Upper bound on the knowable information of a quantum state
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.
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 formula_1, with the i-th state prepared with probability formula_2. Let formula_3 be the classical register containing the choice of state made by Alice. Bob's objective is to recover the value of formula_3 from measurement results on the state he received. Let formula_5 be the classical register containing Bob's measurement outcome. Note that formula_5 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 formula_3 and formula_5, 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 formula_3 and formula_5 as the (classical) mutual information between the two registers maximized over all possible choices of measurements on Bob's side:formula_11where formula_12 is the (classical) mutual information of the joint probability distribution given by formula_13. 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:formula_14where formula_15 is the ensemble of states Alice is using to send information, and formula_16 is the von Neumann entropy. This formula_17 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:formula_18with formula_19 the quantum mutual information of the bipartite state formula_20. 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.
Consider the composite system that describes the entire communication process, which involves Alice's classical input formula_3, the quantum system formula_22, and Bob's classical output formula_5. The classical input formula_3 can be written as a classical register formula_25 with respect to some orthonormal basis formula_26. By writing formula_3 in this manner, the von Neumann entropy formula_28 of the state formula_29 corresponds to the Shannon entropy formula_30 of the probability distribution formula_31:
formula_32
The initial state of the system, where Alice prepares the state formula_33 with probability formula_34, is described by
formula_35
Afterwards, Alice sends the quantum state to Bob. As Bob only has access to the quantum system formula_22 but not the input formula_3, he receives a mixed state of the form formula_38. Bob measures this state with respect to the POVM elements formula_39, and the probabilities formula_40 of measuring the outcomes formula_41 form the classical output formula_5. This measurement process can be described as a quantum instrument
formula_43
where formula_44 is the probability of outcome formula_45 given the state formula_33, while formula_47 for some unitary formula_48 is the normalised post-measurement state. Then, the state of the entire system after the measurement process is
formula_49
Here formula_50 is the identity channel on the system formula_3. Since formula_52 is a quantum channel, and the quantum mutual information is monotonic under completely positive trace-preserving maps, formula_53. Additionally, as the partial trace over formula_54 is also completely positive and trace-preserving, formula_55. These two inequalities give
formula_56
On the left-hand side, the quantities of interest depend only on
formula_57
with joint probabilities formula_58. Clearly, formula_59 and formula_60, which are in the same form as formula_29, describe classical registers. Hence,
formula_62
Meanwhile, formula_63 depends on the term
formula_64
where formula_65 is the identity operator on the quantum system formula_22. Then, the right-hand side is
formula_67
which completes the proof.
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. 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.
References.
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