# Metric for Estimating Congruity between Quantum Images +
## 4 Experimental Validation of the QIFM Metric
### PSNR versus Quantum Image Quality Metrics (QIFM (F), Similarity Score (S) and SIM Values)  
The antinomy that pervades information processing in the digital and quantum realms imposes the need for a different set of rules must be used for quantum state preparation, transformation and recovery [2; 29]. In QIP, once an image is prepared its transformation is only possible via a set of unitary operations as dictated by the algorithmic framework generating the required image application or task [14; 16; 17]. The final step in QIP is quantum state recovery, which is perhaps the most delicate undertaking since, as noted in earlier sections of this paper, the content encoded in the image register could be lost if it is disturbed during any uncontrolled measurement. In addition to the possible collapse of the quantum state, such uncontrolled measurements have inescapable error terms integrated into them. Without necessary refinement, using the digital PSNR image quality metric to assess congruity between two quantum images is both impracticable and non-executable because of the mismatch in terms of its present formulation with the procedures of quantum information processing. This makes the PSNR image quality measure an ill-suited and irrational choice for use in QIP. Notwithstanding this, since, in QIP, classical (digital) copies of images used as references to prepare and initialize their quantum versions [11; 12; 16; 17; 18; 19], it would suffice that the PSNR obtained from the measurement of likeness between classical cover and test image pair should serve as a reference or benchmark to assess fidelity between the quantum versions of the same image pair. Thence, a bivariate statistical analysis could be utilised in order to establish the strength of association or correlation between the quantum metrics (i.e., the QIFM (\(F\)), [PERSON] et al. 's probability-based score (\(S\)) and recalibrated similarity (i.e., the _SIM_ values)) assessing the likeness between a cover and test images pair with their digital counterpart, i.e., the PSNR, obtained for the same pair of images. In trying to establish this relationship, however, we must be mindful of the conflict in the measurement scales used in the PSNR and the trio of quantum image quality metrics that are used in our FPS analysis. While the quantum image quality metrics are measured on a normalised linear scale bounded in the range 0 to 1, the PSNR is measured on a logarithmic scale that spans an infinite length. Therefore, it is imperative that we must find some intermediate yet impartial platform for our correlation analysis. As defined in Equation (15), the PSNR is composed of constant terms having an asymptotic relationship with the mean square error (_MSE_) value with which it varies inversely and, is defined as the squared deviations of a prediction from its true values. This implies that the PSNR and _MSE_ are equivalent measures with asymptotic behaviour in terms of appropriated normalised or linear measurement scales. Moreover, since the _MSE_ can be normalised and reversed (_rNMSE_) as presented in Equation (16) [30], as an equivalence of the PSNR would provide an impartial platform for comparison with the other quantum image quality metrics. \[rNMSE=1-\frac{\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{n-1}1\, \mathnormal{1}\,I_{i,j}-K_{i,j}\,\mathnormal{1}\,^{2}}{\sum\limits_{i=0}^{m-1 }\sum\limits_{j=0}^{n-1}1\,\mathnormal{1}\,I_{i,j}\,\mathnormal{1}\,^{2}} \tag{17}\]  
where _rNMSE_ is the reversed _NMSE_ that fits in the interval of [0, 1] and in terms of quantitative inference a measurement of _rNMSE_ = 1 means that the two images being compared are exact matches. Using Equation (17), _rNMSE_ measurements for the cover and test image pairs in the first column of Table 1 are presented in the rightmost (i.e., the last) column of the same table. Having provided a more practical equivalence for the PSNR values, we have an impartial platform for our analysis between the digital image quality measures and their quantum counterparts.