  The main contribution of our paper is to introduce a number of multivariate
quantum fidelities and show that they satisfy several desirable properties that
are natural extensions of those of the Uhlmann and Holevo fidelities. We
propose three variants that reduce to the average pairwise fidelity for
commuting states: average pairwise $z$-fidelities, the multivariate
semi-definite programming (SDP) fidelity, and a multivariate fidelity inspired
by an existing secrecy measure. The second one is obtained by extending the SDP
formulation of the Uhlmann fidelity to more than two states. All three of these
variants satisfy the following properties: (i) reduction to multivariate
classical fidelities for commuting states, (ii) the data-processing inequality,
(iii) invariance under permutations of the states, (iv) its values are in the
interval $[0,1]$; they are faithful, that is, their values are equal to one if
and only if all the states are equal, and they satisfy orthogonality, that is
their values are equal to zero if and only if the states are mutually
orthogonal to each other, (v) direct-sum property, (vi) joint concavity, and
(vii) uniform continuity bounds under certain conditions. Furthermore, we
establish inequalities relating these different variants, indeed clarifying
that all these definitions coincide with the average pairwise fidelity for
commuting states. Lastly, we introduce another multivariate fidelity called
multivariate log-Euclidean fidelity, which is a quantum generalization of the
Matusita multivariate fidelity. We also show that it satisfies most of the
desirable properties listed above, it is a function of a multivariate
log-Euclidean divergence, and has an operational interpretation in terms of
quantum hypothesis testing with an arbitrarily varying null hypothesis.
