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An Introduction to Schmidt Norms

October 2nd, 2009
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In [1], a family of matrix norms (called Schmidt norms) are studied and some of their uses in quantum information theory are explored. The interested reader is of course welcome to read the results presented in that paper, but for the more casual reader I present here one very crucial preliminary, the Schmidt decomposition theorem, and a proof that the Schmidt norms actually are (as their name suggests) norms.

Schmidt Decomposition Theorem

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

{\bf v}=\sum_{j=1}^k\alpha_j{\bf e_j}\otimes{\bf f_j}

where k ≤ n, {αj} ⊆ R is a family of non-negative real scalars, and {ej}, {fj} ⊆ 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 ∈ Mn is defined to be

\big\|X\big\|_{S(k)}:=\sup_{{\bf v},{\bf w}}\big\{|{\bf w}^*X{\bf v}| : \|{\bf v}\|,\|{\bf w}\|\leq 1,SR({\bf v}),SR({\bf w})\leq k\big\}

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

X=\sum_jA_j\otimes B_j,\ \ {\bf v}={\bf v_1}\otimes{\bf v_2},\text{ and } \ {\bf w}={\bf w_1}\otimes{\bf w_2}.Furthermore, we may write X in this way using matrices Bj 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:

{\bf w_2}^*\Big(\sum_jc_jB_j\Big){\bf v_2}=0 \ \text{ where }c_j={\bf w_1}^*A_j{\bf v_1} \ \ \forall \, j.

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

\sum_jc_jB_j=0.

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

References:

  1. N. Johnston and D. W. Kribs, A family of norms with applications in quantum information theory. Journal of Mathematical Physics 51, 082202 (2010). 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]
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