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Physical Review A

Estimation of SU(2) operation and dense coding: An information geometric approach

Akio Fujiwara

Published Dec 12, 2001• 70 citations
Publisher

Abstract

This paper addresses quantum statistical estimation of operators $U\ensuremath{\in}\mathrm{SU}(2)$ acting on $C{P}^{3}$ as $\ensuremath{\psi}\ensuremath{\mapsto}(U\ensuremath{\bigotimes}I)\ensuremath{\psi}$ where $\ensuremath{\psi}\ensuremath{\in}{C}^{2}\ensuremath{\bigotimes}{C}^{2}.$ This is regarded as a continuous analog of the dense coding. We first prove that the quantum Cram\'er-Rao lower bound takes the minimum, and is achievable, if and only if \ensuremath{\psi} is a maximally entangled state. We next show that an SU(2) orbit on $C{P}^{3}$ equipped with the standard Riemannian structure is isometric to $\mathrm{SU}(2)/{\ifmmode\pm\else\textpm\fi{}I}\ensuremath{\cong}\mathrm{SO}(3)$ if and only if \ensuremath{\psi} is a maximally entangled state. These results provide an alternative view for the optimality of the use of a maximally entangled state.

DOI
10.1103/physreva.65.012316

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This paper addresses quantum statistical estimation of operators U\ensuremath∈SU(2)U\ensuremath{\in}\mathrm{SU}(2)U\ensuremath∈SU(2) acting on CP3C{P}^{3}CP3 as \ensuremathψ\ensuremath↦(U\ensuremath⨂I)\ensuremathψ\ensuremath{\psi}\ensuremath{\mapsto}(U\ensuremath{\bigotimes}I)\ensuremath{\psi}\ensuremathψ\ensuremath↦(U\ensuremath⨂I)\ensuremathψ where \ensuremathψ\ensuremath∈C2\ensuremath⨂C2.\ensuremath{\psi}\ensuremath{\in}{C}^{2}\ensuremath{\bigotimes}{C}^{2}.\ensuremathψ\ensuremath∈C2\ensuremath⨂C2. This is regarded as a continuous analog of the dense coding. We first prove that the quantum Cram'er-Rao lower bound takes the minimum, and is achievable, if and only if \ensuremath{\psi} is a maximally entangled state. We next show that an SU(2) orbit on CP3C{P}^{3}CP3 equipped with the standard Riemannian structure is isometric to SU(2)/\ifmmode±\else\textpm\fiI\ensuremath≅SO(3)\mathrm{SU}(2)/{\ifmmode\pm\else\textpm\fi{}I}\ensuremath{\cong}\mathrm{SO}(3)SU(2)/\ifmmode±\else\textpm\fiI\ensuremath≅SO(3) if and only if \ensuremath{\psi} is a maximally entangled state. These results provide an alternative view for the optimality of the use of a maximally entangled state.
A fundamental starting point in quantum information theory is the consideration of the von Neumann entropy and its generalization to relative quantum entropies. A particular feature of quantum relative entropies is their relation via their Hessian to monotonic Riemannian metrics on the dense set of invertible mixed quantum states (Lesniewski and Ruskai 1999). These metrics are also known as quantum Fisher information metrics and provide a direct link to quantum estimation theory (Helstrom 1969). Quantum Fisher information metrics which are extendable to pure states coincide all with the Fubini Study metric of the projective Hilbert space of complex rays. This theses outlines possible advantages of an inverse approach to quantum information theory, by starting with the Fubini Study metric rather then with the von Neumann entropy. This is done in a first step by associating to the Fubini Study metric a covariant and a contra-variant structure on the punctured Hilbert space as being available in the geometric formulation of quantum mechanics. While the contra-variant structure leads to a quantum version of the Cramér-Rao inequality for general 1-dimensional submanifolds of pure states, the covariant structure provides alternative entanglement monotones by identifying an inner product on the pullback tensor fields on local unitary group orbits of quantum states. It is shown in the case of two qubits that these monotones yield a more efficient estimation of entanglement than standard measures from the literature as those associated with the linearization of the von Neumann entropy.
Quantum error correction allows to actively correct errors occurring in a quantum computation when the noise is weak enough.To make this error correction competitive information about the specific noise is required.Traditionally, this information is obtained by benchmarking the device before operation.We address the question of what can be learned from only the measurements done during decoding.Such estimation of noise models was proposed for surface codes, exploiting their special structure, and in the limit of low error rates, also for other codes.However, so far it has been unclear under what general conditions noise models can be estimated from the syndrome measurements.In this work, we derive a general condition for identifiability of the error rates.For general stabilizer codes, we prove identifiability under the assumption that the rates are small enough.Without this assumption, we prove a result for perfect codes.Finally, we propose a practical estimation method with linear runtime for concatenated codes.We demonstrate that it outperforms other recently proposed methods and that the estimation is optimal in the sense that it reaches the Cramér-Rao bound.Our method paves the way for practical calibration of error corrected quantum devices during operation.
Ideal dense coding protocols allow one to use prior maximal entanglement to send two bits of classical information by the physical transfer of a single encoded qubit. We investigate the case when the prior entanglement is not maximal and the initial state of the entangled pair of qubits being used for the dense coding is a mixed state. We find upper and lower bounds on the capability to do dense coding in terms of the various measures of entanglement and in terms of the average mutual distinguishability of the signal states.
Ideal dense coding protocols allow one to use prior maximal entanglement to send two bits of classical information by the physical transfer of a single encoded qubit. We investigate the case when the prior entanglement is not maximal and the initial state of the entangled pair of qubits being used for the dense coding is a mixed state. We find upper and lower bounds on the capability to do dense coding in terms of the various measures of entanglement. Our results can also be reinterpreted as giving bounds on purification procedures in terms of dense coding capacities.
Ideal dense coding protocols allow one to use prior maximal entanglement to send two bits of classical information by the physical transfer of a single encoded qubit. We investigate the case when the prior entanglement is not maximal and the initial state of the entangled pair of qubits being used for the dense coding is a mixed state. We find upper and lower bounds on the capability to do dense coding in terms of the various measures of entanglement. Our results can also be reinterpreted as giving bounds on purification procedures in terms of dense coding capacities.
Version of 08 July 2026: Technically refined preprint incorporating clarifications discovered post-submission. We derive a fundamental upper bound on the number of particles that can sustain quantum coherence in a maximally entangled state from a unified geometric framework in which the observable spacetime 𝓜 is a four-dimensional continuous embedding within a higher-dimensional substratum 𝓐 with quantized information interface. Three derivations—direct application of action quantization, the quantum Cramér-Rao bound, and the quantum capacity of the 𝓜–𝓐 information channel—converge to the canonical form N_crit = n_max ℏ / (2 |λ| |Ψ_max| V₃ τ); these are the same action-quantization constraint expressed in action, metrological, and channel-capacity language: the Cramér-Rao route additionally supplies the metrological (Heisenberg 1/N) reading, but the ceiling N_crit follows in every case from action quantization, not from three independent principles. The bound predicts sharp coherence saturation independent of environmental isolation, distinguishing the framework from standard environmental decoherence. Discriminating signatures include a conjectured distinction between fermionic and bosonic systems, an abrupt knee in decoherence rate scaling at N_crit, and a trade-off N · τ ≤ K_AFT relevant to quantum computation. A natural reinterpretation of dynamical decoupling as embedding trajectory engineering connects the framework to established experimental techniques.