Joël Abrahams

The game

• Ordinary chess, with no castling for now.
• The engine sees the real board. Every piece has a real state, and "superposition" only means it is hidden from you.
• A piece in superposition is drawn as everything it could be, all at once. The settings can make it cycle through them instead, or add a small red bar of odds under each.

Strictly speaking, this is not superposition. A piece has a real state, and measuring it only reveals what it was all along. What it models is ignorance, a statistical mixture, which is not a superposition. We still say superposition because that word is far better known. It is also a nod to my years of studying quantum information theory and working at a quantum tech startup. And quantum just sounds cool.

We are aware that the engine is not a quantum computer. But even a quantum computer would not make this superposition: real superposition has amplitudes that can interfere, and nothing here does. So this is only an analogy for superposition.

Measuring

• A possibility is a kind of piece on a square, such as a knight on c6.
• Attacking a piece reveals it: the possibilities "collapse" to what it was all along.
• Receiving a check is a partial measurement: it narrows down what a piece may be, though it may still remain in superposition.
• Measuring one piece can tell you about the others, because the back rank ties them together. That is ordinary correlation, not entanglement: nothing here could violate a Bell inequality.
• A piece also "collapses" when only one possibility is left for it: it is then known by deduction, without being measured.

Possibilities

• You are not shown a move of a piece in superposition. It spreads to every square it could have reached without being measured. One possibility is real, the rest are phantoms.
• You measure a piece when you attack it, and partly when it gives check, takes something, or changes your own legal moves. A possibility that would have been measured is gone.
• A new game asks how far pieces spread. With a cap, possibilities are drawn at random, the real one included. What is not drawn counts as ruled out.
• Hyper mode hides which piece moved. After a move that is not measured, every opponent piece spreads, pawns and found pieces too. It has no cap and is significantly harder.

Playing

• Point at a piece to see what it could be. The first column adds up to 100%, with the chance of a phantom, which is the chance that nothing is there. The second assumes the piece is real. These are estimates; see Part I of The odds and the bits.
• A move made illegal by a hidden piece is refused. That possibility of what could have made that move collapses.
• Undo takes back your last move and the reply. What it measured is hidden again, but you still remember it. That makes it a way to cheat.
• Above the board: the pieces you took on the left, the ones you lost on the right. The number is the material lead, in the usual values: pawn 1, knight and bishop 3, rook 5, queen 9.
• Under Game, the uncertainty counts how many opponent pieces are still hidden. It also gives how much you do not know about them in bits, as an upper bound; see Part II of The odds and the bits.

Chess 960

• Chess 960 shuffles the back rank, with the bishops on opposite colours and the king between the rooks. The pawns stay where they are. That gives 960 possible starting positions, and both sides get the same one.

Double Chess 960

• Each side gets its own shuffled back rank, so most games start unbalanced.
• Your opponent's back rank starts hidden: it is drawn at random with the odds you set, which are the prior. Before the game you can change the odds of where their pieces go. If you pin a piece, it will not start in superposition.

The odds and the bits

Part I: The odds

• The odds are ordinary probabilities, not quantum ones, from a simple model of the opponent. It takes any legal move as likely as any other, which is the least informative assumption. A line with a hidden piece on it counts for the chance that the way is clear, and pins and check count once the king is known. The odds never look at the real board, so they cannot give away what the engine knows.
• Stockfish prefers strong moves, so these are estimates, and they can be well off when it has one obvious move. So in principle, yes, Stockfish could have hung its queen. However, the odds make that very unlikely, though never impossible. A later version may weigh each move by how good the engine thinks it is.
• Every possibility starts with a chance of its own: all alike, or the odds you set in Double Chess 960. Ruling something out rescales what is left to 100%. A piece that moved unnoticed was more likely a kind with many quiet moves than one with few. What you learn about one piece says something about the others, because they share one back rank.
• Some shortcuts are taken. In hyper mode each piece is weighed against an average picture of the others, not every combination of them. Two hidden pieces cannot share a square: the chances are worked out by trying many possible worlds (Monte Carlo sampling), so they can be off by a percentage point or so.

Animation

Move animation

Once On repeat

A moved piece is shown travelling to each possibility.