All mutually unbiased bases in dimension six consisting of product states only are constructed. Several continuous families of pairs and two triples of mutually unbiased product bases are found to exist but no quadruple. The exhaustive classification leads to a proof that a complete set of seven mutually unbiased bases, if it exists, cannot contain a triple of mutually unbiased product bases. d ≡ d 1 d 2 ∈ {6, 10, 12, . . .} complete sets of MU bases seem to be absent. In spite of considerable numerical searches [4,5], computer-algebraic efforts [6,7], and numerical calculations with rigorous error bounds, only three MU bases have been found in dimension six, four less than the maximally allowed number [8]. Thus, the sixdimensional state space of a qubit-qutrit system appears to differ structurally from the state space of a pair of qubits (d = 4) or a pair of qutrits (d = 9).One of the few known results in dimension d = 6 is the impossibility to extend, by more that one further MU basis, the pair of MU bases consisting of the standard basis and its dual, the Fourier basis [6]. Thus, triples of MU bases are the largest sets to be found in this way. Another, more recent result [8] states that the Fourier family of Hadamard matrices together with the identity cannot be extended to a MU quadruple. These initial pairs, after non-local equivalence transformations, consist of product states only, a fact which has received little attention.Upon reflection, it seems worthwhile to systematically study MU bases in composite dimensions which contain only product states. In the present paper we carry out a comprehensive study of MU product bases in dimensions six, complementing studies devoted to the entanglement structure of complete sets of MU bases [9][10][11].More specifically, we will derive an exhaustive list of MU product bases in dimension six. The restriction to product states goes hand in hand with local equivalence transformations, or LETs, consisting of local (anti-) unitary transformations. We will find that in the space C 2 ⊗ C 3 , there is a considerable number of inequivalent product bases, a limited set of families of MU product pairs and just two triples of MU product bases. No larger MU product constellations exist. This result effectively limits the number of MU product bases contained in a hypothetical complete set of MU bases in dimension six.The argument will unfold as follows. In Sec. 2 we introduce MU product bases, specify all local (anti-) unitary transformations which map a given set of MU product states to an equivalent one, and summarise relevant properties of MU bases in dimensions two and three. Then, in Sec. 3, we derive all inequivalent product bases in C 4 and C 6 . Sec. 4 has two results on product vectors required to be MU to certain given sets of MU product vectors. These results will be important tools to enumerate all pairs and triples of MU bases in dimension four (Sec. 5) and dimension six (Sec. 6). This classification allows us to conclude, as shown in Sec. 7, that no MU product trip...

We show that a complete set of seven mutually unbiased bases in dimension six, if it exists, cannot contain more than one product basis.

Bifunctional active regions, capable of light generation and detection at the same wavelength, allow a straightforward realization of the integrated mid-infrared photonics for sensing applications. Here, we present a high performance bifunctional device for 8 μm capable of 1 W single facet continuous wave emission at 15 °C. Apart from the general performance benefits, this enables sensing techniques which rely on continuous wave operation, for example, heterodyne detection, to be realized within a monolithic platform and demonstrates that bifunctional operation can be realized at longer wavelength, where wavelength matching becomes increasingly difficult and that the price to be paid in terms of performance is negligible. In laser operation, the device has the same or higher efficiency compared to the best lattice-matched QCLs without same wavelength detection capability, which is only 30% below the record achieved with strained material at this wavelength.

An analytic proof is given which shows that it is impossible to extend any triple of mutually unbiased (MU) product bases in dimension six by a single MU vector. Furthermore, the 16 states obtained by removing two orthogonal states from any MU product triple cannot figure in a (hypothetical) complete set of seven MU bases. These results follow from exploiting the structure of MU product bases in a novel fashion, and they are among the strongest ones obtained for MU bases in dimension six without recourse to computer algebra.