Blog > Approximating the Distribution of Schmidt Vector Norms

Approximating the Distribution of Schmidt Vector Norms

November 6th, 2009

Recently, a family of vector norms [1,2] have been introduced in quantum information theory that are useful for helping classify entanglement of quantum states. In particular, the Schmidt vector k-norm of a vector v ∈ CnCn, for an integer 1 ≤ k ≤ n, is defined by

\|v\|_{s(k)}:=\sup_w\Big\{\big|\langle v,w\rangle\big|:\|w\|=1,SR(w)\leq k\Big\}.

In the above definition, SR(w) refers to the Schmidt rank of the vector w and so these norms are in some ways like a measure of entanglement for pure state vectors. One of the results of [2] shows how to compute these norms efficiently, so with that in mind we can perform all sorts of fun numerical analysis on them. Analytic results are provided in the paper, so I’ll provide more hand-wavey stuff and pictures here. In particular, let’s look at what the distributions of the Schmidt vector norms look like.

Figure 1: The Schmidt 1 and 2 vector norms in 3 ⊗ 3 dimensional space

Figure 1: The distribution of the Schmidt 1 and 2 vector norms in (3 ⊗ 3)-dimensional space

Figure 1 shows the distributions of the Schmidt 1 and 2 norms of unit vectors distributed according to the Haar measure in C3C3, based on 5×105 vectors generated randomly via MATLAB. Note that the Schmidt 3-norm just equals the standard Euclidean norm so it always equals 1 and is thus not shown. Figures 2 and 3 show similar distributions in C4C4 and C5C5.

Figure 2: The distribution of the Schmidt 1, 2, and 3 vector norms in (4 ⊗ 4)-dimensional space

Figure 2: The distribution of the Schmidt 1, 2, and 3 vector norms in (4 ⊗ 4)-dimensional space

Schmidt vector 1, 2, 3, and 4 norms for n = 5

Figure 3: The distribution of the Schmidt 1, 2, 3, and 4 vector norms in (5 ⊗ 5)-dimensional space

The following table shows various basic statistics about the above distributions. I suppose the natural next step is to ask whether or not we can analytically determine the distribution of the Schmidt vector norms. Since these norms are essentially just the singular values of an operator that is associated with the vector, it seems like this might even already be a (partially) solved problem, since many results are known about the distribution of the singular values of random matrices. The difficulty comes in trying to interpret the Haar measure (or any other natural measure on pure states, such as the Hilbert-Schmidt measure) on the associated operators.

Space k Mean Median Std. Dev.
C3C3 1 0.8494 0.8516 0.0554
2 0.9811 0.9860 0.0171
C4C4 1 0.7799 0.7792 0.0501
2 0.9411 0.9435 0.0247
3 0.9921 0.9943 0.0074
C5C5 1 0.7240 0.7225 0.0444
2 0.8976 0.8987 0.0268
3 0.9707 0.9722 0.0129
4 0.9960 0.9971 0.0039

References:

  1. D. Chruscinski, A. Kossakowski, G. Sarbicki, Spectral conditions for entanglement witnesses vs. bound entanglement, Phys. Rev A 80, 042314 (2009). arXiv:0908.1846v2 [quant-ph]
  2. 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]
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