If this was a normal game and the white bishop was on f7, white would have, among other options, a mate in 7 starting with Ba2+. This also collapses the bishops into one so he doesn't have to worry about which of the bishops is real. So you might think that white could simply do this.
But it's not quite that simple. Since this is quantum chess, black could play Rf^g7, which has a 50% chance of blocking the check, attacking the queen, and getting the rook out of danger. White could respond with Bxb1, and now black is in a quandary - if he plays Rxa7 and it works, that proves that his rook is on g7 and not on f5 and white can play Bxh7 and win the game, and if it doesn't work, that proves his rook is on f5 and not on g7 and white can play Qxh7 winning the game. (If I understand the measurement rules correctly, white will know which of these has happened.) So, black can't capture the queen. If he simply moves his king, white can simplify with Qxg7 followed by Bxf5 with an endgame up a bishop and a pawn (in which black will likely clone his king on the dark squares to prevent the white king from safely advancing... and with potential multiple kings it prevents any sort of meaningful opposition... hmm, maybe simplification isn't as good as I thought. But even without simplification, white has won one of the rooks, which is huge.)
If after Ba2+ Rf^g7 Bxb1, black tries something similar to the series of quantum checks I describe later, by starting with R^d2+ Kg3 R^h3+, it doesn't work because of the pin. White can disarm the attack by trying Bxh7, which has a 3/4 possibility of winning outright, and if it doesn't it still measures and fixes the position of black's rook on f5. Black can play R^g7 again, but the position is not the same, and after Bxf5 black loses half a rook and has no quantum checks available.
But black also has the option of Rb^g7. This potentially attacks the white queen, and now white must retreat the queen or correctly guess the location of the rook. But if white guesses wrong, it's a disaster and he loses the queen. So if white wanted to guess, he should have just done so right away, when it didn't involve losing his queen.
So, maybe Ba2 isn't the way to go for white. Let's look at the seemingly simple gxf5. It wins a rook. Black seems to be toast - but, he has some tricks. He can start with R^f1 which may or may not be check, and either taking it or ignoring it gives white a 50% chance at an immediate loss. So, white must respect the threat, and retreats with Kg3. Black can then put a quarter-rook on h3 with R^h3. White responds with Kxh3, and black puts 1/8 rook on h1 with R^h1. White retreats again with Kg3, and black plays R^g5, placing 1/16 of a rook on g5. Now white's king has nowhere that is completely safe. White can try B^g4 to create additional interference - now, black has only a 1/32 chance of Rxg3 winning. But 1/32 is way better than nothing. While black may have better moves, he can force this after gxf5, meaning black has at least a 1/32 chance of winning after gxf5.
Black could also attempt to attack the queen instead of the king. After gxf5 R^b7 Qc5 R1^b5 Qf8, black can attempt Rg7, entangling with the f7 bishop. White can now attempt Qg8 with a 3/4 chance of winning, but there's a 1/4 chance that the g7 rook is real, resulting in Rxg8 Bxg8 (we know it's the a2 bishop because of the entanglement) Kxg8 and now you have a 3 pawn vs 2 endgame. In normal chess this would likely win for white, but here it's a tossup. So that line overall has about a 7/8 chance of winning for white. But this line isn't forced.