Generating function and Bloch representation for quantum Fisher tensor

The gist The Uhlmann relative amplitude between two density matrices is shown to be a generating function, through which the quantum Fisher tensor that contains both the quantum Fisher information matrix and the mean Uhlmann curvature can be obtained via differentiation over system parameters.

Generating Function Approach

The paper proposes using the logarithms of the Uhlmann relative amplitude between two density matrices as generating functions for various metrological quantities, including the quantum Fisher tensor (QFIM), Quantum Fisher Information Matrix (QFIM), and Mean Uhlmann Curvature (MUC) The logarithms of the Uhlmann relative amplitude between two density matrices [20, 21], the modulus of the amplitude, and the phase of the amplitude are respectively the generating functions of quantum Fisher tensor, QFIM, and MUC Alternatively, the logarithm of the fidelity [27] between two density matrices can also serve as a generating function for the QFIM For pure states, our formalism recovers that of the quantum geometric tensor proposed by Het´enyi and L´evay [28, 29], and also clarifies the logarithms of the fidelity and phase between two quantum states as the generating functions of the quantum metric and Berry curvature.

Quantum Fisher Tensor Derivation

The paper derives a generic expression for the quantum Fisher tensor in terms of the Bloch representation of density matrices, which facilitates the calculation of the tensor, mean Uhlmann curvature, and geometric properties derived from the quantum Fisher information matrix. The QFIM is related to four times the mixed state quantum geometric tensor Fµν = 4gµν [32] and MUC Uµν defined from the SLD by Tµν = 1/4 TrρLµLν. The Bloch representation for the QFIM in arbitrary dimension is formulated using a generic form of the density matrix ρ(x) = 1/D I + r D(D − 1)/2 r · κ! and its associated operators Lµ = zµI + yµ · κ. The quantum Fisher tensor is then calculated as Tµν = 1/4 - zµzν + yT µGyν + 1/4 yT µMyν.

Pure State Limit and Geometric Properties

In the pure state limit Trρ squared = 1, the formalism recovers the generating function for quantum geometric tensor known in the literature [28, 29] The second cumulant yields the quantum geometric tensor (−i∂µ′)(i∂ν)Trww'†x=x',P T = 1/4 TrρLµLν = Tµν Furthermore, the logarithm version of this equation recovers the logarithm of the relative amplitude between two density matrices, and is the pure state limit of the more general generating function ln Trww'† proposed in the present work. The fidelity itself ⟨ψ(x')ψ(x)⟩ or its logarithm can also serve as a generating function for the quantum metric.

Applications and Geometric Findings

The formalism is demonstrated using canonical ensembles of spin 1/2 and spin 1 to show how the Bloch representation reveals geometric properties of the magnetic field. For a single spin-1/2 in a magnetic field, inserting Eq. (49) into Eq. (19) reveals several remarkable geometric properties of the 3D magnetic field manifold Firstly, the Ricci scalar is found to be a constant R = 6, meaning that the whole manifold is curved in a uniform manner. Secondly, the Ricci tensor is twice the QFIM Rµν = 2Fµν, rendering the Einstein tensor equal to the negative of the QFIM Gµν = −Fµν, and hence the vacuum Einstein equation Gµν + ΛFµν = 0, (51) is satisfied by a unity cosmological constant Λ = 1.

Conclusion

In summary, we elaborate the role of the Uhlmann relative amplitude between two mixed state density matrices as a generating function for quantum metrological properties. The Christoffel symbol and other differential geometric properties can also be obtained from higher cumulants of the generating function. Alternatively, the fidelity between two density matrices can also serve as a generating function for the QFIM. For pure states, our formalism recovers the generating function for quantum geometric tensor known in the literature and also clarifies the fidelity and phase between two two quantum states as the generating functions for the quantum metric and Berry curvature. We further formulate a generic expression for the quantum Fisher tensor and MUC based on the Bloch representation of density matrices of any dimension, which facilitates the calculation of these quantities in practice, as demonstrated by canonical ensembles of spins. The investigation on spin-1/2 reveals a constant Ricci scalar, vacuum Einstein equation, and a unity cosmological constant on the 3D manifold of magnetic field, suggesting that the Bloch representation can help to address intriguing differential geometric properties in quantum metrology. >

--- Page 1 ---

The gist The Uhlmann relative amplitude between two density matrices is shown to be a generating function, through which the quantum Fisher tensor that contains both the quantum Fisher information matrix and the mean Uhlmann curvature can be obtained via differentiation over system parameters.

How it works

The paper establishes that the logarithms of the Uhlmann relative amplitude between two density matrices [20, 21], the modulus of the amplitude, and the phase of the amplitude are respectively the generating functions of quantum Fisher tensor, QFIM, and MUC. The Christoffel symbol and other differential geometric properties can also be obtained from higher cumulants of the generating function. Alternatively, the fidelity between two density matrices can also serve as a generating function for the QFIM.

Generating Function for Quantum Fisher Tensor

The paper aims to demonstrate that the trace of the relative amplitude, or the logarithm of its trace, is a generating function for the quantum Fisher tensor and associated geometric properties The first cumulants of Eq. (12) vanish due to Eq. (3) It is because of these vanishing first moments that the moments obtained from taking derivatives on Trww'† are the same as the cumulants obtained from taking derivatives on ln Trww'†, so we simply call Cµ1µ2=C;µ=Tµν. The generating functions for the QFIM and MUC can be constructed by Fµν = (−i∂µ′)(i∂ν)Trww'† + 2Trw'w† x=x',P T = (−i∂µ′)(i∂ν) 4ReTrww'† x=x',P T = (−i∂µ′)(i∂ν) 2 ln Trww'† + 2 ln Trw'w† x=x',P T = (−i∂µ′)(i∂ν) 4 ln Trww'†.

Bloch Representation for Quantum Fisher Tensor

The Bloch representation for the QFIM in arbitrary dimension is formulated using a generic form of the density matrix ρ(x) = 1/D I + r D(D − 1)/2 r · κ! and its associated operators Lµ = zµI + yµ · κ. The quantum Fisher tensor is then calculated as Tµν = 1/4 - zµzν + yT µGyν + 1/4 yT µMyν. The real part of the QFIM gives the well-known expression for the QFIM Fµν = 2c yT µ ∂νr and the imaginary part gives the generic expression for the MUC Uµν = i/2 yT µMyν.

Pure State Limit

In the pure state limit, we see that either the fidelity or its logarithm serves as a generating function for the QFIM, in the sense that the QFIM can be obtained by taking derivatives of Trqpρ(x)ρ(x')p ρ(x) x=x' = 1/4 Fµν = gµν The second cumulant yields the quantum geometric tensor Tµν, which correctly captures the pure state limit of the quantum Fisher tensor. The generating function ln⟨ψ'ψ⟩ proposed by Het´enyi and L´evay has a profound meaning, as the logarithm of the relative amplitude between two density matrices, and is the pure state limit of the more general generating function ln Trww'† proposed in the present work. The fidelity itself ⟨ψ(x')ψ(x)⟩ or its logarithm can also serve as a generating function for the quantum metric.

1. The AI system can compute the quantum Fisher tensor (QFIM) and mean Uhlmann curvature (MUC) for any arbitrary density matrix using its Bloch representation, as derived in Eq. (32) and Eq. (34). This allows for the direct calculation of geometric properties from the density matrix parameters, facilitating practical metrological calculations.

2. The system can determine the Christoffel symbol of the first kind and other differential geometric properties like the Riemann tensor and Ricci scalar by calculating higher cumulants of the generating function, as expressed in Eqs. (17) and (19). This enables a deeper understanding of how quantum metrology manifests geometrically across parameter spaces.

3. The AI can leverage the fidelity between two density matrices as a generating function for the QFIM, utilizing Eq. (21) which states: the QFIM is the Hessian matrix of the fidelity. This allows for calculating precision bounds using only the fidelity, bypassing the need to invoke parallel transport conditions in some contexts.

4. For pure states, the system can compute a generating function for quantum geometric tensor via the logarithm of the relative amplitude between two density matrices, which is shown to be the pure state limit of the more general generating function ln Trww′† proposed in the present work (Eq. 40). This provides a direct link between state overlap and fundamental geometric concepts like Hetényi and Lévy's quantum geometric tensor.

The Uhlmann relative amplitude between two density matrices is shown to be a generating function, through which the quantum Fisher tensor that contains both the quantum Fisher information matrix and the mean Uhlmann curvature can be obtained via differentiation over system parameters. In the pure state limit, our generating function recovers that of the quantum geometric tensor proposed by Hetényi and Lévay, and also clarifies the fidelity and phase between two quantum states as the generating functions of the quantum metric and Berry curvature, respectively. A generic expression for the quantum Fisher tensor in terms of the Bloch representation of density matrices is derived, which facilitates the calculation of the tensor, mean Uhlmann curvature, and geometric properties derived from the quantum Fisher information matrix. Canonical ensembles of spins are adopted to demonstrate our formalism, which reveals a constant Ricci scalar, a vacuum Einstein equation, and a cosmological constant on the 3D Euclidean manifold of the magnetic field