2-2

p_⊗q_:=KroneckerProduct[p,q]
adj[x_]:=Conjugate[Transpose[x]]
InnerProd[x_,y_]:=(adj[x].y)[[1]][[1]]
SVD[x_]:=SingularValueDecomposition[x]

Schmidt Decomposition

Nicholas Wheeler
7 February 2013

NOTE: This is an updated (and much expanded) revision of a notebook written in September 2006. I am motivated to revisit this subject because of the central role it plays in a paper of present interest to me: Andreas Albrecht, "Investigating decoherence in a simple system," 
Phys. Rev. D 46, 5504-5520 (1992).

Sources

Erhard Schmidt (1876-1959) was a student of David Hilbert (doctorate awarded in 1905) and participated with Hilbert in the study of integral equations that led to the development of the theory of Hilbert spaces. He is therefore a founding father of functional analysis, and it was from that work that both Gram-Schmidt orthogonalization and Schmidt decomposition devolved (1st decade of the 20th Century).

The Wikipedia article

Hyperlink["Schmidt Decomposition","http://en.wikipedia.org/wiki/Schmidt_decomposition"]

emphasizes that Schmidt decomposition is an implication of (variant formulation of) the Singular Value Decomposition (SVD), the interestingly convolutred history of which

Hyperlink["Singular Value Decomposition", "http://en.wikipedia.org/wiki/Singular_value_decomposition"]

can be traced to discoveries by E. Beltrami and C. Jordan (mid-1870s) and J. J. Sylvester (1889). The theory kept being rediscovered and refined through the 1930s. The algorithms presently used to construct SVDs—which is hardly ever attempted symbolically or by hand, always numerically by computer—date from the 1960s/1970s, when the subject was nearly a century old.  In 1907 Schmidt was led from his integral equations to analogs of the singular values that emerge from matrix theory, without being aware of the connection.

Physicists who have occasion to employ Schmidt decomposition (see, for example, Stephen Barnett, Quantum Information (2009), Appendix D; Maximilian Schlosshauer, Decoherence (2008), §2.15.1) often fail to mention the SVD connection. An exception is T. Heinosaari & M. Ziman, The Mathematical Language of Quantum Theory, from Uncertainty to Entanglement (2012), page 263.

What the Schmidt decomposition (SD) accomplishes

Let ℋ be a complex vector space of composite dimension mn (m ⩽ n), and let ψ ∈ ℋ. Then ψ can be developed

where the (Subscript[λ, k]) are non-negative real numbers, {(Subscript[u, 1]), (Subscript[u, 2]),…,(Subscript[u, m])} are orthonormal in (Subscript[ℋ, m])and {(Subscript[v, 1]),(Subscript[v, 2]),…,(Subscript[v, m]),(Subscript[v, m+1]),…,(Subscript[v, n])} are orthonormal in (Subscript[ℋ, n])(but the red vectors are moot, make no contribution to the Schmidt development of ψ). To illustrate how the SD of ψ arises as an implication of the SVD theorem I proceed by randomly constructed numerical example, since this subject seems to be off-puttingly awkward to treat analytically.

Illustrative case ψ ∈ (Subscript[ℋ, 12])=(Subscript[ℋ, 2])⊗(Subscript[ℋ, 6])

Begin by constructing orthonormal bases in (Subscript[ℋ, 2]) and (Subscript[ℋ, 6]):

(Subscript[𝕀, 2])=IdentityMatrix[2];
(Subscript[𝕀, 6])=IdentityMatrix[6];

Table[(Subscript[e, j])=Transpose[{(Subscript[𝕀, 2])[[j]]}],{j,1,2}];
Table[(Subscript[f, k])=Transpose[{(Subscript[𝕀, 6])[[k]]}],{k,1,6}];

Table[MatrixForm[(Subscript[e, j])],{j,1,2}]
Table[MatrixForm[(Subscript[f, k])],{k,1,6}]

The vectors

Table[(Subscript[g, j,k])=(Subscript[e, j])⊗(Subscript[f, k]),{j,1,2},{k,1,6}];
Table[MatrixForm[(Subscript[g, j,k])],{j,1,2},{k,1,6}]

provide an orthonormal basis in (Subscript[ℋ, 12]), with respect to which arbitrary elements can be developed

ψ=Sum[(Subscript[q, j,k])(Subscript[g, j,k]),{j,1,2},{k,1,6}];
MatrixForm[ψ]

Associate m × n matrices (Subscript[𝔾, j, k]) with the basis vectors (Subscript[g, j, k]) as follows

Table[(Subscript[𝔾, j,k])=(Subscript[e, j]).adj[(Subscript[f, k])],{j,1,2},{k,1,6}];
Table[MatrixForm[(Subscript[𝔾, j,k])],{j,1,2},{k,1,6}]

an associate with the vector ψ (coordinates (Subscript[q, j, k]))  the m × n matrix

ℚ=Sum[(Subscript[q, j,k])(Subscript[𝔾, j,k]),{j,1,2},{k,1,6}];
MatrixForm[ℚ]

That is the matrix of which we propose to construct the SVD. Which is feasible numerically, but not symbolically. So we construct

𝕢=RandomComplex[{-1-I,1+I},{2,6}];
𝕢//MatrixForm

The SingularValueDecomposition command supplies three matrices

𝕌=SVD[𝕢][[1]];
𝔻=SVD[𝕢][[2]];
𝕍=SVD[𝕢][[3]];

of which 𝕌 and 𝕍 are unitary (therefore square, with dimensions m and n, respectively)

𝕌.adj[𝕌]//Chop//MatrixForm
𝕍.adj[𝕍]//Chop//MatrixForm

and 𝔻 is "m × n diagonal" with non-negative real elements

𝔻//MatrixForm

that are precisely the "singular values" (the objects of primary interest to Beltrami, Jordan, Schmidt) of 𝕢:

SingularValueList[𝕢]

The point of the SVD being that 𝕢 = 𝕌.𝔻.adj[𝕍]:

Chop[𝕢-𝕌.𝔻.adj[𝕍]]//MatrixForm

The product 𝕌.𝔻.adj[𝕍] is of the form

to which only the first two rows make a contribution (the elements ■ are irrelevanat/moot).

From the unitarity of 𝕌 we know the columns of 𝕌—call them

Table[(Subscript[u, j])=Chop[Transpose[{Transpose[𝕌][[j]]}]],{j,1,2}];

𝕌//Chop//MatrixForm
Table[MatrixForm[(Subscript[u, j])],{j,1,2}]

are orthonormal:

Table[Chop[InnerProd[(Subscript[u, j]),(Subscript[u, k])]],{j,1,2},{k,1,2}]//MatrixForm

And the same can be said of the columns of adj[𝕍];

Table[(Subscript[v, j])=Chop[Transpose[{adj[𝕍][[j]]}]],{j,1,6}];

or which we have interest, however, only in the first two:

Table[Chop[InnerProd[(Subscript[v, j]),(Subscript[v, k])]],{j,1,2},{k,1,2}]//MatrixForm

Give names to the singular values of 𝕢 (diagonal elements of 𝔻)

Table[(Subscript[λ, j])=SingularValueList[𝕢][[j]],{j,1,2}];

So much by way of preparation.

Now deploy the 2 × 6 = 12 elements of 𝕢 as elements of a ψ vector in (Subscript[ℋ, 12])

ψ=Flatten[Table[{𝕢[[j]][[k]]},{j,1,2},{k,1,6}],1];
MatrixForm[ψ]

and observe finally that ψ = (Subscript[λ, 1])(Subscript[u, 1])⊗(Subscript[v, 1])+(Subscript[λ, 2])(Subscript[u, 2])⊗(Subscript[v, 2]):

Chop[ψ-((Subscript[λ, 1])(Subscript[u, 1])⊗(Subscript[v, 1])+(Subscript[λ, 2])(Subscript[u, 2])⊗(Subscript[v, 2]))]//MatrixForm

We have now in hand the Schmidt decomposition of that ψ. 

But since 12 = 2 × 6 = 3 × 4 is doubly composite, that ψ possesses also a 2nd Schmidt decomposition

Illustrative case ψ ∈ (Subscript[ℋ, 12])=(Subscript[ℋ, 3])⊗(Subscript[ℋ, 4])

Begin afresh by constructing orthonormal bases in (Subscript[ℋ, 3]) and (Subscript[ℋ, 4]):

(Subscript[𝕀, 3])=IdentityMatrix[3];
(Subscript[𝕀, 4])=IdentityMatrix[4];

Table[(Subscript[ℰ, j])=Transpose[{(Subscript[𝕀, 3])[[j]]}],{j,1,3}];
Table[(Subscript[ℱ, k])=Transpose[{(Subscript[𝕀, 4])[[k]]}],{k,1,4}];

Table[MatrixForm[(Subscript[ℰ, j])],{j,1,3}]
Table[MatrixForm[(Subscript[ℱ, k])],{k,1,4}]

The vectors

Table[(Subscript[ℊ, j,k])=(Subscript[ℰ, j])⊗(Subscript[ℱ, k]),{j,1,3},{k,1,4}];
Table[MatrixForm[(Subscript[ℊ, j,k])],{j,1,3},{k,1,4}]

provide a diferently-indexed version of the same orthonormal basis in (Subscript[ℋ, 12]) as we encountered before, and with each of which we can now associate a 3 × 4 matrix

Table[(Subscript[𝒢, j,k])=(Subscript[ℰ, j]).adj[(Subscript[ℱ, k])],{j,1,3},{k,1,4}];
Table[MatrixForm[(Subscript[𝒢, j,k])],{j,1,3},{k,1,4}]

Transition from the 2 × 6 formalism to the 3 × 4 formalism requires modification of the rule by which the elements of ψ ∈ (Subscript[ℋ, 12]) are indexed: where formerly we associated

we have now the association

Accordingly, using the same ψ as before, we construct

𝒬=(({{ψ[[1]][[1]], ψ[[2]][[1]], ψ[[3]][[1]], ψ[[4]][[1]]}, {ψ[[5]][[1]], ψ[[6]][[1]], ψ[[7]][[1]], ψ[[8]][[1]]}, {ψ[[9]][[1]], ψ[[10]][[1]], ψ[[11]][[1]], ψ[[12]][[1]]}}));

𝒰=SVD[𝒬][[1]];
𝒟=SVD[𝒬][[2]];
𝒱=SVD[𝒬][[3]];

We check unitarity

𝒰.adj[𝒰]//Chop//MatrixForm
𝒱.adj[𝒱]//Chop//MatrixForm

and verify that the singular values of 𝒬 are strung along the diagonal of 𝒟:

SingularValueList[𝒬]
𝒟//MatrixForm

Proceding as before, we construct

Table[(Subscript[𝓊, j])=Chop[Transpose[{Transpose[𝒰][[j]]}]],{j,1,3}];
Table[(Subscript[𝓋, j])=Chop[Transpose[{adj[𝒱][[j]]}]],{j,1,4}];
Table[(Subscript[κ, j])=SingularValueList[𝒬][[j]],{j,1,3}];

and observe that

Chop[ψ-((Subscript[κ, 1])(Subscript[𝓊, 1])⊗(Subscript[𝓋, 1])+(Subscript[κ, 2])(Subscript[𝓊, 2])⊗(Subscript[𝓋, 2])+(Subscript[κ, 3])(Subscript[𝓊, 3])⊗(Subscript[𝓋, 3]))]//MatrixForm

We have now in hand a 3-element Schmidt decomposition of ψ, distinct from the  2-element decomposition obtained previously.

DIGRESSION: Number-theoretic question

The preceding discussion raises this question: In how many distinct ways can the composite number N be written as a product of 2 factors? Write

and count the solutions in which {(Subscript[a, 1]),(Subscript[b, 1]),…,(Subscript[c, 1]),(Subscript[a, 2]),(Subscript[b, 2]),…,(Subscript[c, 2])} are non-negative integers. Get

Exclude cases in which either first or second factor is 1, and note that by "distinct" we must consider (Subscript[N, 1])(Subscript[N, 2])=(Subscript[N, 2])(Subscript[N, 1]), so arrive at

EXAMPLE: N = 12 = (2^2)(3^1) gives

(((2+1)(1+1)-2)/2)

which is why in preceding discussion we encountered 2 cases: N = 2 × 6 = 3 × 4.

EXAMPLE: N = 60 = (2^2)(3^1)(5^1)

(((2+1)(1+1)(1+1)-2)/2)

More generally, in how many distinct ways can N be written as a product of n factors? 
The question with n > 2 would become relevant only if one possessed a higher-order generalization of the Schmidt decomposition procedure; i.e., one which pertained to vectors in

REMARK: The same number-theoretic problem arises when one asks: To how many distinct partial trace operations is a square matrix of composite dimension susceptible?

The Schmidt decomposition is standardly considered to pertain to bipartite tensor products. But see

Hyperlink["Tripartite Schmidt","http://arxiv.org/abs/quant-ph/9911073"]

Hyperlink["Multipartite Schmidt","http://arxiv.org/abs/quant-ph/0006125"]

Hyperlink["Multipartite Entanglement","http://en.wikipedia.org/wiki/Multipartite_entanglement"]

Quantum applications

We have heretofore imposed no condition on the vector ψ ∈ (Subscript[ℋ, m])⊗(Subscript[ℋ, n]). But if ψ were normalized it might refer to the pure state of a bipartite composite system. Thus interpreted, ψ refers to an entangled state unless it can be represented

We have, in effect, looked already to the Schmidt decomposition of an entangled random element of (Subscript[ℋ, 2])⊗(Subscript[ℋ, 6]). We look now (using resources already written into Mathematica's memory) to the decomposition of a randomly-constructed disentangled element of (Subscript[ℋ, 2])⊗(Subscript[ℋ, 6]).

Look first to the pre-numeric structure of the situation:

A=(({{(Subscript[a, 1])}, {(Subscript[a, 2])}}));
B=(({{(Subscript[b, 1])}, {(Subscript[b, 2])}, {(Subscript[b, 3])}, {(Subscript[b, 4])}, {(Subscript[b, 5])}, {(Subscript[b, 6])}}));

AB=A⊗B;
AB//MatrixForm

𝔸𝔹=Sum[(Subscript[a, j])(Subscript[b, k])(Subscript[𝔾, j,k]),{j,1,2},{k,1,6}];
𝔸𝔹//MatrixForm

𝔸𝔹

That matrix could have been constructed

Sum[𝔸𝔹[[1]][[k]](Subscript[𝔾, 1,k])+𝔸𝔹[[2]][[k]](Subscript[𝔾, 2,k]),{k,1,6}]

A slight veriant of that procedure will prove indispensible in the numerical work that follows.

Now insert random numbers:

𝔄=RandomComplex[{-1-I,1+I},{2,1}];
𝒜=(𝔄/(Norm[𝔄]));
Norm[𝒜]
𝒜//MatrixForm

𝔅=RandomComplex[{-1-I,1+I},{6,1}];
ℬ=(𝔅/(Norm[𝔅]));
Norm[ℬ]
ℬ//MatrixForm

𝒜ℬ=𝒜⊗ℬ;
Norm[𝒜ℬ]

Proceeding along the now established path, we associate with that 12-vector

𝒜ℬ

the 2 × 6 matrix

𝒬𝒬=Sum[𝒜ℬ[[k]][[1]](Subscript[𝔾, 1,k])+𝒜ℬ[[6+k]][[1]](Subscript[𝔾, 2,k]),{k,1,6}];

𝒬𝒬//MatrixForm

of which we construct the SVD factors

𝒰𝒰=SVD[𝒬𝒬][[1]];
𝒟𝒟=SVD[𝒬𝒬][[2]];
𝒱𝒱=SVD[𝒬𝒬][[3]];

That the state 𝒜ℬ is—because disentangled by construction—"special" shows up when we examine

𝒟𝒟//MatrixForm
SingularValueList[𝒬𝒬]

When (proceeding as formerly) we construct

Table[(Subscript[uu, j])=Chop[Transpose[{Transpose[𝒰𝒰][[j]]}]],{j,1,2}];
Table[(Subscript[vv, j])=Chop[Transpose[{adj[𝒱𝒱][[j]]}]],{j,1,6}];

we obtain

Chop[𝒜ℬ-(1(Subscript[uu, 1])⊗(Subscript[vv, 1])+0(Subscript[uu, 2])⊗(Subscript[vv, 2]))]//MatrixForm

in which the red term is moot: the Schmidt decomposition of 𝒜ℬ = 𝒜⊗ℬ consists of but a single term

…which is hardly surprising. What is somewhat surprising is that

𝒜//MatrixForm
(Subscript[uu, 1])//MatrixForm

are distinct. From

Abs[𝒜[[1]][[1]]]-Abs[(Subscript[uu, 1])[[1]][[1]]]//Chop
Abs[𝒜[[2]][[1]]]-Abs[(Subscript[uu, 1])[[2]][[1]]]//Chop

Arg[((𝒜[[1]][[1]])/((Subscript[uu, 1])[[1]][[1]]))]
Arg[((𝒜[[2]][[1]])/((Subscript[uu, 1])[[2]][[1]]))]

we see that they differ, however, only by a physically irrelevant phase factor:

Chop[(Subscript[uu, 1])Exp[I Arg[((𝒜[[1]][[1]])/((Subscript[uu, 1])[[1]][[1]]))]]-𝒜]//MatrixForm

Schmidt rank and entanglement

Given a pure state vector ψ in (Subscript[ℋ, m])⊗(Subscript[ℋ, n])

construct the m × n matrix

and from the SVD of ψψ extract the m × n matrix

The number of non-zero singular values (or "Schmidt coefficients") λ is called the "Schmidt rank" of ψ  (it is the number of terms that appear in the Schmidt decomposition of ψ). 

The state ψ is disentangled if the Schmidt rank = 1, entangled if the Schmidt rank is  > 1.

One might expect to be able (at least for bipartite systems) to be able to extract from the singular value list (or "Schmidt spectrum") {(Subscript[λ, 1]),(Subscript[λ, 2]),…} a quantified measure of the "degree of entanglement," but from the following elaborate review (31 pages, 177 references, by Martin Plenio & Shashank Virmani—two physicists at Imperial College—in 2006)

Hyperlink["Entanglement Measures","http://arxiv.org/abs/quant-ph/0504163"]

I gather that the entanglement measure problem is not nearly so simple. In the Ensembles section of the following site

Hyperlink["Quantum Entanglement","http://en.wikipedia.org/wiki/Quantum_entanglement"]

it is stated that "In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be NP-Hard. For the 2 × 2 and 2 × 3 cases a necessary and sufficient criterion for separability is given by the famous Positive Partial Transpose (PPT) condition" with reference to a 1996 paper by M, P and R Horodecki. 

Of course, "finding out" mathematically and experimentally are two entirely different matters.

Schmidt decomposition in density matrix theory

The operative notion here is really not Schmidt decomposition per se but the SVD.

Hyperlink["Some Useful Notes","http://www.thphys.may.ie/Notes/MP472/Lecture_9.pdf"]

Hyperlink["Source of Those Notes","http://www.thphys.may.ie/staff/jvala/MP472.htm"]

That material derives from lectures given by Jiri Vala, of the Department of Mathematical Physics at the National University of Ireland/Maynooth in 2008.

We begin by assembling some material to work with. Construct a random 2×2 density matrix:

Table[(Subscript[a, k])=RandomComplex[{-1-I,1+I},{2,1}],{k,1,4}];
Table[(Subscript[α, k])=(Subscript[a, k])/Norm[(Subscript[a, k])],{k,1,4}];
ProbListA=Table[(Subscript[ap, k])=RandomReal[{}],{k,1,4}];
Table[(Subscript[pa, k])=(Subscript[ap, k])/Total[ProbListA],{k,1,4}];
(Subscript[ρ, A])=Chop[Sum[(Subscript[pa, k])(Subscript[α, k]).adj[(Subscript[α, k])],{k,1,4}]];

The following calculations establish that (Subscript[ρ, A]) is hermitian, positive-definite, has unit trace and refers to a mixture:

(Subscript[ρ, A])//MatrixForm
Eigenvalues[(Subscript[ρ, A])]//Chop
Tr[(Subscript[ρ, A])]
Tr[(Subscript[ρ, A]).(Subscript[ρ, A])]//Chop

Were we to use a variant of that same command to construct a random 6×6 density matrix

Table[(Subscript[b, k])=RandomComplex[{-1-I,1+I},{6,1}],{k,1,4}];
Table[(Subscript[β, k])=(Subscript[b, k])/Norm[(Subscript[b, k])],{k,1,4}];
ProbListB=Table[(Subscript[bp, k])=RandomReal[{}],{k,1,4}];
Table[(Subscript[pb, k])=(Subscript[bp, k])/Total[ProbListB],{k,1,4}];
(Subscript[ρ, B])=Chop[Sum[(Subscript[pb, k])(Subscript[β, k]).adj[(Subscript[β, k])],{k,1,4}]];

we would encounter two 0s in the spectrum:

Eigenvalues[(Subscript[ρ, B])]//Chop

LESSON: Use at least N vectors when constructing (by this means) a random N×N density matrix. With this point in mind, I reconstruct (Subscript[ρ, B]):

Table[(Subscript[b, k])=RandomComplex[{-1-I,1+I},{6,1}],{k,1,6}];
Table[(Subscript[β, k])=(Subscript[b, k])/Norm[(Subscript[b, k])],{k,1,6}];
ProbListB=Table[(Subscript[bp, k])=RandomReal[{}],{k,1,6}];
Table[(Subscript[pb, k])=(Subscript[bp, k])/Total[ProbListB],{k,1,6}];
(Subscript[ρ, B])=Chop[Sum[(Subscript[pb, k])(Subscript[β, k]).adj[(Subscript[β, k])],{k,1,6}]];

Again we have hermiticity, positive-definiteness, unit trace and a density matrix that refers to a mixture:

Eigenvalues[(Subscript[ρ, B])]//Chop
Tr[(Subscript[ρ, B])]
Tr[(Subscript[ρ, B]).(Subscript[ρ, B])]//Chop

Construct now the tensor product

ρρ=(Subscript[ρ, A])⊗(Subscript[ρ, B]);

This 12×12 matrix is too large to demonstrate hermiticity by inspection, but these elements provide evidence:

ρρ[[1]][[1]]//Chop
ρρ[[1]][[2]]
ρρ[[2]][[1]]
ρρ[[2]][[2]]//Chop

Because ρρ is a tensor product of matrices that can be independently diagonalized, the spectrum of ρρ is separable (and conversely), as I demonstrate:

(Subscript[Λ, A])=Chop[Eigenvalues[(Subscript[ρ, A])]];
(Subscript[Λ, B])=Chop[Eigenvalues[(Subscript[ρ, B])]];
(Subscript[Λ, ρρ])=Chop[Eigenvalues[ρρ]];

Table[(Subscript[Λ, A])[[j]](Subscript[Λ, B])[[k]],{j,1,2},{k,1,6}]//Flatten//Sort//Reverse
Table[(Subscript[Λ, ρρ])[[j]],{j,1,12}]

We compare those results with the results that emerge when we look to…

A non-separable (entangled) 12 × 12 density matrix

We construct a random instance of such a matrix:

Table[(Subscript[x, k])=RandomComplex[{-1-I,1+I},{12,1}],{k,1,12}];
Table[(Subscript[ξ, k])=(Subscript[x, k])/Norm[(Subscript[x, k])],{k,1,12}];
ProbListX=Table[(Subscript[xp, k])=RandomReal[{}],{k,1,12}];
Table[(Subscript[px, k])=(Subscript[xp, k])/Total[ProbListX],{k,1,12}];
ρ=Chop[Sum[(Subscript[px, k])(Subscript[ξ, k]).adj[(Subscript[ξ, k])],{k,1,12}]];

ρ[[1]][[1]]//Chop
ρ[[1]][[2]]
ρ[[2]][[1]]
ρ[[2]][[2]]//Chop

Eigenvalues[ρ]//Chop
Tr[ρ]
Total[Chop[Eigenvalues[ρ]]]
Tr[ρ.ρ]//Chop
Total[((Chop[Eigenvalues[ρ]])^2)]

We use partial trace operations to construct analogs of (Subscript[ρ, A]) and (Subscript[ρ, B]).

(Subscript[Tr, A])[x_]:=(∑_(j=1)^2)(adj[(Subscript[e, j])]⊗(Subscript[𝕀, 6])).x.((Subscript[e, j])⊗(Subscript[𝕀, 6]))
(Subscript[Tr, B])[x_]:=(∑_(k=1)^6)((Subscript[𝕀, 2])⊗adj[(Subscript[f, k])]).x.((Subscript[𝕀, 2])⊗(Subscript[f, k]))

(Subscript[Reducedρ, A])=Chop[(Subscript[Tr, B])[ρ]];
(Subscript[Reducedρ, A])//MatrixForm
Tr[(Subscript[Reducedρ, A])]
Tr[MatrixPower[(Subscript[Reducedρ, A]),2]]//Chop

(Subscript[Reducedρ, B])=Chop[(Subscript[Tr, A])[ρ]];
(Subscript[Reducedρ, B])//MatrixForm
Tr[(Subscript[Reducedρ, B])]
Tr[MatrixPower[(Subscript[Reducedρ, B]),2]]//Chop

It is now demonstrably NOT the case that

and NOT the case that the eigenvalues of ρ can be developed as products of the eigenvalues of (Subscript[Reducedρ, A]) and (Subscript[Reducedρ, B]), as I illustrate:

ℝ=(Subscript[Reducedρ, A])⊗(Subscript[Reducedρ, B]);

ℝ[[1]][[1]]//Chop
ρ[[1]][[1]]//Chop

ℝ[[1]][[2]]//Chop
ρ[[1]][[2]]//Chop

(Subscript[ReducedΛ, A])=Chop[Eigenvalues[(Subscript[Reducedρ, A])]];
(Subscript[ReducedΛ, B])=Chop[Eigenvalues[(Subscript[Reducedρ, B])]];
(Subscript[Λ, ρ])=Chop[Eigenvalues[ρ]];

Table[(Subscript[ReducedΛ, A])[[j]](Subscript[ReducedΛ, B])[[k]],{j,1,2},{k,1,6}]//Flatten//Sort//Reverse
Table[(Subscript[Λ, ρ])[[j]],{j,1,12}]

REMARK: This, however, leaves open the question: Do there exist some numbers {(Subscript[a, 1]),(Subscript[a, 2])} and {(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4]),(Subscript[b, 5]),(Subscript[b, 6])} such that the list {(Subscript[a, j])(Subscript[b, k])} reproduces the spectrum of ρ? This is precisely the problem which I addressed in a Reed Physics Seminar on 11 October 2006. I made use there (in work motivated by Julia Keller's thesis problem) of an uncommon device, the "generalized matrix inverse" of Moore-Penrose.

I turn now to a variant of the preceeding discussion. Let ϕ ∈ (Subscript[ℋ, 2])⊗ (Subscript[ℋ, 6]). It will then (according to Schmidt) be possible to write

ϕ=(Sqrt[(Subscript[λ, 1])])(Subscript[u, 1])⊗(Subscript[v, 1])+(Sqrt[(Subscript[λ, 2])])(Subscript[u, 2])⊗(Subscript[v, 2])

where the Schmidt coefficients {(Subscript[λ, 1]),(Subscript[λ, 2])} are positive real numbers that sum to unity, {(Subscript[u, 1]),(Subscript[u, 2])} are orthonormal in (Subscript[ℋ, 2]), {(Subscript[v, 1]),(Subscript[v, 2])} are orthonormal in (Subscript[ℋ, 6]). We already possess (as a result of previous work) such vectors written into Mathematica's memory:

Table[Chop[InnerProd[(Subscript[u, j]),(Subscript[u, k])]],{j,1,2},{k,1,2}]//MatrixForm

Table[Chop[InnerProd[(Subscript[v, j]),(Subscript[v, k])]],{j,1,6},{k,1,6}]//MatrixForm

Taking

(Subscript[λ, 1])=RandomReal[{}];
(Subscript[λ, 2])=1-(Subscript[λ, 1]);
{(Subscript[λ, 1]),(Subscript[λ, 2])}
Total[%]

we construct

ϕ=(Sqrt[(Subscript[λ, 1])])(Subscript[u, 1])⊗(Subscript[v, 1])+(Sqrt[(Subscript[λ, 2])])(Subscript[u, 2])⊗(Subscript[v, 2]);
ϕ//MatrixForm
Norm[ϕ]

The associated pure-state density matrix is

𝕣=Chop[ϕ.adj[ϕ]];
𝕣//MatrixForm
Tr[𝕣]
Tr[𝕣.𝕣]//Chop

Which brings me to the point of these remarks: The reduced density matrices obtained from (Subscript[𝕣, pure]) have identical spectra:

Eigenvalues[(Subscript[Tr, B])[𝕣]]//Chop
Eigenvalues[(Subscript[Tr, A])[𝕣]]//Chop