Total variation diminishing

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.

Accessible Information

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 {p₁, p₂, ..., p_n}. Alice then prepares a quantum state, represented by the density matrix ρ_X chosen from a set {ρ₁, ρ₂, ... ρ_n}, 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 measurement on the state ρ_X, 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.

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.

Statement of the theorem

Let {ρ₁, ρ₂, ..., ρ_n} be a set of mixed states and let ρ_X be one of these states drawn according to the probability distribution P = {p₁, p₂, ..., p_n}.

Then, for any measurement described by POVM elements {E_Y} and performed on ρ_X, the amount of accessible information about the variable X knowing the outcome Y of the measurement is bounded from above as follows:

$I(X:Y)\le S(\rho )-\sum _i{p}_{i}S({\rho }_i)$

where $\rho =\sum _i{p}_i{\rho }_i$ and $S(\cdot )$ is the von Neumann entropy.

The quantity on the right hand side of this inequality is called the Holevo information or Holevo χ quantity:

$\chi :=S(\rho )-\sum _i{p}_{i}S({\rho }_i)$.

Proof

The proof can be given using three quantum systems, called $P,Q,M$. $P$ can be intuitively thought as the preparation, $Q$ can be thought as the quantum state prepared by Alice and given to Bob, and $M$ can be thought as Bob's measurement apparatus.

The compound system $P\otimes Q\otimes M$ at the beginning is in the state

${\rho }^{PQM}:=\sum _x{p}_x|x⟩⟨x|\otimes {\rho }_x\otimes |0⟩⟨0|$

This can be thought as Alice having the value $x$ for the random variable $X$. Then the preparation state is the mixed state described by the density matrix $\sum _x{p}_x|x⟩⟨x|$, and the quantum state given to Bob is ${\rho }_x$, and Bob's measurement apparatus is in its initial or rest state $|0⟩$. Using known results of quantum information theory it can be shown that

$S(P^{\prime };M^{\prime })\le S(P;Q)$

which, after some algebraic manipulation, can be shown to be equivalent to the statement of the theorem.

Comments and remarks

In essence, the Holevo bound proves that given n qubits, 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) bits. 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 $2^n-1$ complex numbers to encode the qubits which represent a mere n bits.

Footnotes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (see page 531, subsection 12.1.1 - equation (12.6) )
• Template:Cite arxiv. See in particular Section 11.6 and following. Holevo's theorem is presented as exercise 11.9.1 on page 288.

External links

• Holevo's theorem and its implications for quantum communication and computation, talk by Ashwin Nayak at the Mathematical Sciences Research Institute, 2000