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<title>Kadison–Singer problem</title>

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Kadison–Singer problem

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the Kadison–Singer problem, posed in 1959, was a problem in <a href="Functional_analysis" title="Functional analysis">functional analysis</a> about whether certain extensions of certain <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functionals</a> on certain <a href="C*-algebra" title="C*-algebra">C*-algebras</a> were unique. The uniqueness was proved in 2013.

The statement arose from work on the foundations of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> done by <a href="Paul_Dirac" title="Paul Dirac">Paul Dirac</a> in the 1940s and was formalized in 1959 by <a href="Richard_Kadison" title="Richard Kadison">Richard Kadison</a> and <a href="Isadore_Singer" title="Isadore Singer">Isadore Singer</a>.<a href="#cite_note-1">[1]</a> The problem was subsequently shown to be equivalent to numerous open problems in pure mathematics, applied mathematics, engineering and computer science.<a href="#cite_note-det-2">[2]</a><a href="#cite_note-con-3">[3]</a> Kadison, Singer, and most later authors believed the statement to be false,<a href="#cite_note-det-2">[2]</a><a href="#cite_note-con-3">[3]</a> but, in 2013, it was proven true by <a href="Adam_Marcus_(mathematician)" title="Adam Marcus (mathematician)">Adam Marcus</a>, <a href="Daniel_Spielman" title="Daniel Spielman">Daniel Spielman</a> and <a href="Nikhil_Srivastava" title="Nikhil Srivastava">Nikhil Srivastava</a>,<a href="#cite_note-4">[4]</a> who received the 2014 <a href="George_P%C3%B3lya_Prize" title="George Pólya Prize">Pólya Prize</a> for the achievement.

The solution was made possible by a reformulation provided by Joel Anderson, who showed in 1979 that his "paving conjecture", which only involves operators on finite-dimensional Hilbert spaces, is equivalent to the Kadison–Singer problem. Nik Weaver provided another reformulation in a finite-dimensional setting, and this version was proved true using random polynomials.<a href="#cite_note-disc-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Original_formulation">Original formulation</h2></div>

Consider the <a href="Hilbert_space#Separable_spaces" title="Hilbert space">separable Hilbert space</a> <a href="Sequence_space" title="Sequence space">ℓ2</a> and two related C*-algebras: the algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> of all <a href="Bounded_operator" title="Bounded operator">continuous linear operators</a> from ℓ2 to ℓ2, and the algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> of all diagonal continuous linear operators from ℓ2 to ℓ2.

A <a href="State_(functional_analysis)" title="State (functional analysis)">state</a> on a C*-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a continuous linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :A\to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi :A\to \mathbb {C} }</annotation>

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</math><img src="./1cd3039c5923a75065fefb8f0e05b994d3558935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.493ex; height:2.676ex;" alt="{\displaystyle \varphi :A\to \mathbb {C} }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (I)=1}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (I)=1}</annotation>

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</math><img src="./6a0adf20a1ea775f1802df0b5fb313ce5ac47e16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.762ex; height:2.843ex;" alt="{\displaystyle \varphi (I)=1}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

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</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> denotes the algebra's <a href="Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (T)\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (T)\geq 0}</annotation>

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</math><img src="./2051adf7308e672f4df0afaf8392d911a7fdded0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle \varphi (T)\geq 0}" loading="lazy"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle T\geq 0}</annotation>

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</math><img src="./07251f925ce3c2848edcc56971479a8ac6f5f811.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.897ex; height:2.343ex;" alt="{\displaystyle T\geq 0}" loading="lazy">. Such a state is called <a href="State_(functional_analysis)#Pure_states" title="State (functional analysis)">pure</a> if it is an extremal point in the set of all states on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> (i.e. if it cannot be written as a <a href="Convex_combination" title="Convex combination">convex combination</a> of other states on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">).

By the <a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a>, any functional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> can be extended to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">. Kadison and Singer conjectured that, for the case of pure states, this extension is unique. That is, the Kadison–Singer problem consisted in proving or disproving the following statement:

<dl><dd>to every pure state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> there exists a unique state on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> that extends <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy">.</dd></dl>

This claim is in fact true.

<div class="mw-heading mw-heading2"><h2 id="Paving_conjecture_reformulation">Paving conjecture reformulation</h2></div>

The Kadison–Singer problem has a positive solution <a href="If_and_only_if" title="If and only if">if and only if</a> the following "paving conjecture" is true:<a href="#cite_note-6">[6]</a>

<dl><dd>For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

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</math><img src="./e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"> there exists a natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> so that the following holds: for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> and every linear operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensional Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation>

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</math><img src="./a53b4e76242764d1bca004168353c380fef25258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}" loading="lazy"> with zeros on the diagonal there exists a partition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\dots ,n\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,\dots ,n\}}</annotation>

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</math><img src="./8b98091026e0b10ee08a7c77e229c649f81b0a05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.06ex; height:2.843ex;" alt="{\displaystyle \{1,\dots ,n\}}" loading="lazy"> into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},\dots ,A_{k}}">

<msub>

</mrow>

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<msub>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1},\dots ,A_{k}}</annotation>

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</math><img src="./d3957671cfc6460b7044b5563b789c9fb0670c32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.807ex; height:2.509ex;" alt="{\displaystyle A_{1},\dots ,A_{k}}" loading="lazy"> such that

<msub>

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</mrow>

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</msub>

<msub>

<msub>

</mrow>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \|P_{A_{j}}TP_{A_{j}}\|\leq \varepsilon \|T\|{\text{ for }}j=1,\ldots ,k.}</annotation>

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</math><img src="./7d56dabd9e291819eabea41d6e3c1c0a57e49634.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.649ex; height:3.176ex;" alt="{\displaystyle \|P_{A_{j}}TP_{A_{j}}\|\leq \varepsilon \|T\|{\text{ for }}j=1,\ldots ,k.}" loading="lazy"></dd></dl></dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{A_{j}}}">

<msub>

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle P_{A_{j}}}</annotation>

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</math><img src="./167b5d9a2016ca34059ee542c18ac3343d146366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.671ex; height:3.009ex;" alt="{\displaystyle P_{A_{j}}}" loading="lazy"> denotes the orthogonal projection on the space spanned by the standard unit vectors corresponding to the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j}}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{j}}</annotation>

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</math><img src="./6019bb70c912e59e9d5f442e9217517743ed4831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.653ex; height:2.843ex;" alt="{\displaystyle A_{j}}" loading="lazy">, so that the matrix of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{A_{j}}TP_{A_{j}}}">

<msub>

<msub>

</mrow>

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</mrow>

</msub>

<msub>

<msub>

</mrow>

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</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{A_{j}}TP_{A_{j}}}</annotation>

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</math><img src="./f41582d6b9533c7f0e4931d5d88d89bebab44dff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.979ex; height:3.009ex;" alt="{\displaystyle P_{A_{j}}TP_{A_{j}}}" loading="lazy"> is obtained from the matrix of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> by replacing all rows and columns that don't correspond to the indices in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j}}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{j}}</annotation>

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</math><img src="./6019bb70c912e59e9d5f442e9217517743ed4831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.653ex; height:2.843ex;" alt="{\displaystyle A_{j}}" loading="lazy"> by 0. The matrix norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \|\cdot \|}</annotation>

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</math><img src="./113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}" loading="lazy"> is the <a href="Matrix_norm" title="Matrix norm">spectral norm</a>, i.e. the <a href="Operator_norm" title="Operator norm">operator norm</a> with respect to the Euclidean norm on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}">

<msup>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation>

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</math><img src="./a53b4e76242764d1bca004168353c380fef25258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}" loading="lazy">.

Note that in this statement, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> may only depend on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

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</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy">, not on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Equivalent_discrepancy_statement">Equivalent discrepancy statement</h2></div>

The following "<a href="Discrepancy_theory" title="Discrepancy theory">discrepancy</a>" statement, again equivalent to the Kadison–Singer problem because of previous work by Nik Weaver,<a href="#cite_note-7">[7]</a> was proven by Marcus/Spielman/Srivastava using a technique of random polynomials:

<dl><dd>Suppose vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1},\ldots ,u_{m}\in \mathbb {C} ^{d}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{1},\ldots ,u_{m}\in \mathbb {C} ^{d}}</annotation>

</semantics>

</math><img src="./d74ef34c4c7e3d438db331cd32c9d977b11da115.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.178ex; height:3.009ex;" alt="{\displaystyle u_{1},\ldots ,u_{m}\in \mathbb {C} ^{d}}" loading="lazy"> are given with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{m}u_{i}u_{i}^{*}=I}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{m}u_{i}u_{i}^{*}=I}</annotation>

</semantics>

</math><img src="./86cdac04addac9a8129d613d096f9e1cd10e91e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.526ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{m}u_{i}u_{i}^{*}=I}" loading="lazy"> (the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\times d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\times d}</annotation>

</semantics>

</math><img src="./8832c0aa14719499bb50f640be353abb7f37f069.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.272ex; height:2.176ex;" alt="{\displaystyle d\times d}" loading="lazy"> <a href="Identity_matrix" title="Identity matrix">identity matrix</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u_{i}\|_{2}^{2}\leq \delta }">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|u_{i}\|_{2}^{2}\leq \delta }</annotation>

</semantics>

</math><img src="./1bebc97c78e3ec9c3f1805648a3d5237870b3e16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.656ex; height:3.176ex;" alt="{\displaystyle \|u_{i}\|_{2}^{2}\leq \delta }" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">. Then there exists a partition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\ldots ,m\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{1,\ldots ,m\}}</annotation>

</semantics>

</math><img src="./777012b24f3294235640e1458f22ec30eecd7b7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.706ex; height:2.843ex;" alt="{\displaystyle \{1,\ldots ,m\}}" loading="lazy"> into two sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation>

</semantics>

</math><img src="./5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation>

</semantics>

</math><img src="./1143e284d5f25cef778ab482edf6617a523ddd9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}" loading="lazy"> such that

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

<mfrac>

<msup>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\|\sum _{i\in S_{j}}u_{i}u_{i}^{*}\right\|\leq {\frac {\left(1+{\sqrt {2\delta }}\right)^{2}}{2}}{\text{ for }}j=1,2.}</annotation>

</semantics>

</math><img src="./8152664067b2daf1d0820c3c770a0f58d0b9c255.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:37.889ex; height:8.509ex;" alt="{\displaystyle \left\|\sum _{i\in S_{j}}u_{i}u_{i}^{*}\right\|\leq {\frac {\left(1+{\sqrt {2\delta }}\right)^{2}}{2}}{\text{ for }}j=1,2.}" loading="lazy"></dd></dl></dd></dl>

This statement implies the following:

<dl><dd>Suppose vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1},\ldots ,v_{m}\in \mathbb {R} ^{d}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{1},\ldots ,v_{m}\in \mathbb {R} ^{d}}</annotation>

</semantics>

</math><img src="./0a594b4b3646cfa72f0ceb47addee1cbc87ae48b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.773ex; height:3.009ex;" alt="{\displaystyle v_{1},\ldots ,v_{m}\in \mathbb {R} ^{d}}" loading="lazy"> are given with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|v_{i}\|_{2}^{2}\leq \alpha }">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|v_{i}\|_{2}^{2}\leq \alpha }</annotation>

</semantics>

</math><img src="./9da3236a20e39f1041405a667722e2978f961480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.893ex; height:3.176ex;" alt="{\displaystyle \|v_{i}\|_{2}^{2}\leq \alpha }" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> and

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{m}\langle v_{i},x\rangle ^{2}=1\ {\text{ for all }}x\in \mathbb {R} ^{d}{\text{ with }}\|x\|=1.}</annotation>

</semantics>

</math><img src="./39eb337040043166c0a70efa952634d2d09ab5d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.521ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{m}\langle v_{i},x\rangle ^{2}=1\ {\text{ for all }}x\in \mathbb {R} ^{d}{\text{ with }}\|x\|=1.}" loading="lazy"></dd></dl></dd>

<dd>Then there exists a partition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\ldots ,m\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{1,\ldots ,m\}}</annotation>

</semantics>

</math><img src="./777012b24f3294235640e1458f22ec30eecd7b7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.706ex; height:2.843ex;" alt="{\displaystyle \{1,\ldots ,m\}}" loading="lazy"> into two sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation>

</semantics>

</math><img src="./5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation>

</semantics>

</math><img src="./1143e284d5f25cef778ab482edf6617a523ddd9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}" loading="lazy"> such that, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j=1,2}</annotation>

</semantics>

</math><img src="./fcc932fd3887517325e81803b694ff8b5c1f1b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:7.442ex; height:2.509ex;" alt="{\displaystyle j=1,2}" loading="lazy">:

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|\sum _{i\in S_{j}}\langle v_{i},x\rangle ^{2}-{\frac {1}{2}}\right|\leq 5{\sqrt {\alpha }}\ {\text{ for all }}x\in \mathbb {R} ^{d}{\text{ with }}\|x\|=1.}</annotation>

</semantics>

</math><img src="./14136c4d3e31499092123597d7192b12877e1a1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:53.108ex; height:8.509ex;" alt="{\displaystyle \left|\sum _{i\in S_{j}}\langle v_{i},x\rangle ^{2}-{\frac {1}{2}}\right|\leq 5{\sqrt {\alpha }}\ {\text{ for all }}x\in \mathbb {R} ^{d}{\text{ with }}\|x\|=1.}" loading="lazy"></dd></dl></dd></dl>

Here the "discrepancy" becomes visible when α is small enough: the <a href="Quadratic_form" title="Quadratic form">quadratic form</a> on the unit sphere can be split into two roughly equal pieces, i.e. pieces whose values don't differ much from 1/2 on the unit sphere.

In this form, the theorem can be used to derive statements about certain partitions of graphs.<a href="#cite_note-disc-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFKadisonSinger1959" class="citation journal cs1"><a href="Richard_Kadison" title="Richard Kadison">Kadison, R.</a>; <a href="Isadore_Singer" title="Isadore Singer">Singer, I.</a> (1959). "Extensions of pure states". <a href="American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a>. 81 (2): 383–400. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2372748">10.2307/2372748</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2372748">2372748</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0123922">0123922</a>.</cite>

</li>

<li id="cite_note-det-2">^ <a href="#cite_ref-det_2-0">a</a> <a href="#cite_ref-det_2-1">b</a> <cite id="CITEREFCasazzaFickusTremainWeber2006" class="citation book cs1">Casazza, P. G.; Fickus, M.; Tremain, J. C.; Weber, E. (2006). "The Kadison–Singer problem in mathematics and engineering: a detailed account". Operator theory, operator algebras, and applications. Contemporary Mathematics. Vol. 414. Providence, RI: American Mathematical Society. pp. 299–355. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0510024">math/0510024</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fconm%2F414%2F07820">10.1090/conm/414/07820</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780821839232</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2277219">2277219</a>.</cite>

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<li id="cite_note-con-3">^ <a href="#cite_ref-con_3-0">a</a> <a href="#cite_ref-con_3-1">b</a> <cite id="CITEREFCasazza2015" class="citation arxiv cs1">Casazza, Peter G. (2015). "Consequences of the Marcus/Spielman/Stivastava solution to the Kadison–Singer Problem". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1407.4768">1407.4768</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.FA">math.FA</a>].</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFMarcusSpielmanSrivastava2013" class="citation arxiv cs1"><a href="Adam_Marcus_(mathematician)" title="Adam Marcus (mathematician)">Marcus, Adam</a>; <a href="Daniel_Spielman" title="Daniel Spielman">Spielman, Daniel A.</a>; <a href="Nikhil_Srivastava" title="Nikhil Srivastava">Srivastava, Nikhil</a> (2013). "Interlacing families II: Mixed characteristic polynomials and the Kadison–Singer problem". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1306.3969">1306.3969</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CO">math.CO</a>].</cite>

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<li id="cite_note-disc-5">^ <a href="#cite_ref-disc_5-0">a</a> <a href="#cite_ref-disc_5-1">b</a> <cite id="CITEREFSrivastava2013" class="citation web cs1"><a href="Nikhil_Srivastava" title="Nikhil Srivastava">Srivastava, Nikhil</a> (July 11, 2013). <a rel="nofollow" class="external text" href="http://windowsontheory.org/2013/07/11/discrepancy-graphs-and-the-kadison-singer-conjecture-2/">"Discrepancy, Graphs, and the Kadison–Singer Problem"</a>. Windows on Theory.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFAnderson1979" class="citation journal cs1">Anderson, Joel (1979). <a rel="nofollow" class="external text" href="https://www.ams.org/tran/1979-249-02/S0002-9947-1979-0525675-1/">"Restrictions and representations of states on C∗-algebras"</a>. <a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>. 249 (2): 303–329. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1998793">10.2307/1998793</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1998793">1998793</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0525675">0525675</a>.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFWeaver2004" class="citation journal cs1">Weaver, Nik (2004). "The Kadison-Singer problem in discrepancy theory". Discrete Mathematics. 278 (1–3): 227–239. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0209078">math/0209078</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0012-365X%2803%2900253-X">10.1016/S0012-365X(03)00253-X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5304663">5304663</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFNicholas_J._A._Harvey2013" class="citation web cs1">Nicholas J. A. Harvey (July 11, 2013). <a rel="nofollow" class="external text" href="http://www.cs.ubc.ca/~nickhar/papers/KS/KS.pdf">"An introduction to the Kadison–Singer Problem and the Paving Conjecture"</a> (PDF).</cite></li></ul></div><div><div class="zim-footer">

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