Editing Quantum teleportation (section)

== Certifying quantum teleportation ==
When implementing the quantum teleportation protocol, different experimental noises may arise affecting the state transference.<ref>{{Cite journal |last1=Knoll |first1=Laura T. |last2=Schmiegelow |first2=Christian T. |last3=Larotonda |first3=Miguel A. |date=2014-10-28 |title=Noisy quantum teleportation: An experimental study on the influence of local environments |url=https://link.aps.org/doi/10.1103/PhysRevA.90.042332 |journal=Physical Review A |volume=90 |issue=4 |pages=042332 |doi=10.1103/PhysRevA.90.042332|arxiv=1410.5771 |bibcode=2014PhRvA..90d2332K |hdl=11336/29814 }}</ref> The usual way to benchmark a particular teleportation procedure is by using the average [[Fidelity of quantum states|fidelity]]: Given an arbitrary teleportation protocol producing output states <math>\rho_i</math> with probability <math>p_i</math> for an initial state <math>\rho=|\psi\rangle \langle \psi|</math>, the average fidelity is defined as:<ref>{{Cite journal |last1=Pirandola |first1=S. |last2=Eisert |first2=J. |last3=Weedbrook |first3=C. |last4=Furusawa |first4=A. |last5=Braunstein |first5=S. L. |date=October 2015 |title=Advances in quantum teleportation |url=https://www.nature.com/articles/nphoton.2015.154 |journal=Nature Photonics |language=en |volume=9 |issue=10 |pages=641–652 |doi=10.1038/nphoton.2015.154 |arxiv=1505.07831 |bibcode=2015NaPho...9..641P |issn=1749-4893}}</ref>

<math>\langle \overline{F} \rangle = \int \sum_i p_i F(\rho,\rho_i) d\psi</math>

where the integration is performed over the [[Haar measure]] defined by assuming maximal uncertainty over the initial quantum states <math>|\psi \rangle</math>, and <math>F(\rho,\rho_i)=\left(\text{Tr}\sqrt{\sqrt{\rho}\rho_i\sqrt{\rho}}\right)^2</math> is the Uhlmann-Jozsa [[Fidelity of quantum states|fidelity]]. 

The widely known classical threshold is obtained by optimizing the average fidelity over all classical protocols (i.e. when the sender Alice and the receiver Bob can use just a classical channel to communicate with each other). When teleportation involves qubit states, the maximal classical average fidelity is <math>2/3</math>.<ref>{{Cite journal |last1=Massar |first1=S. |last2=Popescu |first2=S. |date=1995-02-20 |title=Optimal Extraction of Information from Finite Quantum Ensembles |url=https://link.aps.org/doi/10.1103/PhysRevLett.74.1259 |journal=Physical Review Letters |volume=74 |issue=8 |pages=1259–1263 |doi=10.1103/PhysRevLett.74.1259|pmid=10058975 |bibcode=1995PhRvL..74.1259M }}</ref><ref>{{Cite journal |last1=Vidal |first1=G. |last2=Latorre |first2=J. I. |last3=Pascual |first3=P. |last4=Tarrach |first4=R. |date=1999-07-01 |title=Optimal minimal measurements of mixed states |url=http://dx.doi.org/10.1103/physreva.60.126 |journal=Physical Review A |volume=60 |issue=1 |pages=126–135 |doi=10.1103/physreva.60.126 |arxiv=quant-ph/9812068 |bibcode=1999PhRvA..60..126V |issn=1050-2947}}</ref> In this way, a particular protocol with average fidelity <math>\langle \overline{F} \rangle</math> is certified as ''useful'' if <math>\langle \overline{F} \rangle \geq 2/3</math>.<ref name="sat1400" /><ref name="Danube2004" /><ref name="Ma-2012" />

However, using the Uhlmann-Jozsa [[Fidelity of quantum states|fidelity]] as the ''unique'' distance measure for benchmarking teleportation is not justified, and one may choose different distinguishability measures.<ref>{{Cite journal |last=Popescu |first=Sandu |date=1994-02-07 |title=Bell's inequalities versus teleportation: What is nonlocality? |url=https://link.aps.org/doi/10.1103/PhysRevLett.72.797 |journal=Physical Review Letters |volume=72 |issue=6 |pages=797–799 |doi=10.1103/PhysRevLett.72.797|pmid=10056537 |bibcode=1994PhRvL..72..797P }}</ref> For example, there may exist reasons depending on the context in which other measures might be more suitable than fidelity.<ref>{{Cite journal |last1=Ribeiro |first1=G. A. P. |last2=Rigolin |first2=Gustavo |date=2024-06-14 |title=Finite-temperature detection of quantum critical points: A comparative study |url=https://link.aps.org/doi/10.1103/PhysRevB.109.245122 |journal=Physical Review B |volume=109 |issue=24 |pages=245122 |doi=10.1103/PhysRevB.109.245122|arxiv=2406.10178 |bibcode=2024PhRvB.109x5122R }}</ref> In this way, the average distance of teleportation is defined as:<ref name="Bussandri-2024">{{Cite journal |last1=Bussandri |first1=D. G. |last2=Bosyk |first2=G. M. |last3=Toscano |first3=F. |date=2024-03-22 |title=Challenges in certifying quantum teleportation: Moving beyond the conventional fidelity benchmark |url=https://link.aps.org/doi/10.1103/PhysRevA.109.032618 |journal=Physical Review A |volume=109 |issue=3 |pages=032618 |doi=10.1103/PhysRevA.109.032618|arxiv=2403.07994 |bibcode=2024PhRvA.109c2618B }}</ref>

<math>\langle \overline{D} \rangle = \int \sum_i p_i D(\rho,\rho_i) d\psi</math>

being <math>D(\rho,\sigma)</math> a well-behaved (i.e. satisfying identity of indiscernibles and unitary invariance) distinguishability measure between quantum states. Consequently, different classical thresholds exist, depending on the considered distance measure (classical thresholds for [[Trace distance]], quantum [[Jensen–Shannon divergence]], transmission distance, [[Bures distance]], wootters distance, and quantum Hellinger distance, among others, were obtained in Ref. <ref name="Bussandri-2024" />). This points out a particular issue when certifying quantum teleportation: Given a teleportation protocol, its certification is not a universal fact in the sense that depends on the distance used. Then, a particular protocol might be certified as useful for a set of distance quantifiers, and non-useful for other distinguishability measures.<ref name="Bussandri-2024" />