Quantum and classical bounds for two-state overlaps

Abstract

Suppose we have $N$ quantum systems in unknown states $|{\psi }_i⟩$, but know the value of some pairwise overlaps ${\left|⟨{\psi }_k|{\psi }_l⟩\right|}^2$. What can we say about the values of the unknown overlaps? We provide a complete answer to this problem for three pure states and two given overlaps, and a way to obtain bounds for the general case. We discuss how the answer contrasts from that of a classical model featuring only coherence-free, diagonal states, and describe three applications: basis-independent coherence witnesses, dimension witnesses, and characterisation of multi-photon indistinguishability.

arXiv comment8 pages (+8 page Appendix), 6 figures. v3: Expanded content with more examples, figures and applications

References in corpus (17)

• Quantum Machine Learning
• Photonic quantum technologies
• Quantum Coherence as a Resource
• Quantum fingerprinting
• Non-locality and Communication Complexity
• Robustness of coherence: An operational and observable measure of quantum coherence
• Is the quantum state real? An extended review of $\psi$-ontology theorems
• Testing the Hilbert space dimension
• The SWAP test and the Hong-Ou-Mandel effect are equivalent
• Sampling of partially distinguishable bosons and the relation to the multidimensional permanent
• Measurements on the reality of the wavefunction
• No $\psi$-epistemic model can fully explain the indistinguishability of quantum states
• A device for feasible fidelity, purity, Hilbert-Schmidt distance and entanglement witness measurements
• Witnessing genuine multi-photon indistinguishability
• A Survey of Quantum Property Testing
• Experimental quantification of genuine four-photon indistinguishability
• Monogamy of particle statistics in tripartite systems simulating bosons and fermions

Cited by in corpus (20)

• Quantum circuits for measuring weak values, Kirkwood--Dirac quasiprobability distributions, and state spectra
• Aspects of the phenomenology of interference that are genuinely nonclassical
• Unitary-invariant witnesses of quantum imaginarity
• A simple proof that anomalous weak values require coherence
• Quantifying n-photon indistinguishability with a cyclic integrated interferometer
• Experimental certification of contextuality, coherence and dimension in a programmable universal photonic processor
• Coherence and contextuality in a Mach-Zehnder interferometer
• Inequalities witnessing coherence, nonlocality, and contextuality
• Witnesses of coherence and dimension from multiphoton indistinguishability tests
• Contextuality with vanishing coherence and maximal robustness to dephasing
• Many-body coherence and entanglement probed by randomized correlation measurements
• Semi-device independent characterization of multiphoton indistinguishability
• Prepare-and-Magic: Semi-Device Independent Magic Certification in the Prepare-and-Measure Scenario
• Geometry of sets of Bargmann invariants
• Robustness of contextuality under different types of noise as quantifiers for parity-oblivious multiplexing tasks
• Certifying nonstabilizerness in quantum processors
• SWAP Test for an Arbitrary Number of Quantum States
• An Elementary Characterization of Bargmann Invariants
• Measure of set imaginarity
• Estimation of multivariate traces of states given partial classical information