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Trace distance from the viewpoint of quantum operation techniques

Alexey E. Rastegin

Published May 19, 2006• 11 citations
OpenAlex

Abstract

As it is well known, the trace distance is very useful metric on quantum states. In the present paper this measure is exposed within the quantum operations formalism. Consistent proofs of its considerable properties are given. Relations between trace distance and sine distance are discussed.

Related papers

Trace distance from the viewpoint of quantum operation techniques

A E Rastegin

In the present paper, the trace distance is exposed within the quantum operations formalism. The definition of the trace distance in terms of a maximum over all quantum operations is given. It is shown that for any pair of different states there are an uncountably infinite number of maximizing quantum operations. Conversely, for any operation of the described type, there are an uncountably infinite number of those pairs of states that the maximum is reached by the operation. The behaviour of the trace distance under considered operations is studied. Relations and distinctions between the trace distance and the sine distance are discussed.

arXiv:quant-ph/0605172•Journal of Physics A Mathematical and Theoretical•Jul 19, 2007•16 citations

Metrics of quantum states

Zhihao Ma, Jing‐Ling Chen

In this work we study metrics of quantum states, which are natural generalizations of the usual trace metric and Bures metric. Some useful properties of the metrics are proved, such as the joint convexity and contractivity under quantum operations. Our result has a potential application in studying the geometry of quantum states as well as the entanglement detection.

arXiv:0812.3016•Journal of Physics A Mathematical and Theoretical•Apr 14, 2011•5 citations

Comment on “Compare Quantum Operation Sensitivity for Different Distance Measures"

Gao Yun-feng, Jian Qiang Feng

We make comments on the conclusions of the paper: “Compare Quantum Operation Sensitivity for Different Distance Measures" [ Commun. Theor. Phys. 53 (2010) 635 ], and give a physical example to describe the non-equivalence of the fidelity and the trace distance. We prove that the fidelity is not more sensitive to a quantum operation than the trace distance.

Communications in Theoretical Physics•Nov 1, 2011•0 citations

Quantifying the coherence of pure quantum states

Jianxin Chen, Shane Grogan, Nathaniel Johnston +2 more

In recent years, several measures have been proposed for characterizing the coherence of a given quantum state. We derive several results that illuminate how these measures behave when restricted to pure states. Notably, we present an explicit characterization of the closest incoherent state to a given pure state under the trace distance measure of coherence. We then use this result to show that the states maximizing the trace distance of coherence are exactly the maximally coherent states. We define the trace distance of entanglement and show that it coincides with the trace distance of coherence for pure states. Finally, we give an alternate proof to a recent result that the ${\ensuremath{\ell}}_{1}$ measure of coherence of a pure state is never smaller than its relative entropy of coherence.

arXiv:1601.06269•Physical Review A•Oct 7, 2016•34 citations

Optimal estimates of trace distance between bosonic Gaussian states and applications to learning

Lennart Bittel, Francesco Anna Mele, Antonio Anna Mele +2 more

Gaussian states of bosonic quantum systems enjoy numerous technological applications and are ubiquitous in nature. Their significance lies in their simplicity, which in turn rests on the fact that they are uniquely determined by two experimentally accessible quantities, their first and second moments. But what if these moments are only known approximately, as is inevitable in any realistic experiment? What is the resulting error on the Gaussian state itself, as measured by the most operationally meaningful metric for distinguishing quantum states, namely, the trace distance? In this work, we fully resolve this question by demonstrating that if the first and second moments are known up to an error ε , the trace distance error on the state also scales as ε , and this functional dependence is optimal. To prove this, we establish tight bounds on the trace distance between two Gaussian states in terms of the norm distance of their first and second moments. As an application, we improve existing bounds on the sample complexity of tomography of Gaussian states.

arXiv:2411.02368•Quantum•Jun 12, 2025•6 citations

Distance-Independent Universality of Clifford+T

Jens Palsberg, Keli Huang, Abdullah Almanei

A well-known theorem states that the Clifford+T gate set is universal for quantum computing. The theorem combines approximation using a distance measure on unitary matrices with equality up to global phase. Previous proofs combine these two notions for specific distance measures, but do not identify the properties that govern how they work together. Which properties are sufficient to state and prove the theorem? We answer this question by defining projective distance measures using four axioms. We prove that the universality theorem holds for every distance measure satisfying these axioms. Thus, our formulation of the theorem is independent of any particular choice of distance measure. We show that the Hilbert-Schmidt distance is already a projective distance measure. We also develop a general construction that transforms a large class of distance measures into projective distance measures and apply it to obtain projective versions of the operator-norm distance, the Frobenius distance, and the trace distance. Finally, we formalize the proof of the universality theorem in Lean.

quant-pharXiv:2610.03364•Oct 5, 2026•0 citations
In the present paper, the trace distance is exposed within the quantum operations formalism. The definition of the trace distance in terms of a maximum over all quantum operations is given. It is shown that for any pair of different states there are an uncountably infinite number of maximizing quantum operations. Conversely, for any operation of the described type, there are an uncountably infinite number of those pairs of states that the maximum is reached by the operation. The behaviour of the trace distance under considered operations is studied. Relations and distinctions between the trace distance and the sine distance are discussed.
In recent years, several measures have been proposed for characterizing the coherence of a given quantum state. We derive several results that illuminate how these measures behave when restricted to pure states. Notably, we present an explicit characterization of the closest incoherent state to a given pure state under the trace distance measure of coherence. We then use this result to show that the states maximizing the trace distance of coherence are exactly the maximally coherent states. We define the trace distance of entanglement and show that it coincides with the trace distance of coherence for pure states. Finally, we give an alternate proof to a recent result that the ${\ensuremath{\ell}}_{1}$ measure of coherence of a pure state is never smaller than its relative entropy of coherence.
Gaussian states of bosonic quantum systems enjoy numerous technological applications and are ubiquitous in nature. Their significance lies in their simplicity, which in turn rests on the fact that they are uniquely determined by two experimentally accessible quantities, their first and second moments. But what if these moments are only known approximately, as is inevitable in any realistic experiment? What is the resulting error on the Gaussian state itself, as measured by the most operationally meaningful metric for distinguishing quantum states, namely, the trace distance? In this work, we fully resolve this question by demonstrating that if the first and second moments are known up to an error ε , the trace distance error on the state also scales as ε , and this functional dependence is optimal. To prove this, we establish tight bounds on the trace distance between two Gaussian states in terms of the norm distance of their first and second moments. As an application, we improve existing bounds on the sample complexity of tomography of Gaussian states.
A well-known theorem states that the Clifford+T gate set is universal for quantum computing. The theorem combines approximation using a distance measure on unitary matrices with equality up to global phase. Previous proofs combine these two notions for specific distance measures, but do not identify the properties that govern how they work together. Which properties are sufficient to state and prove the theorem? We answer this question by defining projective distance measures using four axioms. We prove that the universality theorem holds for every distance measure satisfying these axioms. Thus, our formulation of the theorem is independent of any particular choice of distance measure. We show that the Hilbert-Schmidt distance is already a projective distance measure. We also develop a general construction that transforms a large class of distance measures into projective distance measures and apply it to obtain projective versions of the operator-norm distance, the Frobenius distance, and the trace distance. Finally, we formalize the proof of the universality theorem in Lean.