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| '''Holevo's theorem''' is an important limitative theorem in [[quantum computing]], an interdisciplinary field of [[physics]] and [[computer science]]. It is sometimes called '''Holevo's bound''', since it establishes an [[upper bound]] to the amount of information which can be known about a [[quantum state]] (accessible information). It was published by [[Alexander Holevo]] in 1973.
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| ==Accessible Information==
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| As for several concepts in quantum information theory, accessible information is best understood in terms of a 2-party communication. So we introduce two parties, Alice and Bob. Alice has a ''classical'' [[random variable]] ''X'', which can take the values {1, 2, ..., ''n''} with corresponding probabilities {''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n''</sub>}. Alice then prepares a [[quantum state]], represented by the [[density matrix]] ''ρ<sub>X</sub>'' chosen from a set {''ρ''<sub>1</sub>, ''ρ''<sub>2</sub>, ... ''ρ''<sub>''n''</sub>}, and gives this state to Bob. Bob's goal is to find the value of ''X'', and in order to do that, he performs a [[Quantum measurement|measurement]] on the state ''ρ''<sub>''X''</sub>, obtaining a classical outcome, which we denote with ''Y''. In this context, the amount of accessible information, that is, the amount of information that Bob can get about the variable ''X'', is the maximum value of the [[mutual information]] ''I''(''X'' : ''Y'') between the random variables ''X'' and ''Y'' over all the possible measurements that Bob can do.<ref name=NielsenChuang>{{harvtxt|Nielsen|Chuang|2000}}</ref>
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| There is currently no known formula to compute the accessible information. There are however several upper bounds, the best-known of which is the Holevo bound, which is specified in the following theorem.<ref name=NielsenChuang />
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| ==Statement of the theorem==
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| Let {''ρ''<sub>1</sub>, ''ρ''<sub>2</sub>, ..., ''ρ''<sub>''n''</sub>} be a set of mixed states and let ''ρ''<sub>''X''</sub> be one of these states drawn according to the probability distribution ''P'' = {''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n''</sub>}.
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| Then, for any measurement described by [[POVM]] elements {''E''<sub>''Y''</sub>} and performed on ''ρ''<sub>''X''</sub>, the amount of accessible information about the variable ''X'' knowing the outcome ''Y'' of the measurement is bounded from above as follows:
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| :<math>I(X:Y) \leq S(\rho) - \sum_i p_i S(\rho_i)</math>
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| where <math>\rho = \sum_i p_i \rho_i</math> and <math>S(\cdot)</math> is the [[von Neumann entropy]].
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| The quantity on the right hand side of this inequality is called the '''Holevo information''' or '''Holevo ''χ'' quantity''':
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| :<math>\chi := S(\rho) - \sum_i p_i S(\rho_i) </math>.
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| ==Proof==
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| The proof can be given using three quantum systems, called <math>P, Q, M</math>. <math>P</math> can be intuitively thought as the ''preparation'', <math>Q</math> can be thought as the quantum state prepared by Alice and given to Bob, and <math>M</math> can be thought as Bob's measurement apparatus.
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| The compound system <math>P \otimes Q \otimes M</math> at the beginning is in the state
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| :<math>\rho^{PQM} := \sum_x p_x |x\rangle \langle x| \otimes \rho_x \otimes |0\rangle \langle0|</math>
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| This can be thought as Alice having the value <math>x</math> for the random variable <math>X</math>. Then the ''preparation state'' is the [[Quantum_state#Mixed_states|mixed state]] described by the [[density matrix]] <math> \sum_x p_x |x\rangle \langle x|</math>, and the quantum state given to Bob is <math>\rho_x</math>, and Bob's ''measurement apparatus'' is in its ''initial'' or ''rest'' state <math>|0\rangle </math>.
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| Using known results of quantum information theory it can be shown that
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| :<math> S(P';M') \leq S(P;Q)</math>
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| which, after some algebraic manipulation, can be shown to be equivalent to the statement of the theorem.<ref name=NielsenChuang />
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| ==Comments and remarks==
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| In essence, the Holevo bound proves that given ''n'' [[qubit]]s, although they can "carry" a larger amount of (classical) information (thanks to quantum superposition), the amount of classical information that can be ''retrieved'', i.e. ''accessed'', can be only up to ''n'' classical (non-quantum encoded) [[bit]]s. This is surprising, for two reasons: (1) quantum computing is so often more powerful than classical computing, that results which show it to be only as good or inferior to conventional techniques are unusual, and (2) because it takes <math>2^n-1</math> [[complex number]]s to encode the qubits which represent a mere ''n'' bits.
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| ==Footnotes==
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| {{Reflist}}
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| ==References==
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| * {{cite journal|first=Alexander S.|last=Holevo|authorlink=Alexander Holevo|title=Bounds for the quantity of information transmitted by a quantum communication channel|journal=Problems of Information Transmission|volume=9|pages=177–183|year=1973}}
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| * {{cite book |title=Quantum Computation and Quantum Information |last1=Nielsen |first1=Michael A. |authorlink1=Michael_Nielsen |last2=Chuang |first2=Isaac L. |authorlink2=Isaac_Chuang |year=2000 |publisher=Cambridge University Press |location=Cambridge, UK |isbn=978-0-521-63235-5 |oclc=43641333}} (see page 531, subsection 12.1.1 - equation (12.6) )
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| * {{cite arxiv | first = Mark M. | last = Wilde | title= From Classical to Quantum Shannon Theory |year= 2011| arxiv= 1106.1445v2 |ref=harv}}. See in particular Section 11.6 and following. Holevo's theorem is presented as exercise 11.9.1 on page 288.
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| ==External links==
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| *[http://www.msri.org/publications/ln/msri/2000/qcomputing/nayak/1/ Holevo's theorem and its implications for quantum communication and computation], talk by Ashwin Nayak at the [[Mathematical Sciences Research Institute]], 2000
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| [[Category:Quantum mechanical entropy]]
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| [[Category:Quantum information theory]]
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Hi there. My title is Sophia Meagher although it is not the name on my birth certificate. What me and my family members adore is doing ballet but I've been using on new things lately. For a whilst I've been in Alaska but I will have to transfer in a year or two. My day job is an information officer but I've already utilized for another one.
Here is my web page ... clairvoyance