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IEEE International Symposium on Information Theory (ISIT)

Universal non-destructive estimation of quantum relative entropy

Alexei Kaltchenko

Published Dec 23, 2004• 2 citations
Publisher

Abstract

This paper describes the universal nondestructive estimation of quantum relative entropy. The sequences with the independent realizations of two i.i.d sources over a finite alphabet and the corresponding marginal distributions with the universal estimation is used to define the function quantum relative entropy. A product state emitted by a quantum i.i.d source with density matrix is subjected to a unitary transformation with a computational basis. The quantum entropy of a source and the eigenbasis without compromising the fidelity is learnt. With the arbitrary high accuracy the fidelity arbitrary is close to the unity

DOI
10.1109/isit.2004.1365392

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This paper describes the universal nondestructive estimation of quantum relative entropy. The sequences with the independent realizations of two i.i.d sources over a finite alphabet and the corresponding marginal distributions with the universal estimation is used to define the function quantum relative entropy. A product state emitted by a quantum i.i.d source with density matrix is subjected to a unitary transformation with a computational basis. The quantum entropy of a source and the eigenbasis without compromising the fidelity is learnt. With the arbitrary high accuracy the fidelity arbitrary is close to the unity
Motivated by the importance of entropy functions in quantum data compression, entanglement theory, and various quantum information-processing tasks, this study demonstrates how classical algorithms for entropy estimation can effectively contribute to the construction of quantum algorithms for universal quantum entropy estimation. Given two quantum i.i.d. sources with completely unknown density matrices, algorithms are developed for estimating quantum cross entropy and quantum relative entropy. These estimation techniques represent a quantum generalization of the classical algorithms by Lempel, Ziv, and Merhav.
We describe a universal information–compression scheme that compresses any pure quantum independent identically distributed (i.i.d.) source asymptotically to its von Neumann entropy, with no prior knowledge of the structure of the source. We introduce a diagonalization procedure that enables any classical compression algorithm to be used in a quantum context. Our scheme is then based on the corresponding quantum translation of the classical Lempel–Ziv algorithm. Our methods lead to a conceptually simple way of estimating the entropy of a source in terms of the measurement of an associated length parameter while maintaining high fidelity for long blocks. As a byproduct we also estimate the eigenbasis of the source. Since our scheme is based on the Lempel–Ziv method, it can potentially be applied also to non–i.i.d. sources that satisfy some further regularity conditions.
Consider a source of pure quantum states with density matrix ρ and von Neumann entropy S . It is known that the quantum information of the source may be faithfully compressed to 5 qubits/signal (asymptotically for long sequences) and that this limit is optimal if we demand high fidelity for long strings of signals. The usual method for achieving this compression, based on the typical subspace of the source, requires knowledge of the density matrix. We describe a new “universal” compression scheme which does not even require knowledge of ρ. More precisely, suppose that the source is known to have von Neumann entropy less than some given bound S 0 but is otherwise completely unspecified. Our method will faithfully compress the quantum information of any such source to S 0 qubits/signal. It is known that S is always less than or equal to the Shannon entropy H of the prior distribution of the source. According to Shannon’s theorem, H provides the limit for compression of classical signals, so one may ask the question: what is the physical origin of the extra compression available in the quantum case compared to the classical case (where the prior probabilities are the same)? One may suggest that this effect is a consequence of possible non-orthogonality of signal states. However we demonstrate that it is possible to make each pair of signal states of a source more parallel (keeping the prior probabilities the same) while increasing the von Neumann entropy. This casts doubt on the conventional wisdom that overlap of quantum states is a measure of their (non-)distinguishability. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The action of a NOT gate on a classical bit results in a change of its value from a 0 to a 1 and vice versa. The action of the classical NOT gate is in principle perfect because with fidelity equal to unity it complements the value of a bit. The action of the quantum NOT gate in a computational basis |0⟩ and |1⟩ is very similar to the action of the classical NOT gate. However, a more general quantum mechanical operation which corresponds to a classical NOT gate would take a qubit in an arbitrary state |Ψ⟩ and produce a qubit in the state |Ψ⊥⟩ orthogonal to |Ψ⟩. This operation is anti-unitary and therefore, cannot be realized exactly. So how well we can do? We find a unitary transformation acting on an input qubit and some auxiliary qubits, which represent degrees of freedom of the quantum NOT gate itself, which approximately realizes the NOT operation on the state of the original qubit. We call this ‘device’ a universal-NOT gate because the size of the error it produces is independent of the input state. We show that an optimal U-NOT gate which has as its input N identical qubits and produces M outputs achieves a fidelity of F = (N + 1)/ (N + 2), which is equal to the fidelity of estimation of the input qubits. We also show that when a priori information about the state of the input qubit is available, the fidelity of a quantum NOT gate can be much better than the fidelity of estimation.