For pure states, the quantum Berry curvature was well studied. However, the quantum curvature for mixed states has received less attention. From the concept of symmetric logarithmic derivative, we introduce a mixed-state quantum curvature and find that it plays a key role in the field of multi-parameter precision estimations. Through spectral decomposition, we derive the mixed-state Berry curvature for both the full-rank and non-full-rank density matrices. As an example, we obtain the exact expression of the Berry curvature for an arbitrary qubit state.For pure states, the quantum Berry curvature was well studied. However, the quantum curvature for mixed states has received less attention. From the concept of symmetric logarithmic derivative, we introduce a mixed-state quantum curvature and find that it plays a key role in the field of multi-parameter precision estimations. Through spectral decomposition, we derive the mixed-state Berry curvature for both the full-rank and non-full-rank density matrices. As an example, we obtain the exact expression of the Berry curvature for an arbitrary qubit state.

quant-pharXiv:2511.16215Communications in Theoretical PhysicsNov 21, 20250 citations

Parameter estimation for mixed states from a single copy

Thomas Konrad, Otfried Gühne, Jürgen Audretsch +1 more

Given a single copy of a mixed state of the form $\ensuremath{\rho}=\ensuremath{\lambda}{\ensuremath{\rho}}_{1}+(1\ensuremath{-}\ensuremath{\lambda}){\ensuremath{\rho}}_{2}$, what is the optimal measurement to estimate the parameter $\ensuremath{\lambda}$ if ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$ are known? We present a general strategy to obtain the optimal measurements employing a Bayesian estimator. The measurements are chosen to minimize the deviation between the estimated value and the true value of $\ensuremath{\lambda}$. We explicitly determine the optimal measurements for a general two-dimensional system and for important higher dimensional cases.Given a single copy of a mixed state of the form $\ensuremath{\rho}=\ensuremath{\lambda}{\ensuremath{\rho}}_{1}+(1\ensuremath{-}\ensuremath{\lambda}){\ensuremath{\rho}}_{2}$, what is the optimal measurement to estimate the parameter $\ensuremath{\lambda}$ if ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$ are known? We present a general strategy to obtain the optimal measurements employing a Bayesian estimator. The measurements are chosen to minimize the deviation between the estimated value and the true value of $\ensuremath{\lambda}$. We explicitly determine the optimal measurements for a general two-dimensional system and for important higher dimensional cases.

arXiv:quant-ph/0702211Physical Review AJun 1, 20072 citations

Quantum State Estimation

F. De Martini, AND SCIARRINO F., MARCO RICCI

arXiv:1707.03783Lecture notes in physicsJan 1, 2004822 citations

Preliminaries of Quantum-State Estimation

WORLD SCIENTIFIC eBooksAug 12, 20150 citations

Mixed state tomography reduces to pure state tomography

Angelos Pelecanos, Jack Spilecki, Ewin Tang +1 more

A longstanding belief in quantum tomography is that estimating a mixed state is far harder than estimating a pure state. This is borne out in the mathematics, where mixed state algorithms have always required more sophisticated techniques to design and analyze than pure state algorithms. We present a new approach to tomography demonstrating that, contrary to this belief, state-of-the-art mixed state tomography follows easily and naturally from pure state algorithms. We analyze the following strategy: given $n$ copies of an unknown state $ρ$ , convert them into copies of a purification $|ρ\rangle$ ; run a pure state tomography algorithm to produce an estimate of $|ρ\rangle$ ; and output the resulting estimate of $ρ$ . The purification subroutine was recently discovered via the "acorn trick" of Tang, Wright, and Zhandry. With this strategy, we obtain the first tomography algorithm which is sample-optimal in all parameters. For a rank- $r$ $d$ -dimensional state, it uses $n = O((rd + \log(1/δ))/\varepsilon)$ samples to output an estimate which is $\varepsilon$ -close in fidelity with probability at least $1-δ$ . This algorithm also uses poly $(n)$ gates, making it the first gate-efficient tomography algorithm which is sample-optimal even in terms of the dimension $d$ alone. Moreover, with this method we recover essentially all results on mixed state tomography, including its applications to tomography with limited entanglement, classical shadows, and quantum metrology. Our proofs are simple, closing the gap in conceptual difficulty between mixed and pure tomography. Our results also clarify the role of entangled measurement in mixed state tomography: the only step of the algorithm which requires entanglement across copies is the purification step, suggesting that, for tomography, the reason entanglement is useful is for consistent purification.A longstanding belief in quantum tomography is that estimating a mixed state is far harder than estimating a pure state. This is borne out in the mathematics, where mixed state algorithms have always required more sophisticated techniques to design and analyze than pure state algorithms. We present a new approach to tomography demonstrating that, contrary to this belief, state-of-the-art mixed state tomography follows easily and naturally from pure state algorithms. We analyze the following strategy: given $n$ copies of an unknown state $ρ$, convert them into copies of a purification $|ρ\rangle$; run a pure state tomography algorithm to produce an estimate of $|ρ\rangle$; and output the resulting estimate of $ρ$. The purification subroutine was recently discovered via the "acorn trick" of Tang, Wright, and Zhandry. With this strategy, we obtain the first tomography algorithm which is sample-optimal in all parameters. For a rank-$r$ $d$-dimensional state, it uses $n = O((rd + \log(1/δ))/\varepsilon)$ samples to output an estimate which is $\varepsilon$-close in fidelity with probability at least $1-δ$. This algorithm also uses poly$(n)$ gates, making it the first gate-efficient tomography algorithm which is sample-optimal even in terms of the dimension $d$ alone. Moreover, with this method we recover essentially all results on mixed state tomography, including its applications to tomography with limited entanglement, classical shadows, and quantum metrology. Our proofs are simple, closing the gap in conceptual difficulty between mixed and pure tomography. Our results also clarify the role of entangled measurement in mixed state tomography: the only step of the algorithm which requires entanglement across copies is the purification step, suggesting that, for tomography, the reason entanglement is useful is for consistent purification.

quant-pharXiv:2511.15806Nov 21, 20250 citations

n

copies of an unknown state

ρ

, convert them into copies of a purification

|ρ\rangle

; run a pure state tomography algorithm to produce an estimate of

|ρ\rangle

; and output the resulting estimate of

ρ

. The purification subroutine was recently discovered via the "acorn trick" of Tang, Wright, and Zhandry. With this strategy, we obtain the first tomography algorithm which is sample-optimal in all parameters. For a rank-

r

d

-dimensional state, it uses

n = O((rd + \log(1/δ))/\varepsilon)

samples to output an estimate which is

\varepsilon

-close in fidelity with probability at least

1-δ

. This algorithm also uses poly

(n)

gates, making it the first gate-efficient tomography algorithm which is sample-optimal even in terms of the dimension

d

alone. Moreover, with this method we recover essentially all results on mixed state tomography, including its applications to tomography with limited entanglement, classical shadows, and quantum metrology. Our proofs are simple, closing the gap in conceptual difficulty between mixed and pure tomography. Our results also clarify the role of entangled measurement in mixed state tomography: the only step of the algorithm which requires entanglement across copies is the purification step, suggesting that, for tomography, the reason entanglement is useful is for consistent purification.