Schmidt Decomposition Theorem

The Schmidt decomposition theorem says that any complex vector v ∈ Cⁿ ⊗ Cⁿ can be written as

where k ≤ n, {α_j} ⊆ R is a family of non-negative real scalars, and {e_j}, {f_j} ⊆ Cⁿ are two orthonormal sets of vectors. I won’t prove the theorem here — a proof can be found on its Wikipedia page (it’s basically the singular value decomposition in disguise). For our purposes the most important thing to realize is that, for some vectors v, we can write v in its Schmidt decomposition with k < n. The least k such that v can be written in the form above is called the Schmidt rank of v, and we denote it by SR(v). Every vector v has SR(v) ≤ n.

Schmidt Matrix Norms

The Schmidt k-norm of a matrix X ∈ M_n is defined to be

That might look like a horribly complex definition upon first glance, but it’s not so hard to get your head around when you realize that the Schmidt k-norm for k = n is simply the standard operator norm of X. It is clear then that the Schmidt k-norm for k < n must be a smaller quantity. Indeed, from a quantum information perspective, the norm measures how much the operator represented by X can stretch pure states that “aren’t very entangled.” The interested reader can learn about the various properties and applications of these norms in [1] — what I present here is simply a proof that the Schmidt k-norm is indeed a norm (since this is not explicitly done in the paper).

Proof that the Schmidt k-norm is a norm. It is clear from the definition that the absolute value of a constant pulls out of the Schmidt norms and that the Schmidt norms satisfy the triangle inequality. The only challenging property of the norm to verify is that the Schmidt norm of X being zero implies X = 0.

To prove this, assume that we are in the k = 1 case (if we can show that this property holds for k = 1, it immediately follows that the same property must hold for k > 1). Then recall that we can write X as the sum of elementary tensors, so we can write

Furthermore, we may write X in this way using matrices B_j that are linearly independent (see, for example, Proposition 24 of [2], or simply note that you could choose them to be a family of matrix units). Thus, if the Schmidt 1-norm of X equals zero, then it follows that for any v₁, v₂, w₁, and w₂:

Since this holds for any v₂ and w₂, it follows that

Because we chose the B_j matrices to be linearly independent, it follows that c_j = 0 for all j. By referring back to the definition of c_j, we see that this then implies A_j = 0 for all j, so X = 0 as desired. QED.

References:

1. N. Johnston and D.W. Kribs, Schmidt norms for quantum states. Preprint (2009). arXiv:0909.3907 [quant-ph]
2. Johnston, N., Kribs, D. W., and Paulsen, V., Computing stabilized norms for quantum operations. Quantum Information & Computation 9 1 & 2, 16-35 (2009). arXiv:0711.3636v1 [quant-ph]

Roughly speaking, MD(E) is the largest subalgebra of M_n on which E behaves multiplicatively. It will be useful to make this notion precise:

Definition. Let A be a subalgebra of M_n and let π : A → M_n. Then π is said to be a *-homomorphism if π(ab) = π(a)π(b) and π(a^*) = π(a)^* for all a,b ∈ A.

Thus, MD(E) is roughly the largest subalgebra of M_n such that, when E is restricted to it, E is a *-homomorphism (I keep saying “roughly speaking” because of the “∀b ∈ M_n” in the definition of MD(E) — the definition of a *-homomorphism only requires that the multiplicativity hold ∀b ∈ A). Probably the most well-known result about the multiplicative domain is the following theorem of Choi [1,2], which shows how the multiplicative domain simplifies when E is such that E(I) = I (i.e., when E is unital):

Theorem [Choi]. Let E: M_n → M_n be a completely positive map such that E(I) = I. Then

One thing in particular that this theorem shows is that, when E(I) = I, the multiplicative domain of E only needs to be multiplicative within MD(E) (i.e., we can remove the “roughly speaking” that I spoke of earlier).

MD(E) in Quantum Error Correction

Before moving onto how MD(E) plays a role in quantum error correction, let’s consider some examples to get a better feeling for what the multiplicative domain looks like.

• If E is the identity map (that is, it is the map that takes a matrix to itself) then MD(E) = M_n, the entire matrix algebra.
• If E(a) = Diag(a) (i.e., E simply erases all of the off-diagonal entries of the matrix a), then MD(E) = {Diag(a)}, the set of diagonal matrices.

Notice that in the first example, the map E is very well-behaved (as well-behaved as a map ever could be); it preserves all of the information that is put into it. We also see that MD(E) is as large as possible. In the second example, the map E does not preserve information put into it (indeed, one nice way to think about matrices in the quantum information setting is that the diagonal matrices are “classical” and rest of the matrices are “quantum” — thus the map E(a) = Diag(a) is effectively removing all of the “quantumness” of the input data). We also see that MD(E) is tiny in this case (too small to put any quantum data into).

The above examples then hint that if the map E preserves quantum data, then MD(E) should be large enough to store some quantum information safely. This isn’t quite true, but the intuition is right, and we get the following result, which was published as Theorem 11 in this paper:

Theorem. Let E: M_n → M_n be a quantum channel (i.e., a completely positive map such that Tr(E(a)) = Tr(a) for all a ∈ M_n) such that E(I) = I. Then MD(E) = UCC(E), the algebra of unitarily-correctable codes for E.

What this means is that, when E is unital, its multiplicative domain encodes exactly the matrices that we can correct via a unitary operation. This doesn’t tell us anything about correctable codes that are not unitarily-correctable, though (i.e., matrices that can only be corrected by a more complicated correction operation). To capture these codes, we have to generalize a bit.

Generalized Multiplicative Domains

In order to generalize the multiplicative domain, we can require that the map E be multiplicative with another map π that is already a *-homomorphism, rather than require that it be multiplicative with itself. This is the main theme of this paper, which was submitted for publication this week. We define generalized multiplicative domains as follows:

Definition. Let A be a subalgebra of M_n, let E : M_n → M_n be completely positive, and let π : A → M_n be a *-homomorphism. Then the multiplicative domain of E with respect to π, denoted MD_π(E), is the algebra given by

It turns out that these generalized multiplicative domains are reasonably well-behaved and generalize the standard multiplicative domain in exactly the way that we wanted: they capture all correctable codes for arbitrary quantum channels (see Theorem 11 of the last paper I mentioned). Furthermore, there are even some characterizations of MD_π(E) analogous to the theorem of Choi above (see Theorems 5 and 7, as well as Corollary 12).

References:

1. M.-D. Choi, A Schwarz inequality for positive linear maps on C*-algebras. Illinois Journal of Mathematics, 18 (1974), 565-574.
2. V. I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, 2003.