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arXiv:2602.14732

Projections with Respect to Bures Distance and Fidelity: Closed-Forms and Applications

A. Afham, Marco Tomamichel

quant-phSubmitted Feb 16, 2026• Announced Feb 17, 2026• 0 citations
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Abstract

We derive simple and unified closed-form expressions for projections with respect to fidelity (equivalently, the Bures and purified distances) onto several sets of interest. These include projections of bipartite positive semidefinite (PSD) matrices onto the set of PSD matrices with a given marginal, and projections of ensembles of PSD matrices onto the set of PSD decompositions of a given matrix, with important special cases corresponding to projections onto the set of quantum channels (via the Choi isomorphism) and onto the set of measurements. We introduce prior-channel decompositions of completely positive (CP) maps, which uniquely decompose any CP map into a prior PSD matrix and a quantum channel. This decomposition generalizes the Choi-Jamiolkowski isomorphism by establishing a bijective correspondence between arbitrary bipartite PSD matrices and channel-state pairs, and we show that it arises naturally from the fidelity projections developed here. As applications, we show that the pretty good measurement - associated with a weighted ensemble - is the fidelity projection of the ensemble onto the set of measurements, and that the Petz recovery map - associated with a reference state and forward channel - is the projection of a CP map (constructed from the channel-state pair) onto the set of reverse quantum channels, thereby recasting the well-known identification of the Petz map with quantum Bayes' rule in information-geometric terms. Our results also provide an information-geometric underpinning of the Leifer-Spekkens quantum state over time formalism [Leifer and Spekkens, Phys. Rev. A 88, 052130 (2013)].

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40 pages. Comments welcome!

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Quantum process tomography (QPT) protocols are often designed by lifting quantum state tomography (QST) protocols through the Choi-Jamiolkowski isomorphism, at the cost of an extra step enforcing the trace-preserving (TP) constraint. Existing approaches fall into two classes. The folklore partial normalization, which sandwiches the density estimate between inverse square roots of its input marginal, has a closed form but is used as a heuristic, its only analyses being asymptotic and losing a factor polynomial in the dimension. Projection-based methods are principled but have no closed forms and scale poorly with dimension. We resolve this dichotomy by showing that the partial normalization is the exact projection onto the Choi states of quantum channels in fidelity (equivalently in Bures or purified distance). Consequently, any QST algorithm with guarantees in fidelity lifts to a QPT algorithm that inherits its sample complexity, through a single closed-form congruence that preserves the Choi rank and is computed in Kraus form without any eigendecomposition of Choi dimension. We lift two single-copy protocols and prove that the non-adaptive one (Fidelity Projected Least Squares) recovers the Choi rank and reaches infidelity $\varepsilon$ from a number of channel uses proportional to $1/\varepsilon$. Numerically, for channels of low Choi rank, it improves on the state-of-the-art projected least squares in accuracy, and by orders of magnitude in the time and memory of the TP-regularization step and of the returned estimate.

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Matrix Results And Techniques In Quantum Information Science And Related Topics

Diane Christine Pelejo

In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at $2^n\times 2^n$ unitary matrices, which describe operations on a closed $n$-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general $2^n\times 2^n$ unitary matrix to the product of no more than $2^{n-1}(2^n-1)$ single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element $\Phi$ among a given set of quantum channels $\mathcal{S}$ that will produce the optimal value of a scalar function $D(\rho_1,\Phi(\rho_2))$, on two fixed quantum states $\rho_1$ and $\rho_2$. Some of the functions we considered for $D(\cdot,\cdot)$ are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when $\mathcal{S}$ is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers $a_1\geq\ldots\geq a_{2n}\geq 0$, we want to determine if there exist a $2n\times 2n$ density matrix $\rho$ having eigenvalues $a_1,\ldots,a_{2n}$ and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on $a_1,\ldots,a_{2n}$ for $n\leq 6$ and state some observations on general values of $n$. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that $\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}$ for given $\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_n$ and $\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_m$, (b) constructing a multipartite state $\rho$ having a prescribed set of reduced states $\rho_1,\ldots, \rho_r$ on $r$ of its subsystems, (c) constructing a multipartite state$\rho$ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix $A$ can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets $K_1$ and $K_2$ of $\IC$ given by $K_1K_2 = \{ab: a \in K_1, b\in K_2\}$, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that $K_1K_2$ is star-shaped when $K_1$ and $K_2$ are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set $K_1K_2$ is star-shaped if $K_1$ is a line segment or a circular disk.

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Choi-Jamiołkowski isomorphism(s) and channel/state duality, naturally!

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Provided a maximally entangled state vector in ${\mathcal{H}}\otimes{\mathcal{H}}$, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of ${\mathsf CP}({\mathcal{H}},{\mathcal{K}})$ (completely positive maps of density operators on ${\mathcal{H}}$ tothose on ${\mathcal{K}}$) onto the density operators on ${\mathcal{K}}\otimes{\mathcal{H}}$, which extends to a unitary map. Here, ${\mathcal{H}}$ and ${\mathcal{K}}$ are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both ${\mathcal{H}}$ and ${\mathcal{K}}$ which produces density operators on either ${\mathsf L}({\mathcal{H}},{\mathcal{K}})$ (Hilbert-Schmidt inner product) or ${\mathcal{K}}\otimes{\mathcal{H}}^*$ instead of ${\mathcal{K}}\otimes{\mathcal{H}}$. In particular, no entangled state input is required. There are no such binatural maps carried on ${\mathcal{K}}\otimes{\mathcal{H}}$, but those natural in only ${\mathcal{K}}$ are canonically associated with unitary maps ${\mathcal{H}}\rightarrow{\mathcal{H}}^*$ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.

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Continuity of certain quantum channel divergences

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Random density matrices: Analytical results for mean root fidelity and the mean-square Bures distance

Aritra Laha, Agrim Aggarwal, Santosh Kumar

The Bures distance holds a special place among various distance measures due to its several distinguishing features and has applications in diverse problems in quantum information theory. It is related to fidelity, and among other things, it serves as a bona fide measure for quantifying the separability of quantum states. In this work, we calculate exact analytical results for the mean root fidelity and mean-square Bures distance between a fixed density matrix and a random density matrix and also between two random density matrices. In the course of derivation, we also obtain the spectral density for the product of the above pairs of density matrices. We corroborate our analytical results using Monte Carlo simulations. Moreover, we compare these results with the mean-square Bures distance between reduced density matrices generated using coupled kicked tops and find very good agreement.

arXiv:2105.02743•Physical Review A•Aug 31, 2021•9 citations
We derive simple and unified closed-form expressions for projections with respect to fidelity (equivalently, the Bures and purified distances) onto several sets of interest. These include projections of bipartite positive semidefinite (PSD) matrices onto the set of PSD matrices with a given marginal, and projections of ensembles of PSD matrices onto the set of PSD decompositions of a given matrix, with important special cases corresponding to projections onto the set of quantum channels (via the Choi isomorphism) and onto the set of measurements. We introduce prior-channel decompositions of completely positive (CP) maps, which uniquely decompose any CP map into a prior PSD matrix and a quantum channel. This decomposition generalizes the Choi-Jamiolkowski isomorphism by establishing a bijective correspondence between arbitrary bipartite PSD matrices and channel-state pairs, and we show that it arises naturally from the fidelity projections developed here. As applications, we show that the pretty good measurement - associated with a weighted ensemble - is the fidelity projection of the ensemble onto the set of measurements, and that the Petz recovery map - associated with a reference state and forward channel - is the projection of a CP map (constructed from the channel-state pair) onto the set of reverse quantum channels, thereby recasting the well-known identification of the Petz map with quantum Bayes' rule in information-geometric terms. Our results also provide an information-geometric underpinning of the Leifer-Spekkens quantum state over time formalism [Leifer and Spekkens, Phys. Rev. A 88, 052130 (2013)].
Quantum process tomography (QPT) protocols are often designed by lifting quantum state tomography (QST) protocols through the Choi-Jamiolkowski isomorphism, at the cost of an extra step enforcing the trace-preserving (TP) constraint. Existing approaches fall into two classes. The folklore partial normalization, which sandwiches the density estimate between inverse square roots of its input marginal, has a closed form but is used as a heuristic, its only analyses being asymptotic and losing a factor polynomial in the dimension. Projection-based methods are principled but have no closed forms and scale poorly with dimension. We resolve this dichotomy by showing that the partial normalization is the exact projection onto the Choi states of quantum channels in fidelity (equivalently in Bures or purified distance). Consequently, any QST algorithm with guarantees in fidelity lifts to a QPT algorithm that inherits its sample complexity, through a single closed-form congruence that preserves the Choi rank and is computed in Kraus form without any eigendecomposition of Choi dimension. We lift two single-copy protocols and prove that the non-adaptive one (Fidelity Projected Least Squares) recovers the Choi rank and reaches infidelity ε\varepsilonε from a number of channel uses proportional to 1/ε1/\varepsilon1/ε. Numerically, for channels of low Choi rank, it improves on the state-of-the-art projected least squares in accuracy, and by orders of magnitude in the time and memory of the TP-regularization step and of the returned estimate.
In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at 2n×2n2^n\times 2^n2n×2n unitary matrices, which describe operations on a closed nnn-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general 2n×2n2^n\times 2^n2n×2n unitary matrix to the product of no more than 2n−1(2n−1)2^{n-1}(2^n-1)2n−1(2n−1) single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element Φ\PhiΦ among a given set of quantum channels S\mathcal{S}S that will produce the optimal value of a scalar function D(ρ1,Φ(ρ2))D(\rho_1,\Phi(\rho_2))D(ρ1​,Φ(ρ2​)), on two fixed quantum states ρ1\rho_1ρ1​ and ρ2\rho_2ρ2​. Some of the functions we considered for D(⋅,⋅)D(\cdot,\cdot)D(⋅,⋅) are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when S\mathcal{S}S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers a1≥…≥a2n≥0a_1\geq\ldots\geq a_{2n}\geq 0a1​≥…≥a2n​≥0, we want to determine if there exist a 2n×2n2n\times 2n2n×2n density matrix ρ\rhoρ having eigenvalues a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ for n≤6n\leq 6n≤6 and state some observations on general values of nnn. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k)\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k) for given ρ(1),…,ρ(k)∈Dn\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_nρ(1),…,ρ(k)∈Dn​ and σ(1),…,σ(k)∈Dm\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_mσ(1),…,σ(k)∈Dm​, (b) constructing a multipartite state ρ\rhoρ having a prescribed set of reduced states ρ1,…,ρr\rho_1,\ldots, \rho_rρ1​,…,ρr​ on rrr of its subsystems, (c) constructing a multipartite stateρ\rhoρ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix AAA can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets K1K_1K1​ and K2K_2K2​ of $\IC$ given by K1K2={ab:a∈K1,b∈K2}K_1K_2 = \{ab: a \in K_1, b\in K_2\}K1​K2​={ab:a∈K1​,b∈K2​}, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that K1K2K_1K_2K1​K2​ is star-shaped when K1K_1K1​ and K2K_2K2​ are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set K1K2K_1K_2K1​K2​ is star-shaped if K1K_1K1​ is a line segment or a circular disk.
Provided a maximally entangled state vector in H⊗H{\mathcal{H}}\otimes{\mathcal{H}}H⊗H, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of CP(H,K){\mathsf CP}({\mathcal{H}},{\mathcal{K}})CP(H,K) (completely positive maps of density operators on H{\mathcal{H}}H tothose on K{\mathcal{K}}K) onto the density operators on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, which extends to a unitary map. Here, H{\mathcal{H}}H and K{\mathcal{K}}K are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both H{\mathcal{H}}H and K{\mathcal{K}}K which produces density operators on either L(H,K){\mathsf L}({\mathcal{H}},{\mathcal{K}})L(H,K) (Hilbert-Schmidt inner product) or K⊗H∗{\mathcal{K}}\otimes{\mathcal{H}}^*K⊗H∗ instead of K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H. In particular, no entangled state input is required. There are no such binatural maps carried on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, but those natural in only K{\mathcal{K}}K are canonically associated with unitary maps H→H∗{\mathcal{H}}\rightarrow{\mathcal{H}}^*H→H∗ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.
The geometric channel fidelity, the geometric Rényi divergences and the Belavkin--Staszewski relative entropy give converse bounds for capacities, discrimination and estimation. Channels are known only approximately, yet two pairs of channels with equal sums can be arbitrarily close in the diamond norm and have fidelities 000 and 111. The remedy is the Arveson--Radon--Nikodym derivative TTT of one map with respect to the sum ΩΩΩ, whose transpose is the environment effect selecting that map in a Stinespring dilation of ΩΩΩ. In TTT, these quantities and all maximal fff-divergences of operator convex fff are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most 2∥T1−T2∥∞1/22\|T_1-T_2\|_\infty^{1/2}2∥T1​−T2​∥∞1/2​, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.
Abstract The Choi representation of completely positive trace preserving (CPTP) maps, i.e. quantum channels is often used in the context of quantum information and computation as it is easy to work with. It is a correspondence between CPTP maps and quantum states also termed as the Choi–Jamiołkowski isomorphism. It is especially useful if a parametrization of the set of CPTP maps is needed in order to consider a general map or optimize over the set of these. Here we provide a brief introduction to this topic, focusing on certain useful calculational techniques which are presented in full detail.
The Bures distance holds a special place among various distance measures due to its several distinguishing features and has applications in diverse problems in quantum information theory. It is related to fidelity, and among other things, it serves as a bona fide measure for quantifying the separability of quantum states. In this work, we calculate exact analytical results for the mean root fidelity and mean-square Bures distance between a fixed density matrix and a random density matrix and also between two random density matrices. In the course of derivation, we also obtain the spectral density for the product of the above pairs of density matrices. We corroborate our analytical results using Monte Carlo simulations. Moreover, we compare these results with the mean-square Bures distance between reduced density matrices generated using coupled kicked tops and find very good agreement.