<p>QUANTUM INFORMATION, <br>ENTANGLEMENT AND ENTROPY </p><p>Siddharth Sharma </p><p>Abstract </p><p>In this review I had given an introduction to axiomatic Quantum Mechanics, Quantum Information and its measure entropy. Also, I had given an introduction to Entanglement and its measure as the minimum of relative quantum entropy of a density matrix of a Hilbert space with respect to all separable density matrices of the same Hilbert space. I also discussed geodesic in the space of density matrices, their orthogonality and Pythagorean theorem of density matrix. </p><p>Postulates of Quantum Mechanics </p><p>The basic postulate of quantum mechanics is about the Hilbert space formalism: <br>• <strong>Postulates 0: </strong>To each quantum mechanical system, has a complex Hilbert space ℍ associated to it: </p><p>ℍ<br>⁄<br>⟦ ⟧&nbsp;⟦ ⟧ </p><p>{</p><p>}</p><p><strong>Set of all pure state, </strong></p><p>≔ {&nbsp;푓 | 푓&nbsp;≔ ℎ:&nbsp;ℎ ∼ 푓&nbsp;} and ℎ ∼ 푓 ⇔ ∃ 푧 ∈ ℂ such that <br>∼</p><p>ℎ = 푧 푓 where 푧 is called phase <br>• <strong>Postulates 1: </strong>The physical states of a quantum mechanical system are described by statistical operators acting on the Hilbert space. </p><p>A <strong>density matrix </strong>or <strong>statistical operator, </strong>휌 is a positive operator of trace 1 on the </p><p></p><ul style="display: flex;"><li style="flex:1">〈</li><li style="flex:1">〉</li><li style="flex:1">〈</li><li style="flex:1">〉</li></ul><p></p><p>Hilbert space. Where 휌 is called positive if&nbsp;푥, 휌푥&nbsp;for all 푥 ∈ ℍ and ⋇,⋇ . <br>• <strong>Postulates 2: </strong>The observables of a quantum mechanical system are described by selfadjoint operators acting on the Hilbert space. </p><p>A self-adjoint operator A on a Hilbert space ℍ is a linear operator 퐴: ℍ → ℍ which satisfies </p><p></p><ul style="display: flex;"><li style="flex:1">〈</li><li style="flex:1">〉</li><li style="flex:1">〈</li><li style="flex:1">〉</li></ul><p>퐴푥, 푦&nbsp;= 푥,&nbsp;퐴푦 for 푥, 푦 ∈ ℍ. </p><p><strong>Lemma: </strong>The density matrices acting on a Hilbert space form a convex set whose extreme points are the pure states. <strong>Proof</strong>. Denote by <strong>Σ </strong>the set of density matrices. It is obvious that a convex combination of density matrices is positive and of trace one. Therefore, <strong>Σ </strong>is a convex set. </p><p><strong>Lemma: </strong>Let </p><p></p><ul style="display: flex;"><li style="flex:1">|</li><li style="flex:1">⟩⟨ | </li><li style="flex:1">|</li><li style="flex:1">|</li><li style="flex:1">⟩⟨ </li></ul><p>휌 = ∑ 푥<sub style="top: 0.2em;">푒 </sub>푥<sub style="top: 0.2em;">푒 </sub>= ∑ 푦<sub style="top: 0.2em;">ℎ </sub>푦<sub style="top: 0.2em;">ℎ </sub></p><p></p><ul style="display: flex;"><li style="flex:1">푒∈픹 </li><li style="flex:1">ℎ∈픹 </li></ul><p></p><p>be decompositions of a density matrix and where 픹 is the basis of Hilbert space, ℍ. </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><p>푒ℎ∈픹 </p><p></p><ul style="display: flex;"><li style="flex:1">Then there exists a unitary matrix&nbsp;푈<sub style="top: 0.2em;">푒ℎ </sub></li><li style="flex:1">such that </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">|</li><li style="flex:1">⟩</li><li style="flex:1">|</li><li style="flex:1">⟩</li><li style="flex:1">∑ 푈<sub style="top: 0.2em;">푒ℎ </sub>푥<sub style="top: 0.2em;">푒 </sub>= 푦<sub style="top: 0.2em;">ℎ </sub></li></ul><p></p><p>푒∈픹 </p><p>Where, </p><p></p><ul style="display: flex;"><li style="flex:1">⟨</li><li style="flex:1">|</li><li style="flex:1">⟩</li><li style="flex:1">푥<sub style="top: 0.2em;">푒 </sub>푦<sub style="top: 0.2em;">ℎ </sub></li></ul><p></p><p>푈<sub style="top: 0.2em;">푒ℎ </sub></p><p>≔</p><ul style="display: flex;"><li style="flex:1">⟨</li><li style="flex:1">|</li><li style="flex:1">⟩</li><li style="flex:1">√ 푥<sub style="top: 0.19em;">푒 </sub>푥<sub style="top: 0.19em;">ℎ </sub></li></ul><p></p><p>• <strong>Postulates 3: </strong>Let 풳 be a finite set and for 푥 ∈ 풳 an operator 푂<sub style="top: 0.2em;">ꢀ </sub>∈ ℬ(ℍ) be given such that </p><p>∑ 푂<sub style="top: 0.2em;">ꢀ</sub><sup style="top: -0.37em;">⋇</sup>푂<sub style="top: 0.2em;">ꢀ </sub>= 푖푑<sub style="top: 0.2em;">ℍ </sub></p><p>ꢀ∈풳 </p><p>where 푂<sub style="top: 0.2017em;">ꢀ</sub><sup style="top: -0.37em;">⋇ </sup>is conjugate of 푂<sub style="top: 0.2017em;">ꢀ</sub>. Such an indexed family of operators is a model of a measurement with values in 풳. If the measurement is performed in a state 휌, then the </p><p>⋇</p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><p>outcome 푥 ∈ 풳 appears with probability tr 푂<sub style="top: 0.2em;">ꢀ </sub>휌푂<sub style="top: 0.2em;">ꢀ </sub>and after the measurement the state of the system is </p><p>푂<sub style="top: 0.2em;">ꢀ</sub><sup style="top: -0.37em;">⋇</sup>휌푂<sub style="top: 0.2em;">ꢀ </sub></p><p>⋇</p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">푡푟 푂<sub style="top: 0.2em;">ꢀ </sub>휌푂<sub style="top: 0.2em;">ꢀ </sub></li></ul><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><p>If 휑 ∶ ℬ(ℍ) → ℂ is a linear functional such that 휑 퐴&nbsp;≥ 0 if 퐴 is positive and 휑 푖푑&nbsp;= 1, Then there exists a density matrix 휌<sub style="top: 0.2em;">ꢁ </sub>such that, </p><p>( ) <br>휑 퐴&nbsp;= tr(휌<sub style="top: 0.2em;">ꢁ</sub>퐴) </p><p>The functional 휑 associates the expectation value to the observables 퐴. The density matrices 휌 and 휌′ are called <strong>orthogonal </strong>if any eigenvector of 휌 is orthogonal to any eigenvector of 휌′. </p><p>Let 픹 be an orthonormal basis in a Hilbert space ℍ . The unit vector 휉 ∈ ℍ&nbsp;is </p><p><strong>complementary </strong>to the given basis if </p><p>1</p><ul style="display: flex;"><li style="flex:1">|〈 〉| </li><li style="flex:1">ꢂ, 휉 </li><li style="flex:1">=</li></ul><p>dim ℍ </p><p>√</p><p>For all ꢂ ∈ 픹, where dim ℍ is dimension of ℍ. Two orthonormal bases are called complementary if all vectors in the first basis are complementary to the other basis. </p><p>• <strong>Postulates 4: </strong>The composite system is described by the tensor product Hilbert space </p><p>⨂ ℍ<sub style="top: 0.2em;">휗 </sub></p><p>휗</p><p>Given a density matrix 휌 on ⨂<sub style="top: 0.45em;">휗 </sub>ℍ<sub style="top: 0.2em;">휗 </sub>there are density matrices 휌<sub style="top: 0.2em;">휗 </sub>∈ ℬ(ℍ<sub style="top: 0.2em;">휗</sub>) such that </p><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">(</li><li style="flex:1">)</li></ul><p>tr (휌 ⨂ 풜<sub style="top: 0.2em;">훾 </sub>ꢃ ) = tr&nbsp;휌<sub style="top: 0.2em;">휗</sub>퐴<sub style="top: 0.2em;">휗 </sub></p><p>훾</p><p>Where, </p><p>퐴<sub style="top: 0.2em;">휗</sub>, ꢄ&nbsp;= ꢃ </p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">풜<sub style="top: 0.2em;">훾 </sub>ꢃ ≔&nbsp;{ </li></ul><p>푖푑<sub style="top: 0.2em;">ℍ</sub><sub style="top: 0.36em;">ꢅ</sub>, ꢄ&nbsp;≠ ꢃ </p><p>and 퐴<sub style="top: 0.2em;">휗 </sub>∈ ℬ(ℍ<sub style="top: 0.2em;">휗</sub>), here 휌<sub style="top: 0.2em;">휗 </sub>is called <strong>reduced density matrices</strong>. </p><p>Let, </p><p>휌 = ⨂ 휌<sub style="top: 0.2em;">휗 </sub></p><p>휗</p><p>( ) </p><p>Then we define 퐭퐫<sub style="top: 0.2em;">휸 </sub>흆 as follows: </p><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">tr<sub style="top: 0.2em;">훾 </sub>휌 ≔&nbsp;tr(휌<sub style="top: 0.2em;">훾</sub>) ⨂ 휌<sub style="top: 0.2em;">휗 </sub></li></ul><p></p><p>휗≠훾 </p><p>Information and its Measures </p><p>• <strong>Shannon entropy: </strong>In his revolutionary paper Shannon proposed a statistical approach and he posed the problem in the following way: “Suppose we have a set of possible events whose probabilities of occurrence are 푝<sub style="top: 0.2em;">1</sub>, 푝<sub style="top: 0.2em;">2</sub>, . . . , 푝<sub style="top: 0.2em;">푛</sub>. These probabilities are known but that is all we know concerning which event will occur. Can we find a measure of how much “choice” is involved in the selection of the event or how uncertain we are of the outcome?” Denoting such a measure by 퐻(푝<sub style="top: 0.2em;">1</sub>, 푝<sub style="top: 0.2em;">2</sub>, . . . , 푝<sub style="top: 0.2em;">푛</sub>), he listed three very reasonable requirements which should be satisfied, those three postulates are as follows: </p><p>a. <strong>Continuity</strong>: 퐻(푝, 1 −&nbsp;푝) is a continuous function of 푝. b. <strong>Symmetry: </strong>퐻(푝<sub style="top: 0.2em;">1</sub>, 푝<sub style="top: 0.2em;">2</sub>, . . . , 푝<sub style="top: 0.2em;">푛</sub>), is&nbsp;a symmetric function of its variables. </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">)</li></ul><p>c. <strong>Recursion: </strong>For every 0 ≤ 휆 &lt; 1 the recursion 퐻 푝<sub style="top: 0.2017em;">1</sub>, . , 휆푝<sub style="top: 0.2017em;">ꢆ</sub>, . ,&nbsp;1 − 휆&nbsp;푝<sub style="top: 0.2017em;">ꢆ</sub>. , 푝<sub style="top: 0.2017em;">푛 </sub><br>=</p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">퐻 푝<sub style="top: 0.2em;">1</sub>, 푝<sub style="top: 0.2em;">2</sub>, . , 푝<sub style="top: 0.2em;">ꢆ</sub>. . , 푝<sub style="top: 0.2em;">푛 </sub>+ 푝<sub style="top: 0.2em;">ꢆ</sub>퐻(휆, 1 − 휆), holds </li></ul><p></p><p>According to him there is only one function satisfying all this postulate is: </p><p>푛</p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">( ) </li><li style="flex:1">퐻 푝<sub style="top: 0.2em;">1</sub>, 푝<sub style="top: 0.2em;">2</sub>, . , 푝<sub style="top: 0.2em;">ꢆ</sub>. . , 푝<sub style="top: 0.2em;">푛 </sub>= −휅 ∑ 푝<sub style="top: 0.2em;">ꢇ</sub>푙ꢈ 푝<sub style="top: 0.2em;">ꢇ </sub></li></ul><p></p><p>ꢇ=1 </p><p>푛</p><p>Let (푋<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>be random variables with values in the set 풳 ≔&nbsp;<sub style="top: 0.24em;">ꢇ=1 </sub>풳<sub style="top: 0.2em;">ꢇ</sub>. The following </p><p>∏</p><p>notation will be used. a. 푝(푥<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>is probability of (푋<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>at (푥<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>b. 푝(푥<sub style="top: 0.2em;">ꢇ</sub>|푥 ) is probability of 푋<sub style="top: 0.2em;">ꢇ </sub>at 푥<sub style="top: 0.2em;">ꢇ </sub>knowing the probability of 푋 . at 푥 </p><p></p><ul style="display: flex;"><li style="flex:1">푗</li><li style="flex:1">푗</li><li style="flex:1">푗</li></ul><p></p><p>Then, </p><p></p><ul style="display: flex;"><li style="flex:1">퐻(푋<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>≔ − </li><li style="flex:1">∑</li><li style="flex:1">푝(푥<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>ln 푝(푥<sub style="top: 0.2em;">ꢇ</sub>)<sub style="top: 0.24em;">ꢇ</sub><sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">=1 </sub></li></ul><p></p><p>ꢊ</p><p>(ꢀ )&nbsp;∈풳 </p><p>ꢉ</p><p>ꢉ=1 </p><p>Where if ꢈ = 2 </p><p>퐻(푋<sub style="top: 0.2em;">ꢇ</sub>, 푋 ) ≔ −&nbsp;∑ ∑&nbsp;푝(푥<sub style="top: 0.2em;">ꢇ</sub>, 푥&nbsp;) ln 푝(푥<sub style="top: 0.2em;">ꢇ</sub>, 푥&nbsp;) </p><p></p><ul style="display: flex;"><li style="flex:1">푗</li><li style="flex:1">푗</li><li style="flex:1">푗</li></ul><p>ꢀ ∈풳&nbsp;ꢀ ∈풳 </p><p></p><ul style="display: flex;"><li style="flex:1">ꢉ</li><li style="flex:1">ꢉ</li><li style="flex:1">ꢋ</li><li style="flex:1">ꢋ</li></ul><p></p><p>and </p><p>푝(푥<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>= 푝(푥<sub style="top: 0.2em;">2</sub>)푝(푥<sub style="top: 0.2em;">1</sub>|푥<sub style="top: 0.2em;">2</sub>, 푥<sub style="top: 0.2em;">3</sub>, … . 푥<sub style="top: 0.2em;">푛</sub>) </p><p>Which leads to: </p><p>푛</p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">푝 ((( 푥<sub style="top: 0.2em;">1</sub>|푥<sub style="top: 0.2em;">2 </sub>|푥<sub style="top: 0.2em;">3</sub>)| … . ) |푥<sub style="top: 0.2em;">푛</sub>) </li></ul><p></p><ul style="display: flex;"><li style="flex:1">푝(푥<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub></li><li style="flex:1">=</li><li style="flex:1">∏ 푝(푥<sub style="top: 0.2em;">2</sub>) </li></ul><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">푝 푥<sub style="top: 0.2em;">1 </sub></li></ul><p></p><p>ꢇ=1 </p><p><strong>Theorem: </strong>If (푋<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>are random variables of finite range, then </p><p>푛</p><p>퐻(푋<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>≤ ∑ 퐻(푋<sub style="top: 0.2em;">ꢇ</sub>) </p><p>ꢇ=1 </p><p>Proof: </p><p></p><ul style="display: flex;"><li style="flex:1">퐻(푋<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>≔ − </li><li style="flex:1">∑</li><li style="flex:1">푝(푥<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>ln 푝(푥<sub style="top: 0.2em;">ꢇ</sub>)<sub style="top: 0.24em;">ꢇ</sub><sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">=1 </sub></li></ul><p></p><p>ꢊ</p><p>(ꢀ )&nbsp;∈풳 </p><p>ꢉ</p><p>ꢉ=1 </p><p>And </p><p></p><ul style="display: flex;"><li style="flex:1">푛</li><li style="flex:1">푛</li><li style="flex:1">푛</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">∑ 퐻(푋<sub style="top: 0.2em;">ꢇ</sub>) ≔ − </li><li style="flex:1">∑</li><li style="flex:1">∏ 푝(푥<sub style="top: 0.2em;">2</sub>) ln ∏ 푝(푥<sub style="top: 0.2em;">2</sub>) </li></ul><p></p><p>ꢊ</p><p></p><ul style="display: flex;"><li style="flex:1">ꢇ=1 </li><li style="flex:1">ꢇ=1 </li><li style="flex:1">ꢇ=1 </li></ul><p></p><p>(ꢀ )&nbsp;∈풳 </p><p>ꢉ</p><p>ꢉ=1 </p><p>And </p><p>푛</p><p>∏ 푝(푥<sub style="top: 0.2em;">2</sub>) ≤ 푝(푥<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub></p><p>ꢇ=1 </p><p>As </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">푝 푥<sub style="top: 0.2em;">1 </sub>≤ 푝 ((( 푥<sub style="top: 0.2em;">1</sub>|푥<sub style="top: 0.2em;">2 </sub>|푥<sub style="top: 0.2em;">3</sub>)| … . ) |푥<sub style="top: 0.2em;">푛</sub>) </li></ul><p></p><p>Hence, </p><p>푛</p><p>퐻(푋<sub style="top: 0.2em;">ꢇ</sub>)<sup style="top: -0.41em;">푛</sup><sub style="top: 0.24em;">ꢇ=1 </sub>≤ ∑ 퐻(푋<sub style="top: 0.2em;">ꢇ</sub>) </p><p>ꢇ=1 </p><p><strong>Fano’s inequality: </strong>Let 푋 and 푌 be random variables such that their range is in a set of cardinality d and let 푝 ≔&nbsp;푃푟표푏(푋 ≠ 푌) .Then </p><p>퐻(푋|푌) ≤ 푝 푙표푔(푑 − 1) + 퐻(푝, 1 − 푝) </p><p>• <strong>von Neumann Entropy: </strong>Let 휌 be density matrix of a quantum system then von <br>Neumann Entropy of that quantum system: </p><p></p><ul style="display: flex;"><li style="flex:1">( )&nbsp;) </li><li style="flex:1">(</li><li style="flex:1">푆 휌&nbsp;≔ −휅 tr&nbsp;휌 ln 휌 </li></ul><p></p><p><strong>Theorem: </strong>Let 휌 and 휎 be densities on a 푑 −dimensional Hilbert space and let </p><p></p><ul style="display: flex;"><li style="flex:1">‖</li><li style="flex:1">‖</li></ul><p></p><p>1</p><p>휌 − 휎 <br>푝 ≔ </p><p>2</p><p>where Then, <br><sub style="top: 0.2em;">1</sub>: ℬ(ℍ) → ℝ<sup style="top: -0.39em;">+</sup><sub style="top: 0.22em;">0 </sub>, is the norm on operator space ℬ(ℍ) of Hilbert space ℍ. </p><p>‖ ‖ <br>⋇<br>|푆(휌) − 푆(휎)|&nbsp;≤ 푝&nbsp;푙표푔(푑 −&nbsp;1) +&nbsp;퐻(푝, 1 −&nbsp;푝) </p><p>Holds. </p><p>• <strong>Quantum Relative Entropy: </strong></p><p>The relative entropy, or I-divergence of the probability distributions 푝(푥) and 푞(푥), is defined as </p><p>∞</p><p>( ) <br>푝 푥 </p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">( ) </li><li style="flex:1">퐷 푝||&nbsp;푞 ≔&nbsp;∫ 푝&nbsp;푥 ln </li><li style="flex:1">푑푥 </li></ul><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">푞 푥 </li></ul><p></p><p>−∞ </p><p>Assume that 휌 and 휎 are density matrices on a Hilbert space ℍ , then </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">( ) </li><li style="flex:1">( ) </li><li style="flex:1">tr 휌&nbsp;ln 휌 − ln 휎&nbsp;if supp&nbsp;ρ ≤&nbsp;supp 휎 </li></ul><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p>푆 휌||&nbsp;휎 ≔&nbsp;{ <br>+∞ 표푡ℎꢂ푟푤푖푠ꢂ </p><p>Hilbert–Schmidt inner product </p><p>훥푎 = 휌푎휎<sup style="top: -0.37em;">−1 </sup><br>( ) <br>For all 푎 ∈ ℬ&nbsp;ℍ </p><p>12<br>12</p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">〈</li><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">〉</li><li style="flex:1">tr 휌&nbsp;ln 휌 − ln 휎&nbsp;= −&nbsp;휌 ,&nbsp;ln 훥&nbsp;휌 </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">⋇</li><li style="flex:1">⋇</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">〈</li><li style="flex:1">〉</li><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">where, 퐴, 퐵&nbsp;≔ tr&nbsp;퐴 퐵&nbsp;and 퐴 is conjugate of 퐴 </li></ul><p></p><p>⋇</p><p>12<br>12</p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">(</li><li style="flex:1">)</li></ul><p>tr 휌&nbsp;ln 휌 − ln 휎&nbsp;= −tr ((휌 ) (휌 ) ln 훥&nbsp;) </p><p>Let 휌 ≡ ꢂ<sup style="top: -0.37em;">ꢌ</sup>, 휔 and 휎 be three invertible densities. The ꢂ − 푔ꢂ표푑ꢂ푠푖푐 connecting 휌 and 휎 is the curve </p><p>ꢂ<sup style="top: -0.37em;">ꢌ+ꢍꢎ </sup></p><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">ꢄ<sub style="top: 0.2em;">푒 </sub>푡 ≔ </li></ul><p></p><p>ꢌ+ꢍꢎ </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">tr ꢂ </li></ul><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">( ) </li></ul><p>for all 푡 ∈ [0,1], where 퐴 ≔ ln 휎 − ln 휌. Then, ꢄ<sub style="top: 0.2em;">푒 </sub>0 =&nbsp;휌 and ꢄ<sub style="top: 0.2em;">푒 </sub>1 =&nbsp;휎. The 푚 − </p><p>푔ꢂ표푑ꢂ푠푖푐 connecting 휌 and 휔 is the curve. </p><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">ꢄ<sub style="top: 0.2em;">ꢏ </sub>푡 ≔&nbsp;휌 + 푡퐵 </li></ul><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">( ) </li></ul><p>for all 푡 ∈ [0,1] where 퐵 ≔ 휔 − 휌. Then&nbsp;ꢄ<sub style="top: 0.2em;">ꢏ </sub>0 =&nbsp;휌 and ꢄ<sub style="top: 0.2em;">ꢏ </sub>1 =&nbsp;ω </p><p>Assume that the ꢂ − 푔ꢂ표푑ꢂ푠푖푐 connecting 휌 and 휎 is orthogonal to the 푚 − 푔ꢂ표푑ꢂ푠푖푐 connecting 휌 and 휔 with respect to the inner product </p><p>∞</p><p>−1 </p><p>⋇</p><p>−1 </p><p></p><ul style="display: flex;"><li style="flex:1">〈</li><li style="flex:1">〉</li><li style="flex:1">(( </li><li style="flex:1">)</li><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">)</li></ul><p>퐸, 퐹&nbsp;≔ ∫ tr 푠핀&nbsp;+ 휌&nbsp;퐸 푠핀&nbsp;+ 휌&nbsp;퐹 푑푠 </p><p>ꢐ</p><p>0</p><p>A plain computation yields </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">〈</li><li style="flex:1">〉</li><li style="flex:1">푆(휔||휌) +&nbsp;푆(휌||휎) − 푆(휔||휎) = tr&nbsp;퐴퐵 = 퐴,&nbsp;퐵 </li></ul><p></p><p>ꢌꢑ <br>∞</p><p>−1 </p><p></p><ul style="display: flex;"><li style="flex:1">푇 ∶&nbsp;푋 ↦ ∫ 푠핀 + 휌&nbsp;푋 푠핀&nbsp;+ 휌&nbsp;<sup style="top: -0.37em;">−1</sup>푑푠 </li><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><p>ꢐ</p><p>0</p><p>Them According Dénes Petz And, </p><p></p><ul style="display: flex;"><li style="flex:1">〈</li><li style="flex:1">〉</li><li style="flex:1">〈</li><li style="flex:1">( )&nbsp;〉 </li><li style="flex:1">= 푇 푋&nbsp;, 푌 </li></ul><p></p><p>ꢐ</p><p>푋, 푌 </p><p></p><ul style="display: flex;"><li style="flex:1">ꢌꢑ </li><li style="flex:1">ꢐ</li></ul><p></p><p>( ) <br>푇 퐴 =&nbsp;ꢄ<sub style="top: 0.2em;">푒 </sub><br>( ) </p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">̇</li></ul><p></p><p>ꢐ</p><p>It is obvious that: Therefore </p><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">0</li><li style="flex:1">퐵 = ꢄ<sub style="top: 0.2em;">ꢏ </sub></li><li style="flex:1">̇</li></ul><p></p><ul style="display: flex;"><li style="flex:1">〈</li><li style="flex:1">〉</li><li style="flex:1">〈</li><li style="flex:1">( ) </li><li style="flex:1">〉</li><li style="flex:1">〈 (&nbsp;) (&nbsp;)〉 </li><li style="flex:1">0 ,&nbsp;ꢄ<sub style="top: 0.2em;">ꢏ </sub></li></ul><p></p><p>ꢐ</p><p></p><ul style="display: flex;"><li style="flex:1">퐴, 퐵 </li><li style="flex:1">= 푇 퐴&nbsp;, 퐵&nbsp;= ꢄ<sub style="top: 0.2em;">푒 </sub></li><li style="flex:1">̇</li><li style="flex:1">̇</li><li style="flex:1">0</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">ꢌꢑ </li><li style="flex:1">ꢐ</li><li style="flex:1">ꢐ</li></ul><p></p><p>and we can conclude that if our assumption holds then that implies: </p><p></p><ul style="display: flex;"><li style="flex:1">〈 (&nbsp;) (&nbsp;)〉 |&nbsp;) </li><li style="flex:1">ꢄ<sub style="top: 0.2em;">푒 </sub>0 ,&nbsp;ꢄ<sub style="top: 0.2em;">ꢏ </sub>= 0 ⇒ 푆(휔||휌)&nbsp;+ 푆(휌| 휎&nbsp;= 푆(휔||휎) </li><li style="flex:1">̇</li><li style="flex:1">̇</li><li style="flex:1">0</li></ul><p></p><p>ꢐ</p><p>This is some time called Pythagorean theorem of density matrix. </p><p>• <strong>Renyi Entropy, Quantum Renyi Entropy and Quantum Relative Renyi Entropy: </strong></p><p>The Renyi entropy of order 훼 ≠ 1&nbsp;of the probability distribution (푝<sub style="top: 0.2em;">1</sub>, 푝<sub style="top: 0.2em;">2</sub>, . . . , 푝<sub style="top: 0.2em;">푛</sub>) is defined by </p><p>푛</p><p>1</p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">퐻<sub style="top: 0.2em;">ꢒ </sub>푝<sub style="top: 0.2em;">1</sub>, 푝<sub style="top: 0.2em;">2</sub>, . . . , 푝<sub style="top: 0.2em;">푛 </sub></li><li style="flex:1">=</li><li style="flex:1">ln ∑ 푝<sub style="top: 0.23em;">푘</sub><sup style="top: -0.4em;">ꢒ </sup></li></ul><p>1 − 훼 </p><p>푘=1 </p><p>The Quantum Renyi entropy of order 훼 ≠ 1 of the density matrix 휌 is defined by </p><p>1</p><p>ꢒ</p><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">푆<sub style="top: 0.2em;">ꢒ </sub>휌 = </li><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">ln tr&nbsp;휌 </li></ul><p>1 − 훼 </p><p>The Quantum Renyi entropy of order 훼 ≠ 1 of the density matrix 휌 with respect 휎 to is defined by </p><p>1</p><p>ꢒ</p><p>1−ꢒ </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">푆<sub style="top: 0.2em;">ꢒ </sub>휌||휎 = </li><li style="flex:1">ln tr&nbsp;휌 휎 </li></ul><p>훼 − 1 </p><p>Entanglement </p><p>Let ℬ(ℍ<sub style="top: 0.2em;">ꢎ</sub>) and ℬ(ℍ<sub style="top: 0.2em;">ꢓ</sub>) be the algebras of bounded operators acting on the Hilbert spaces ℍ<sub style="top: 0.2em;">ꢎ </sub>and ℍ<sub style="top: 0.2em;">ꢓ</sub>. The Hilbert space of the composite system is ℍ<sub style="top: 0.2em;">ꢎꢓ </sub>≔ ℍ<sub style="top: 0.2em;">ꢎ </sub>⊗ ℍ<sub style="top: 0.2em;">ꢓ</sub>. The </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><p>algebra of the operators acting on ℍ<sub style="top: 0.2em;">ꢎꢓ </sub>is ℬ ℍ<sub style="top: 0.2em;">ꢎꢓ </sub>≔ ℬ(ℍ<sub style="top: 0.2em;">ꢎ</sub>) ⊗ ℬ(ℍ<sub style="top: 0.2em;">ꢓ</sub>). </p>