<p>The geometric measure of entanglement <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E">
<mml:semantics>
<mml:mi>E</mml:mi>
<mml:annotation encoding="application/x-tex">E</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> of an <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m">
<mml:semantics>
<mml:mi>m</mml:mi>
<mml:annotation encoding="application/x-tex">m</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> qubit quantum state has maximum value bounded above by <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m">
<mml:semantics>
<mml:mi>m</mml:mi>
<mml:annotation encoding="application/x-tex">m</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. In previous work of Gross, Flammia, and Eisert, it was shown that <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E greater-than-or-equal-to m minus upper O left-parenthesis log m right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>≥<!-- ≥ --></mml:mo>
<mml:mi>m</mml:mi>
<mml:mo>−<!-- − --></mml:mo>
<mml:mi>O</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>log</mml:mi>
<mml:mo><!-- --></mml:mo>
<mml:mi>m</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">E \ge m-O(\log m)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> with high probability as <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m right-arrow normal infinity">
<mml:semantics>
<mml:mrow>
<mml:mi>m</mml:mi>
<mml:mo stretchy="false">→<!-- → --></mml:mo>
<mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">m\to \infty</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. They showed, as a consequence, that the vast majority of states are too entangled to be computationally useful. In this paper, we show that for <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m">
<mml:semantics>
<mml:mi>m</mml:mi>
<mml:annotation encoding="application/x-tex">m</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> qubit <italic>Boson</italic> quantum states, the maximal possible geometric measure of entanglement is bounded above by <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="log Subscript 2 Baseline m">
<mml:semantics>
<mml:mrow>
<mml:msub>
<mml:mi>log</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mspace width="negativethinmathspace" />
<mml:mi>m</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\log _2\! m</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, opening the door to many computationally universal states. We further show the corresponding concentration result that <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E greater-than-or-equal-to log Subscript 2 Baseline m minus upper O left-parenthesis log log m right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>≥<!-- ≥ --></mml:mo>
<mml:msub>
<mml:mi>log</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mspace width="negativethinmathspace" />
<mml:mi>m</mml:mi>
<mml:mo>−<!-- − --></mml:mo>
<mml:mi>O</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>log</mml:mi>
<mml:mo><!-- --></mml:mo>
<mml:mi>log</mml:mi>
<mml:mo><!-- --></mml:mo>
<mml:mi>m</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">E \ge \log _2\! m - O(\log \log m)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> with high probability as <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m right-arrow normal infinity">
<mml:semantics>
<mml:mrow>
<mml:mi>m</mml:mi>
<mml:mo stretchy="false">→<!-- → --></mml:mo>
<mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">m\to \infty</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. We extend these results also to <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m">
<mml:semantics>
<mml:mi>m</mml:mi>
<mml:annotation encoding="application/x-tex">m</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-mode <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-bit Boson quantum states.</p>
[1]
Jonathan Novak.
Truncations of Random Unitary Matrices and Young Tableaux
,
2007,
Electron. J. Comb..
[2]
Shmuel Friedland,et al.
Remarks on the Symmetric Rank of Symmetric Tensors
,
2015,
SIAM J. Matrix Anal. Appl..
[3]
Dénes Petz,et al.
On asymptotics of large Haar distributed unitary matrices
,
2004,
Period. Math. Hung..
[4]
S. Friedland,et al.
Best approximation on semi-algebraic sets and k-border rank approximation of symmetric tensors
,
2013,
1311.1561.
[5]
D. Gross,et al.
Most quantum States are too entangled to be useful as computational resources.
,
2008,
Physical review letters.
[6]
Renato Renner,et al.
Security of quantum key distribution
,
2005,
Ausgezeichnete Informatikdissertationen.
[7]
Shmuel Friedland,et al.
Variation of tensor powers and spectrat
,
1982
.
[8]
M. Ledoux.
The concentration of measure phenomenon
,
2001
.
[9]
S. Banach.
Über homogene Polynome in ($L^{2}$)
,
1938
.
[10]
B. Lanyon,et al.
Experimental demonstration of a compiled version of Shor's algorithm with quantum entanglement.
,
2007,
Physical review letters.
[11]
Shmuel Friedland,et al.
Computational Complexity of Tensor Nuclear Norm
,
2014,
ArXiv.
[12]
C. D. Boor,et al.
Polynomial interpolation in several variables
,
1994
.
[13]
Tomas Sauer,et al.
Polynomial interpolation in several variables
,
2000,
Adv. Comput. Math..
[14]
Shmuel Friedland,et al.
Geometric measure of entanglement of symmetric d-qubits is polynomial-time computable
,
2016,
1608.01354.
[15]
F. Hiai,et al.
The semicircle law, free random variables, and entropy
,
2006
.
[16]
Fredrik Meyer,et al.
Representation theory
,
2015
.