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<title>Projective Hilbert space</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Projective Hilbert space</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a> and the foundations of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, the <b>projective Hilbert space</b> or <b>ray space</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} (H)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} (H)}</annotation>

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</math></span><img src="./df9ab9357902a5f4ee6c4be4e4eae7896c676de2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.7ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} (H)}" loading="lazy"></span> of a <a href="Complex_number" title="Complex number">complex</a> <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> is the set of <a href="Equivalence_class" title="Equivalence class">equivalence classes</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [v]}">

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<annotation encoding="application/x-tex">{\displaystyle [v]}</annotation>

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</math></span><img src="./03fbb9b8f28912a0f0fffa45ef4e6562c5788fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.421ex; height:2.843ex;" alt="{\displaystyle [v]}" loading="lazy"></span> of non-zero vectors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in H}">

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<annotation encoding="application/x-tex">{\displaystyle v\in H}</annotation>

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</math></span><img src="./6007e16cc49606df770456e3a47909f8573e717d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.032ex; height:2.176ex;" alt="{\displaystyle v\in H}" loading="lazy"></span>, for the <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }">

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

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</math></span><img src="./afcc42adfcfdc24d5c4c474869e5d8eaa78d1173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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

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<annotation encoding="application/x-tex">{\displaystyle w\sim v}</annotation>

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</math></span><img src="./b9adc2d03cfc4c2cadba46d5532a261ffdd2bd00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.89ex; height:1.676ex;" alt="{\displaystyle w\sim v}" loading="lazy"></span> if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=\lambda w}">

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<annotation encoding="application/x-tex">{\displaystyle v=\lambda w}</annotation>

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</math></span><img src="./d306abdcc23e61463b326e527358fa746ff5c978.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.245ex; height:2.176ex;" alt="{\displaystyle v=\lambda w}" loading="lazy"></span> for some non-zero complex number <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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<p>This is the usual construction of <a href="Projectivization" title="Projectivization">projectivization</a>, applied to a complex Hilbert space.<sup id="cite_ref-FOOTNOTEMiranda199594_1-0" class="reference"><a href="#cite_note-FOOTNOTEMiranda199594-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In quantum mechanics, the equivalence classes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [v]}">

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<annotation encoding="application/x-tex">{\displaystyle [v]}</annotation>

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</math></span><img src="./03fbb9b8f28912a0f0fffa45ef4e6562c5788fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.421ex; height:2.843ex;" alt="{\displaystyle [v]}" loading="lazy"></span> are also referred to as <b>rays</b> or <b>projective rays</b>. Each such projective ray is a copy of the nonzero complex numbers, which is topologically a two-dimensional plane after one point has been removed.

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

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Wigner's_theorem#Rays_and_ray_space" title="Wigner's theorem">Wigner's theorem § Rays and ray space</a></div>

<p>The physical significance of the projective Hilbert space is that in <a href="Quantum_mechanics" title="Quantum mechanics">quantum theory</a>, the <a href="Wave_function" title="Wave function">wave functions</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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

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</math></span><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \psi }">

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

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</math></span><img src="./53d2ea83bc244e81f29149eb121ce0e5076e12b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.868ex; height:2.509ex;" alt="{\displaystyle \lambda \psi }" loading="lazy"></span> represent the same <i>physical state</i>, for any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \neq 0}">

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

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</math></span><img src="./dedcbbf652c56a75cb890f36b4ff1f3bd984af45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.616ex; height:2.676ex;" alt="{\displaystyle \lambda \neq 0}" loading="lazy"></span>. The <a href="Born_rule" title="Born rule">Born rule</a> demands that if the system is physical and measurable, its wave function has unit <a href="Normed_vector_space" title="Normed vector space">norm</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi |\psi \rangle =1}">

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<annotation encoding="application/x-tex">{\displaystyle \langle \psi |\psi \rangle =1}</annotation>

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</math></span><img src="./e0004edb3b03f76513b7511c9e8b04bb47230c0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.743ex; height:2.843ex;" alt="{\displaystyle \langle \psi |\psi \rangle =1}" loading="lazy"></span>, in which case it is called a <a href="Normalized_wave_function" class="mw-redirect" title="Normalized wave function">normalized wave function</a>. The unit norm constraint does not completely determine <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

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

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</math></span><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"></span> within the ray, since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

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

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</math></span><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"></span> could be multiplied by any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> with <a href="Absolute_value" title="Absolute value">absolute value</a> 1 (the <a href="Circle_group" title="Circle group">circle group</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}">

</mstyle>

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

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</math></span><img src="./e62b00d74ee0cefb86cc052365625abff56d43e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}" loading="lazy"></span> action) and retain its normalization. Such a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

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

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> can be written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =e^{i\phi }}">

<msup>

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</mstyle>

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

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</math></span><img src="./77611eedfcccc911deadf22576d50178bce64b14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.317ex; height:2.676ex;" alt="{\displaystyle \lambda =e^{i\phi }}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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

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</math></span><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> called the global <a href="Phase_factor" title="Phase factor">phase</a>.

</p><p>Rays that differ by such a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

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

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> correspond to the same state (cf. <a href="Quantum_state#Mathematical_generalizations" title="Quantum state">quantum state (algebraic definition)</a>, given a <a href="C*-algebra" title="C*-algebra">C*-algebra</a> of observables and a representation on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

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

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span>). No measurement can recover the phase of a ray; it is not observable. One says that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}">

</mstyle>

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

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</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

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

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> is an <a href="Irreducible_representation" title="Irreducible representation">irreducible representation</a> of the algebra of observables then the rays induce <a href="Quantum_state#Formalism_in_quantum_physics" title="Quantum state">pure states</a>. Convex linear combinations of rays naturally give rise to <a href="Density_matrix" title="Density matrix">density matrix</a> which (still in case of an irreducible representation) correspond to mixed states.

</p><p>In the case <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

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

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> is finite-dimensional, i.e., <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=H_{n}}">

<msub>

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

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</math></span><img src="./5a22a2bfc662beb3921c8b488798cbb569a2699a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.312ex; height:2.509ex;" alt="{\displaystyle H=H_{n}}" loading="lazy"></span>, the Hilbert space reduces to a finite-dimensional <a href="Inner_product_space" title="Inner product space">inner product space</a> and the set of projective rays may be treated as a <a href="Complex_projective_space" title="Complex projective space">complex projective space</a>; it is a <a href="Homogeneous_space" title="Homogeneous space">homogeneous space</a> for a <a href="Unitary_group" title="Unitary group">unitary group</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {U} (n)}">

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</mstyle>

</mrow>

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

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</math></span><img src="./a32fa84df5de5dfa91b6bdc88fb03fc8792c9f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle \mathrm {U} (n)}" loading="lazy"></span>. That is,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} (H_{n})=\mathbb {C} \mathbf {P} ^{n-1}}</annotation>

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</math></span><img src="./784fc379af3ae99e085605c1cddd2cb25e04ee41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.708ex; height:3.176ex;" alt="{\displaystyle \mathbf {P} (H_{n})=\mathbb {C} \mathbf {P} ^{n-1}}" loading="lazy"></span>,</dd></dl>

<p>which carries a <a href="K%C3%A4hler_metric" class="mw-redirect" title="Kähler metric">Kähler metric</a>, called the <a href="Fubini%E2%80%93Study_metric" title="Fubini–Study metric">Fubini–Study metric</a>, derived from the Hilbert space's norm.<sup id="cite_ref-FOOTNOTEKongLiu20219_2-0" class="reference"><a href="#cite_note-FOOTNOTEKongLiu20219-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTECirelliLanzavecchiaMania1983_3-0" class="reference"><a href="#cite_note-FOOTNOTECirelliLanzavecchiaMania1983-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>As such, the projectivization of, e.g., two-dimensional complex Hilbert space (the space describing one <a href="Qubit" title="Qubit">qubit</a>) is the <a href="Complex_projective_line" class="mw-redirect" title="Complex projective line">complex projective line</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \mathbf {P} ^{1}}">

</mrow>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \mathbf {P} ^{1}}</annotation>

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</math></span><img src="./3aed37c85a6044ca9e9f9e782558a173c56076f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.559ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} \mathbf {P} ^{1}}" loading="lazy"></span>. This is known as the <a href="Bloch_sphere" title="Bloch sphere">Bloch sphere</a> or, equivalently, the <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a>. See <a href="Hopf_fibration" title="Hopf fibration">Hopf fibration</a> for details of the projectivization construction in this case.

</p>

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

<p>The <a href="Cartesian_product" title="Cartesian product">Cartesian product</a> of projective Hilbert spaces is not a projective space. The <a href="Segre_mapping" class="mw-redirect" title="Segre mapping">Segre mapping</a> is an embedding of the Cartesian product of two projective spaces into the projective space associated to the <a href="Tensor_product" title="Tensor product">tensor product</a> of the two Hilbert spaces, given by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} (H)\times \mathbf {P} (H')\to \mathbf {P} (H\otimes H'),([x],[y])\mapsto [x\otimes y]}">

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</msup>

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<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} (H)\times \mathbf {P} (H')\to \mathbf {P} (H\otimes H'),([x],[y])\mapsto [x\otimes y]}</annotation>

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</math></span><img src="./956e99717c31d52a6f251bbd279fbd6601be60e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.09ex; height:3.009ex;" alt="{\displaystyle \mathbf {P} (H)\times \mathbf {P} (H')\to \mathbf {P} (H\otimes H'),([x],[y])\mapsto [x\otimes y]}" loading="lazy"></span>. In quantum theory, it describes how to make states of the composite system from states of its constituents. It is only an <a href="Embedding" title="Embedding">embedding</a>, not a surjection; most of the tensor product space does not lie in its <a href="Range_of_a_function" title="Range of a function">image</a> and represents <i><a href="Quantum_entanglement" title="Quantum entanglement">entangled states</a></i>.

</p>

<ul><li><a href="Complex_projective_space" title="Complex projective space">Complex projective space</a></li>

<li><a href="Projective_representation" title="Projective representation">Projective representation</a></li>

<li><a href="Projective_space" title="Projective space">Projective space</a>, for the concept in general</li></ul>

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

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<li id="cite_note-FOOTNOTEMiranda199594-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMiranda199594_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMiranda1995">Miranda 1995</a>, p. 94.</span>

</li>

</li>

<li id="cite_note-FOOTNOTECirelliLanzavecchiaMania1983-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECirelliLanzavecchiaMania1983_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCirelliLanzavecchiaMania1983">Cirelli, Lanzavecchia & Mania 1983</a>.</span>

</li>

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

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

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</style><cite id="CITEREFAshtekarSchilling1999" class="citation book cs1">Ashtekar, Abhay; Schilling, Troy A. (1999). "Geometrical Formulation of Quantum Mechanics". <i>On Einstein's Path</i>. New York, NY: Springer New York. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/9706069">gr-qc/9706069</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-1422-9_3">10.1007/978-1-4612-1422-9_3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-7137-6</bdi>.</cite></li>

<li><cite id="CITEREFCirelliLanzavecchiaMania1983" class="citation journal cs1">Cirelli, R; Lanzavecchia, P; Mania, A (1983). "Normal pure states of the von Neumann algebra of bounded operators as Kahler manifold". <i>Journal of Physics A: Mathematical and General</i>. <b>16</b> (16). IOP Publishing: <span class="nowrap">3829–</span>3835. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1983JPhA...16.3829C">1983JPhA...16.3829C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F16%2F16%2F020">10.1088/0305-4470/16/16/020</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0305-4470">0305-4470</a>.</cite></li>

<li><cite id="CITEREFKongLiu2021" class="citation journal cs1">Kong, Otto C. W.; Liu, Wei-Yin (2021). "Noncommutative Coordinate Picture of the Quantum Phase Space". <i>Chinese Journal of Physics</i>. <b>71</b>. Elsevier BV: 418. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1903.11962">1903.11962</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021ChJPh..71..418K">2021ChJPh..71..418K</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%2Fj.cjph.2021.03.014">10.1016/j.cjph.2021.03.014</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:85543324">85543324</a>.</cite></li>

<li><cite id="CITEREFMiranda1995" class="citation book cs1">Miranda, Rick (1995). <i>Algebraic Curves and Riemann Surfaces</i>. Providence (R.I.): American Mathematical Soc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-0268-2</bdi>.</cite></li></ul></div><div><div class="zim-footer">

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