (Subscript[c, 1])}, {(Subscript[a, 1]) (Subscript[b, 2]) (Subscript[c, 2])}, {(Subscript[a, 2]) (Subscript[b, 1]) (Subscript[c, 1])}, {(Subscript[a, 2]) (Subscript[b, 1]) (Subscript[c, 2])}, {(Subscript[a, 2]) (Subscript[b, 2]) (Subscript[c, 1])}, {(Subscript[a, 2]) (Subscript[b, 2]) (Subscript[c, 2])}}))=(({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}, {(Subscript[ψ, 3])}, {(Subscript[ψ, 4])}, {(Subscript[ψ, 5])}, {(Subscript[ψ, 6])}, {(Subscript[ψ, 7])}, {(Subscript[ψ, 8])}}))

Immediately

(({{1, 0, 1, 0, 1, 0}, {1, 0, 1, 0, 0, 1}, {1, 0, 0, 1, 1, 0}, {1, 0, 0, 1, 0, 1}, {0, 1, 1, 0, 1, 0}, {0, 1, 1, 0, 0, 1}, {0, 1, 0, 1, 1, 0}, {0, 1, 0, 1, 0, 1}}))(({{Log[(Subscript[a, 1])]}, {Log[(Subscript[a, 2])]}, {Log[(Subscript[b, 1])]}, {Log[(Subscript[b, 2])]}, {Log[(Subscript[c, 1])]}, {Log[(Subscript[c, 2])]}}))=Log[(({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}, {(Subscript[ψ, 3])}, {(Subscript[ψ, 4])}, {(Subscript[ψ, 5])}, {(Subscript[ψ, 6])}, {(Subscript[ψ, 7])}, {(Subscript[ψ, 8])}}))]

and from

NullSpace[(({{1, 0, 1, 0, 1, 0}, {1, 0, 1, 0, 0, 1}, {1, 0, 0, 1, 1, 0}, {1, 0, 0, 1, 0, 1}, {0, 1, 1, 0, 1, 0}, {0, 1, 1, 0, 0, 1}, {0, 1, 0, 1, 1, 0}, {0, 1, 0, 1, 0, 1}}))]

we are led to write

(({{Log[(Subscript[a, 1])]}, {Log[(Subscript[a, 2])]}, {Log[(Subscript[b, 1])]}, {Log[(Subscript[b, 2])]}, {Log[(Subscript[c, 1])]}, {Log[(Subscript[c, 2])]}}))+(({{λ}, {λ}, {0}, {0}, {-λ}, {-λ}}))+(({{μ}, {μ}, {-μ}, {-μ}, {0}, {0}}))=PseudoInverse[(({{1, 0, 1, 0, 1, 0}, {1, 0, 1, 0, 0, 1}, {1, 0, 0, 1, 1, 0}, {1, 0, 0, 1, 0, 1}, {0, 1, 1, 0, 1, 0}, {0, 1, 1, 0, 0, 1}, {0, 1, 0, 1, 1, 0}, {0, 1, 0, 1, 0, 1}}))].Log[(({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}, {(Subscript[ψ, 3])}, {(Subscript[ψ, 4])}, {(Subscript[ψ, 5])}, {(Subscript[ψ, 6])}, {(Subscript[ψ, 7])}, {(Subscript[ψ, 8])}}))]

Writing

(({{λ}, {λ}, {0}, {0}, {-λ}, {-λ}}))=(({{Log[k]}, {Log[k]}, {Log[1]}, {Log[1]}, {Log[1/k]}, {Log[1/k]}}))
(({{μ}, {μ}, {-μ}, {-μ}, {0}, {0}}))=(({{Log[m]}, {Log[m]}, {Log[1/m]}, {Log[1/m]}, {Log[1]}, {Log[1]}}))

we would proceed essentially as before, paying no heed to the pseudo-inverse is now even goofier:

PseudoInverse[(({{1, 0, 1, 0, 1, 0}, {1, 0, 1, 0, 0, 1}, {1, 0, 0, 1, 1, 0}, {1, 0, 0, 1, 0, 1}, {0, 1, 1, 0, 1, 0}, {0, 1, 1, 0, 0, 1}, {0, 1, 0, 1, 1, 0}, {0, 1, 0, 1, 0, 1}}))]//MatrixForm

I am encouraged to think that the method described here works even in the n-partite case. The Schmidt decomposition method and such other methods for resolving the decision problem as are presently known to me fail if n > 2.

Note finally that such "ambituity" as is present in the tripartite case can be traced to the obvious facts (1) that

((p α)⊗(q β))⊗(r γ)==(p q r)(α⊗β)⊗γ

and (2) that the expression on the right reduces to

(α⊗β)⊗γ

when the constants p, q, r are constrained to satisfy

p  q r=1

The n-partite generalization of this remark is evident. As was previously remarked, Schmidt decomposition and many of the other procedures described in the literature are special to bipartite systems.

Important ADDENDUM

It was as a matter of (I thought) mere expository convenience that I have heretofore assumed all vector components to be REAL. Here I discuss the consequence of relaxing that assumption. It is (really this time!) as a matter of mere expository convenience that I return to the particulars of my  FIRST EXAMPLE

Ψ=(({{(Sqrt[3/8])}, {(Sqrt[3/8])}, {(Sqrt[1/8])}, {(Sqrt[1/8])}}));

in which we make now the following adjustment:

Ψ=(({{(Sqrt[3/8])(E^(I (Subscript[ϕ, 1])))}, {(Sqrt[3/8])(E^(I (Subscript[ϕ, 2])))}, {(Sqrt[1/8])(E^(I (Subscript[ϕ, 3])))}, {(Sqrt[1/8])(E^(I (Subscript[ϕ, 4])))}}));

Our algorithm then provides

(({{(Subscript[a, 1])}, {(Subscript[a, 2])}, {(Subscript[b, 1])}, {(Subscript[b, 2])}}))=(({{1/(Subscript[k, a]), 0, 0, 0}, {0, 1/(Subscript[k, a]), 0, 0}, {0, 0, (Subscript[k, b]), 0}, {0, 0, 0, (Subscript[k, b])}})).Exp[PseudoInverse[(({{1, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}}))].Log[Ψ]]

PowerExpand[%]

So we have

α={{(((3^(3/8)) (E^(((3 I (Subscript[ϕ, 1]))/8)+((3 I (Subscript[ϕ, 2]))/8)-((I (Subscript[ϕ, 3]))/8)-((I (Subscript[ϕ, 4]))/8))))/((2^(3/4)) (Subscript[k, a])))},{((E^(-((I (Subscript[ϕ, 1]))/8)-((I (Subscript[ϕ, 2]))/8)+((3 I (Subscript[ϕ, 3]))/8)+((3 I (Subscript[ϕ, 4]))/8)))/((2^(3/4)) (3^(1/8)) (Subscript[k, a])))}};
β={{(((3^(1/8)) (E^(((3 I (Subscript[ϕ, 1]))/8)-((I (Subscript[ϕ, 2]))/8)+((3 I (Subscript[ϕ, 3]))/8)-((I (Subscript[ϕ, 4]))/8))) (Subscript[k, b]))/(2^(3/4)))},{(((3^(1/8)) (E^(-((I (Subscript[ϕ, 1]))/8)+((3 I (Subscript[ϕ, 2]))/8)-((I (Subscript[ϕ, 3]))/8)+((3 I (Subscript[ϕ, 4]))/8))) (Subscript[k, b]))/(2^(3/4)))}};

To properly manage the complex numbers I introduce this construction (due to Joel Franklin):

CompConj[A_] := A /.Complex[0,n_]->-Complex[0,n]

The squared norms of α and β are given now by expressions

Transpose[CompConj[α]].α
Transpose[CompConj[β]].β

to which the phase factors make no contribution, so we are not surprised todiscover that

Solve[(1/(2 (Sqrt[2]) (3^(1/4)) (Subscript[k, a]^2)))+((3^(3/4))/(2 (Sqrt[2]) (Subscript[k, a]^2)))==1, (Subscript[k, a])]
Solve[(((3^(1/4)) (Subscript[k, b]^2))/(Sqrt[2]))==1,(Subscript[k, b])]

yield exactly the results that we obtained prior to the introduction of phase factors—results that led us then to the conclusion that Ψ was in this instance separable.

We have arrived now at the normalized complex vectors

α={{(((3^(3/8)) (E^(((3 I (Subscript[ϕ, 1]))/8)+((3 I (Subscript[ϕ, 2]))/8)-((I (Subscript[ϕ, 3]))/8)-((I (Subscript[ϕ, 4]))/8))))/((2^(3/4)) (Subscript[k, a])))},{((E^(-((I (Subscript[ϕ, 1]))/8)-((I (Subscript[ϕ, 2]))/8)+((3 I (Subscript[ϕ, 3]))/8)+((3 I (Subscript[ϕ, 4]))/8)))/((2^(3/4)) (3^(1/8)) (Subscript[k, a])))}}/.(Subscript[k, a])->((2^(1/4))/(3^(1/8)))
β={{(((3^(1/8)) (E^(((3 I (Subscript[ϕ, 1]))/8)-((I (Subscript[ϕ, 2]))/8)+((3 I (Subscript[ϕ, 3]))/8)-((I (Subscript[ϕ, 4]))/8))) (Subscript[k, b]))/(2^(3/4)))},{(((3^(1/8)) (E^(-((I (Subscript[ϕ, 1]))/8)+((3 I (Subscript[ϕ, 2]))/8)-((I (Subscript[ϕ, 3]))/8)+((3 I (Subscript[ϕ, 4]))/8))) (Subscript[k, b]))/(2^(3/4)))}}/.(Subscript[k, b])->((2^(1/4))/(3^(1/8)))

We now construct

α⊗β//MatrixForm

and find that to recover the given 4-vector

Ψ//MatrixForm

it must be the case that

((3 I (Subscript[ϕ, 1]))/4)+((I (Subscript[ϕ, 2]))/4)+((I (Subscript[ϕ, 3]))/4)-((I (Subscript[ϕ, 4]))/4)==I (Subscript[ϕ, 1])
((I (Subscript[ϕ, 1]))/4)+((3 I (Subscript[ϕ, 2]))/4)-((I (Subscript[ϕ, 3]))/4)+((I (Subscript[ϕ, 4]))/4)==I  (Subscript[ϕ, 2])
((I (Subscript[ϕ, 1]))/4)-((I (Subscript[ϕ, 2]))/4)+((3 I (Subscript[ϕ, 3]))/4)+((I (Subscript[ϕ, 4]))/4)==I (Subscript[ϕ, 3])
-((I (Subscript[ϕ, 1]))/4)+((I (Subscript[ϕ, 2]))/4)+((I (Subscript[ϕ, 3]))/4)+((3 I (Subscript[ϕ, 4]))/4)==I (Subscript[ϕ, 4])

These equations can be formulated

(({{-(1/4), (1/4), (1/4), -(1/4)}, {(1/4), -(1/4), -(1/4), (1/4)}, {(1/4), -(1/4), -(1/4), (1/4)}, {-(1/4), (1/4), (1/4), -(1/4)}})).(({{(Subscript[ϕ, 1])}, {(Subscript[ϕ, 2])}, {(Subscript[ϕ, 3])}, {(Subscript[ϕ, 4])}}))==(({{0}, {0}, {0}, {0}}))

where, as it happens,

Det[(({{-(1/4), (1/4), (1/4), -(1/4)}, {(1/4), -(1/4), -(1/4), (1/4)}, {(1/4), -(1/4), -(1/4), (1/4)}, {-(1/4), (1/4), (1/4), -(1/4)}}))]

Looking to

NullSpace[(({{-(1/4), (1/4), (1/4), -(1/4)}, {(1/4), -(1/4), -(1/4), (1/4)}, {(1/4), -(1/4), -(1/4), (1/4)}, {-(1/4), (1/4), (1/4), -(1/4)}}))]

we conclude that the ϕs can be described

(({{(Subscript[ϕ, 1])}, {(Subscript[ϕ, 2])}, {(Subscript[ϕ, 3])}, {(Subscript[ϕ, 4])}}))=x(({{1}, {1}, {0}, {0}}))+y(({{1}, {0}, {1}, {0}}))+z(({{1}, {0}, {0}, {-1}}))

where x, y and z  are arbitrary constants. The vector α⊗β  therefore becomes

(({{(1/2) (Sqrt[(3/2)]) (E^(((3 I (Subscript[ϕ, 1]))/4)+((I (Subscript[ϕ, 2]))/4)+((I (Subscript[ϕ, 3]))/4)-((I (Subscript[ϕ, 4]))/4)))}, {(1/2) (Sqrt[(3/2)]) (E^(((I (Subscript[ϕ, 1]))/4)+((3 I (Subscript[ϕ, 2]))/4)-((I (Subscript[ϕ, 3]))/4)+((I (Subscript[ϕ, 4]))/4)))}, {((E^(((I (Subscript[ϕ, 1]))/4)-((I (Subscript[ϕ, 2]))/4)+((3 I (Subscript[ϕ, 3]))/4)+((I (Subscript[ϕ, 4]))/4)))/(2 (Sqrt[2])))}, {((E^(-((I (Subscript[ϕ, 1]))/4)+((I (Subscript[ϕ, 2]))/4)+((I (Subscript[ϕ, 3]))/4)+((3 I (Subscript[ϕ, 4]))/4)))/(2 (Sqrt[2])))}}))/.{(Subscript[ϕ, 1])->x+y+z,(Subscript[ϕ, 2])->x,(Subscript[ϕ, 3])->y,(Subscript[ϕ, 4])->-z}

Simplify[%]

which with the restoration of ϕ-notation becomes

{{(1/2) (Sqrt[(3/2)]) (E^(I (x+y+z)))},{(1/2) (Sqrt[(3/2)]) (E^(I x))},{((E^(I y))/(2 (Sqrt[2])))},{((E^(-I z))/(2 (Sqrt[2])))}}/.{x->(Subscript[ϕ, 2]), y->(Subscript[ϕ, 3]),z->-(Subscript[ϕ, 4])}//MatrixForm

(({{(1/2) (Sqrt[(3/2)]) (E^(I ((Subscript[ϕ, 2])+(Subscript[ϕ, 3])-(Subscript[ϕ, 4]))))}, {(1/2) (Sqrt[(3/2)]) (E^(I (Subscript[ϕ, 2])))}, {    (1/(2 (Sqrt[2])))(E^(I (Subscript[ϕ, 3])))}, {    (1/(2 (Sqrt[2])))(E^(I (Subscript[ϕ, 4])))}}))

while—by construction, but gratifyingly—from the given 4-vector Ψ we now recover the same thing:

Ψ/.{(Subscript[ϕ, 1])->(Subscript[ϕ, 2])+(Subscript[ϕ, 3])-(Subscript[ϕ, 4])}//MatrixForm

We conclude that the separability decision problem splits into two independent problems:
	• the DECISION PROBLEM WITH RESPECT TO AMPLITUDES, and
	• the DECISION PROBLEM WITH RESPECT TO PHASES.
The former is resolved by the procedure discussed in the main body of the text, while the latter is resolved by the statement that for

(({{(Subscript[Ψ, 1])(E^(I (Subscript[ϕ, 1])))}, {(Subscript[Ψ, 2])(E^(I (Subscript[ϕ, 2])))}, {(Subscript[Ψ, 3])(E^(I (Subscript[ϕ, 3])))}, {(Subscript[Ψ, 4])(E^(I (Subscript[ϕ, 4])))}}))

to be separable it must be the case that

(Subscript[ϕ, 1])=(Subscript[ϕ, 2])+(Subscript[ϕ, 3])-(Subscript[ϕ, 4])

It then follows (I revert now to my EXAMPLE)

α={{(1/2) (Sqrt[3]) (E^(((3 I (Subscript[ϕ, 1]))/8)+((3 I (Subscript[ϕ, 2]))/8)-((I (Subscript[ϕ, 3]))/8)-((I (Subscript[ϕ, 4]))/8)))},{(1/2) (E^(-((I (Subscript[ϕ, 1]))/8)-((I (Subscript[ϕ, 2]))/8)+((3 I (Subscript[ϕ, 3]))/8)+((3 I (Subscript[ϕ, 4]))/8)))}}/.(Subscript[ϕ, 1])->(Subscript[ϕ, 2])+(Subscript[ϕ, 3])-(Subscript[ϕ, 4])//Simplify
β={{((E^(((3 I (Subscript[ϕ, 1]))/8)-((I (Subscript[ϕ, 2]))/8)+((3 I (Subscript[ϕ, 3]))/8)-((I (Subscript[ϕ, 4]))/8)))/(Sqrt[2]))},{((E^(-((I (Subscript[ϕ, 1]))/8)+((3 I (Subscript[ϕ, 2]))/8)-((I (Subscript[ϕ, 3]))/8)+((3 I (Subscript[ϕ, 4]))/8)))/(Sqrt[2]))}}/.(Subscript[ϕ, 1])->(Subscript[ϕ, 2])+(Subscript[ϕ, 3])-(Subscript[ϕ, 4])//Simplify

α//MatrixForm
β//MatrixForm

Notice finally that from

(Subscript[ϕ, 1])=.

(Subscript[ϕ, 1])[ϕ2_,ϕ3_,ϕ4_]:=ϕ2+ϕ3-ϕ4

it follows that if

(Subscript[ϕ, 2])->(Subscript[ϕ, 2])+χ
(Subscript[ϕ, 3])->(Subscript[ϕ, 3])+χ
(Subscript[ϕ, 4])->(Subscript[ϕ, 4])+χ

then by

(Subscript[ϕ, 1])[(Subscript[ϕ, 2])+χ,(Subscript[ϕ, 3])+χ,(Subscript[ϕ, 4])+χ]

it is automatic that

(Subscript[ϕ, 1])->(Subscript[ϕ, 1])+χ

and in net effect we have the unphysical phase adjustment

Ψ->Ψ (E^(I χ))

Looking to the induced response of the phases that enter into the descriptions of α and β, we have

(1/4)(3 (Subscript[ϕ, 2])+(Subscript[ϕ, 3])-2 (Subscript[ϕ, 4]))->(1/4)(3 (Subscript[ϕ, 2])+(Subscript[ϕ, 3])-2 (Subscript[ϕ, 4]))+(1/2)χ
-(1/4)((Subscript[ϕ, 2])-(Subscript[ϕ, 3])-2 (Subscript[ϕ, 4]))->-(1/4)((Subscript[ϕ, 2])-(Subscript[ϕ, 3])-2 (Subscript[ϕ, 4]))+(1/2)χ

(1/4)((Subscript[ϕ, 2])+3 (Subscript[ϕ, 3])-2 (Subscript[ϕ, 4]))->(1/4)((Subscript[ϕ, 2])+3 (Subscript[ϕ, 3])-2 (Subscript[ϕ, 4]))+(1/2)χ
(1/4)((Subscript[ϕ, 2])-(Subscript[ϕ, 3])+2 (Subscript[ϕ, 4]))->(1/4)((Subscript[ϕ, 2])-(Subscript[ϕ, 3])+2 (Subscript[ϕ, 4]))+(1/2)χ

We have here traced  Ψ->Ψ (e^(i χ))  back to the statements

α->α (E^(I χ/2))
β->β (E^(I χ/2))

We note belatedly that the multipliers (Subscript[k, a])  and (Subscript[k, b])  should, in the complex formalism, be endowed with phase factors, which (as an examination of the argument would show) remain subject to no restriction. Thus are we led to the unphysical phase adjustments

α->α (E^(I (Subscript[χ, a])/2))
β->β (E^(I (Subscript[χ, b])/2))

which induce this equally unphysical composite companion:

Ψ->Ψ (E^(I((Subscript[χ, a])+(Subscript[χ, b]))))