Holevo's theorem

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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 that can be known about a quantum state (accessible information). It was published by Alexander Holevo in 1973.

Statement of the theorem

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Suppose Alice wants to send a classical message to Bob by encoding it into a quantum state, and suppose she can prepare a state from some fixed set  , with the i-th state prepared with probability  . Let   be the classical register containing the choice of state made by Alice. Bob's objective is to recover the value of   from measurement results on the state he received. Let   be the classical register containing Bob's measurement outcome. Note that   is therefore a random variable whose probability distribution depends on Bob's choice of measurement.

Holevo's theorem bounds the amount of correlation between the classical registers   and  , regardless of Bob's measurement choice, in terms of the Holevo information. This is useful in practice because the Holevo information does not depend on the measurement choice, and therefore its computation does not require performing an optimization over the possible measurements.

More precisely, define the accessible information between   and   as the (classical) mutual information between the two registers maximized over all possible choices of measurements on Bob's side: where   is the (classical) mutual information of the joint probability distribution given by  . There is currently no known formula to analytically solve the optimization in the definition of accessible information in the general case. Nonetheless, we always have the upper bound: where   is the ensemble of states Alice is using to send information, and   is the von Neumann entropy. This   is called the Holevo information or Holevo χ quantity.

Note that the Holevo information also equals the quantum mutual information of the classical-quantum state corresponding to the ensemble: with   the quantum mutual information of the bipartite state  . It follows that Holevo's theorem can be concisely summarized as a bound on the accessible information in terms of the quantum mutual information for classical-quantum states.

Proof

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Consider the composite system that describes the entire communication process, which involves Alice's classical input  , the quantum system  , and Bob's classical output  . The classical input   can be written as a classical register   with respect to some orthonormal basis  . By writing   in this manner, the von Neumann entropy   of the state   corresponds to the Shannon entropy   of the probability distribution  :

 

The initial state of the system, where Alice prepares the state   with probability  , is described by

 

Afterwards, Alice sends the quantum state to Bob. As Bob only has access to the quantum system   but not the input  , he receives a mixed state of the form  . Bob measures this state with respect to the POVM elements  , and the probabilities   of measuring the outcomes   form the classical output  . This measurement process can be described as a quantum instrument

 

where   is the probability of outcome   given the state  , while   for some unitary   is the normalised post-measurement state. Then, the state of the entire system after the measurement process is

 

Here   is the identity channel on the system  . Since   is a quantum channel, and the quantum mutual information is monotonic under completely positive trace-preserving maps,[1]  . Additionally, as the partial trace over   is also completely positive and trace-preserving,  . These two inequalities give

 

On the left-hand side, the quantities of interest depend only on

 

with joint probabilities  . Clearly,   and  , which are in the same form as  , describe classical registers. Hence,

 

Meanwhile,   depends on the term

 

where   is the identity operator on the quantum system  . Then, the right-hand side is

 

which completes the proof.

Comments and remarks

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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. It was also established, both theoretically and experimentally, that there are computations where quantum bits carry more information through the process of the computation than is possible classically.[2]

See also

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References

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  1. ^ Preskill, John (June 2016). "Chapter 10. Quantum Shannon Theory" (PDF). Quantum Information. pp. 23–24. Retrieved 30 June 2021.
  2. ^ Maslov, Dmitri; Kim, Jin-Sung; Bravyi, Sergey; Yoder, Theodore J.; Sheldon, Sarah (2021-06-28). "Quantum advantage for computations with limited space". Nature Physics. 17 (8): 894–897. arXiv:2008.06478. Bibcode:2021NatPh..17..894M. doi:10.1038/s41567-021-01271-7. S2CID 221136153.

Further reading

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