Stuck at home with a flu (great!, this way I miss the Neutel 2019 conference in Venice, among other things...), I spend an inordinate amount of hours in bed, with my brains boiling. This is no good, as typically I come out with weird ideas that, if later developed when I recover, would end up sucking more of my time.
The example I can bring from last night is a potentially groundbreaking extension of the game of chess (or maybe not). During a couple of sleepless hours I came up with part of what I am writing up here; the rest was filled in during the following daily hours...
In a chess game, every player makes a move in turn. The game is deterministic, and there is no chance factor involved whatsoever. This is one of its most appreciated qualities, in fact: the player who makes the best moves wins. But what would happen if we hacked the game by subtracting from it its quintessential logical, deterministic nature?
Enter quantum chess. In quantum chess you still alternate with your opponent in moving pieces. However, you get to decide whether the move you make will be a deterministic or a probabilistic one. In the first case you play a normal move; in the second, you instead declare two different moves (both have to be legal by the standard rules of chess, but more on that later, as we will need to hack the concept of legality a bit), starting from the position after your opponent's move(s). Once you declare them, each of the chosen piece moves will have a 50% probability of having actually being made. So, from the starting position, e.g., you can make the move
1. Ng1-f3(0.5) + e2-e4(0.5)
This means that the result on the board is now a superposition of two quantum states, each with a 50% probability. Neither you, nor your opponent, have a chance to know which of the two moves have actually been made, and hence determine what the position on the board actually is (modulo other entangled pairs of moves already having taken place), until a "measurement" of the relevant board squares is made. This can happen in several ways.
To shed some light in the apparently confusing situation, let us see some of the ways to bring the position into a certain state. Let us suppose that the game continued, after the quantum opening move by white, with the following "normal" moves:
1. ... d7-d5 2. d2-d3 e7-e5 3. Nb1-c3 Bc8-g4 4.Bc1-d2
At this point, e.g., among other choices (normal or quantum moves not affected by the real positions of the "entangled" white g1/f3 knight and e2/e4 king pawn) black has two interesting options:
Black can choose to make a deterministic move whereby one of her pieces lands in a square which has a 50% chance of being occupied by a white piece: 4. ... Bg4-(x)f3. In that case, a coin toss (or a random number throw if, as I think would be advisable, the game is played on a device capable of doing that for you) determines if the piece was actually there. The measuring act of the probing black move causes the wavefunction of the white knight, until then split between g1 and f3, to "collapse" into one of the two states. If the knight was indeed in f3 it is now captured. Otherwise, the bishop just lands in f3, and the knight "collapses" into g1. The black move has clarified where the knight really is (or was).
Either way, the piece entangled with the Nf3 by the first white move, the pawn in e2/e4, also collapses into the relevant square: if the coin toss determines that the knight had been moved to f3 on white's move 1, then the pawn is to be placed on e2; if on the other hand the knight is still in g1, it is the pawn which must have moved forward on move 1, so it collapses in e4.
Note that the possibility of collapsing the wavefunction by making a deterministic move is not shared by all pieces: a pawn, e.g., captures by moving in a way that is illegal if there is no piece to capture there. So, if black had a pawn on d5 and there were an entangled white pawn on e2/e4, the move d5xe4 would only be possible as part of a quantum move, as described below.
Black can make a move conditional to one of the two possible outcomes of the previous quantum move by white: e.g., 4....Bg4-xd1, capturing the white queen! This is of course only a legal move if the knight is _not_ in f3, and if the pawn originally in square e2 has moved away (in this case, i.e., if it is in e4), as the bishop cannot jump any white piece: in other words, the g4-d1 diagonal must be free of obstacles. Since the two conditions are entangled with one another by white's move 1 (i.e. either we have Nf3 and pawn e2, so two blockers of the diagonal, or we have Ng1 and pawn e4, and a free diagonal), the move can indeed be played conditional to the first white move having been 1.e2-e4!
When a move has a chance of being illegal, as in this case, it can be proposed by a player only as part of a quantum move, the other part being either (1) unaffected in its legality by indetermined positions of other pieces on the board, or interestingly, (2) entangled with the other proposed move in such a way that there is a guarantee that one of the two will be legal. Hence black may attempt to capture 4. Bg4-xd1 only if (1) the move is proposed together with a 100% legal move, e.g. 4. ... Nb8-c6; or (2) together with 4. ...e5-e4, a move that is legal if the alternative is not legal!
The three readers that got this far down the post will, I am sure, appreciate the surprising nature of this construction, and the endless intricacies that it may give rise to. And we have not even started, as there are many more things to consider. The most important one is related to the king.
In regular chess, the king cannot step into a square controlled by an enemy piece. In quantum chess, it is logical to postulate that it can, provided the enemy piece has a less than 100% probability of actually being in the controlling square. So, for example, in the case of the Nf3/pe4 entangled pair, and assuming now that the black king is on g6, a black move Kg6-g5 would be legal, as the knight could, with 50% chance, still be on g1 and the g5 square be free from white control; the same goes for a move like Kd6-d5, when the square d5 has a 50% chance of being controlled by a white pawn on e4. Of course such a Russian roulette kind of move could only be justified in extreme situations; but it could also allow for surprising narrow escapes!
Another rule of regular chess that requires our attention in constructing quantum chess is the en-passant capture. The en-passant capture occurs when a pawn moves forward by two squares from its original square, e.g. a white pawn e2-e4. If an enemy pawn is situated in either the d4 or f4 square, it can take the white pawn while it moves through the in-between square, with the en-passant capture d4xe3 (or f4xe3). This is the only case in chess when you can capture a piece by moving yours in an empty square.
What happens in quantum chess in that case? We need to have a look. Consider the position in which black has a pawn in f4, a pawn in g2 and a king in b6. If the move e2-e4 is a quantum move, the opponent can propose to take en-passant (e.g. f4xe3) only (with an exception we discuss below) by proposing it as part of a quantum move, since f4xe3 as a regular move would risk being illegal. Her en-passant capture must, as before, be paired with (1) a 100% legal move (say, Kb6-c6, if the square c6 is 100% free from white control) or (2) with a move involving the probing of the quantum piece entangled with the e2-e4 move, if this has the attribute of guaranteeing legality. Suppose the move entangled with e2-e4 was again the move Ng1-f3: the knight is in f3 only if the pawn has not moved e2-e4, therefore black can propose f4xe3 together with g2-g1=Q, too, as the pawn can advance to g1 if the knight has freed that square. Note that proposing g2-g1=Q together with f4xe3 would not be allowed if the g1 square were not certain to be free: but it is, as either f4xe3 is legal, or the knight has vacated g1.
The exception I mentioned above is simple to explain. Suppose white had proposed the quantum pair e2-e4 (0.5) + e2-e3 (0.5). The f4 pawn can now confidently make the move f4xe3, and this need not be a member of a pair of quantum moves, but a fully deterministic one; the black move in this case collapses the wave function of the white pawn in e3 with 100% probability!
Another interesting possibility arises when we give a check. If we do so with a quantum move whose entangled other move is not one that threatens the enemy king in its current location, the opponent needs not take action by necessity, as the threat has a 50% chance of not being real. But if the quantum move entangles two moves that both give check, that is a certainty of attack. The usual ways to parry the check of regular chess then apply, with the difference that any move that decreases below 100% the chance of leaving the king en prise is then considered legal. We are adults in the room, and we can take these risks.
Promotion and under-promotion pose no real challenge, as the cases above should clarify what happens in all cases. Of course, a quantum move can involve the promotion to the same square of the same pawn to two different pieces, just like it can involve the move of the same piece to two different squares. In both cases we end up with an entangled state; the only difference is that there is certainty about the location of the piece! This has the consequence that the promoted piece can be captured by a regular move. E.g. in the sequence 1. ... g2-g1=Q + g2-g1=N 2. Nxg1, the white move 2.Nxg1 is not a quantum move - there is no conditionality necessary.
So far we have explored specific cases of interest in dealing with quantum entanglement on the board. We have not yet discussed, however, what happens when a player wants to continue moving a piece that is entangled with another one.
First of all, note that a piece can be entangled with itself. For example, if on move 1 I propose the quantum move 1. Nf3 + Nh3, the knight is now certainly not any more in g1, and it has a 50% chance of being in either f3 and h3. Now, if white on move 2 wants to play 2.Ng5, this is a perfectly deterministic move, playable and legal in all cases, as the square g5 can be reached by both f3 and h3.
Different is the case of a move like 4.Ng5 in the sequence we examined at the beginning (1. Ng1-f3+e2-e4, d7-d5 2. d2-d3 e7-e5 3. Nb1-c3 Bc8-g4 4.Nf3-g5). Here, this of course cannot be a deterministic move - there is no certainty we can play the knight there as the knight could still be in g1. It is part of a quantum move, but then what could be the other part? Any legal move, as always. A partner in the pair can be a move like Nb8-c6, or a move by the piece originally entangled with the knight, pawn e4-xd5. Note that the pawn is on e4 if the knight is still on g1, so e4xd5 is legal if Nf3-g5 isn't.
In summary, both players have, at any point during a quantum chess game, chances to muddle waters by introducing uncertainty in the position - or if you want, multiplying the possible universes that the game lives in. This "superposition of states" of quantum-mechanical flavor should allow for a wide variety of possible swindles, traps, and gambles. Note, however, that making a quantum move does not multiply your options. By playing, e.g., 1.Ng1-f3+Nb1-c3 at the first move you are not doubling the influence of your knights: they do, in a sense, have a chance to jump, at your next move, to more squares than they would if you had only moved one of them; but each of these chances is weighted by a 50% probability. In other words, quantum moves do not increase the control of the board, in an absolute sense. What they do, though, is to increase the possibilities.
Are quantum moves "correct" in the general sense of making moves that do not worsen your position? Can they be beneficial to your position? These two are basically the same question, and I simply answer it like this: double threats!
Of course a quantum move composed of two moves that both produce a direct threat is beneficial! The opponent may be unable to parry both threats at the same time, in which case there is a chance that one of the two threat will be carried out to execution. Take for instance the following situation: you are black, and you have a rook in a4 and a rook in h4. White proposes the move pair 1.Ng1-f3 + Nb1-c3: both knights now attack a rook.You can now decide to either move to safety the rook a4, or the rook h4, or do a quantum move in turn. Of course, if you deterministically move away either rook, there is a 50% chance that white will capture the other one at the next move. But if you reply with a quantum move "saving" both rooks - 1....Ra4-a5 + Rh4-h5, white can still, at his next move, try to grab one of them. She can, in fact, propose the quantum pair 2.Nf3-(x)h4 + Nc3-(x)a4. A coin toss will determine which knight is in the actual starting position, hence which move gets played; a second coin toss will determine if the black rook is in that landing square, or if it is the one that has moved away.... White has thus a 50% chance, still, of bagging one of your rooks.
I think this is enough for an explanation of my non-dream of last night. I intend to come back to this subject in another post quite soon; and even sooner, to embellish this post with some board graphics, to ease its reading. I hope you find quantum chess as intersting as I seem to!