Joël Abrahams

Why two editions

• Quantum key distribution rests on this difference.
• BB84 uses superposition. An eavesdropper who measures a state disturbs it. The two parties notice.
• Ekert's protocol uses entanglement. The parties test a Bell inequality. If it is violated, no local hidden variable fixed the outcomes in advance. So nobody could have known the key beforehand.

Playing

• Above the board: the pieces you took on the left, the ones you lost on the right. The number is the material lead. Pawn 1, knight and bishop 3, rook 5, queen 9.
• Under Game, the uncertainty counts the opponent's pieces you are unsure of. It also gives your missing information in bits. See The bits.
• Point at a piece to see its chances. The settings can turn the card off.

The bits

• Entropy counts how unsure you are, in bits. One bit is one fair coin flip you cannot call.
• Shannon's formula takes the average surprise: $$H=-\sum _i{p}_i{\log }_2{p}_i$$ A possibility with chance $p_i$ has surprise ${\log }_2(1/p_i)$ bits. The average over all possibilities is H.
• It depends on the model of the opponent. It measures your uncertainty, not a property of the position.
• Measuring a piece takes its entropy to 0.

Chess 960

• Chess 960 shuffles the back rank. Bishops stand on opposite colours. The king stands between the rooks. The pawns stay put.
• There are 960 starting positions. Both sides get the same one.

Double Chess 960

• Each side gets its own shuffled back rank. Most games start unbalanced.
• In the hidden variable edition your opponent's back rank starts hidden. It is drawn from odds you can set before the game, the prior. Pinning a piece reveals it.
• In the quantum edition nothing starts hidden.

The opponent's turn

1. Collapse. Each cloud that your pieces can see is measured. It is found on a square, or it stays a cloud. The king is always measured.
2. Move. Stockfish plays on the pieces that are on the board.
3. Spread. Every piece spreads into a cloud again. That is what you see on your turn.

Clouds

• Every piece of the opponent is a cloud, the king and the pawns too.
• A cloud has an amplitude on every square. The chance of finding it there is the squared magnitude of that amplitude. This is the Born rule.
• At the end of its turn each such piece spreads over the squares it could move to. This is a quantum walk. Amplitudes interfere: a cloud piles up on some squares and thins out on others. A mixture of chances cannot do that.
• So a piece that just moved already fades. Over a game, pieces dissolve and are found again.
• A cloud never goes onto a piece or onto a square that gives check. Those squares are cut out and the rest is rescaled to 1. That step is a measurement, not part of the walk.
• A pawn only moves forward, but a walk goes both ways. So the cloud of a pawn can spread back along its file. It never goes to the first or the last rank.
• The king is the exception in a turn. It is always measured before the engine plays.
• Growth, in the settings, sets how far a cloud spreads.

Measuring

• At the start of its turn, each cloud is measured against what your pieces attack. The question is: are you on one of these squares? With the chance it has there, it is found. Then it lands on one of them, drawn by those chances. Otherwise it is not there, and it goes on with those squares cut out.
• Your moves measure too. Every square your piece passes over and lands on is measured.
• A piece found on the target is taken. Take the king and you win.
• A piece found on the way blocks the move. The move fails and you lose the turn.
• Nothing decides an outcome in advance. It is drawn when the measurement happens.

Pawns get a strange power here. A pawn can move diagonally without capturing. Attacking a cloud is an attack, even though the piece may not be there when it collapses. If it is not, the pawn just moves to the empty square. This only works where a cloud has a chance.

The engine

• Stockfish never sees a cloud. It plays the pieces on the board: yours and the clouds just found. That includes the king, which is always measured, whatever settle says. The engine can not play without it. So its position is new every turn.
• Settle is an extra rule. A cloud whose likeliest square has at least the chosen chance is measured anyway. That gives the opponent pieces to play. Without it, only pieces you attack land on the board, and the king. The opponent has little to move.
• Mate and stalemate count the pieces on the board, as the engine sees them.
• There is no undo, replay or export. The engine plays a new position every turn. That is not one game you could replay.

The bits

• The bits are the Shannon entropy of where each cloud would be found. The chances are the squared amplitudes.
• The clouds are independent, so the total is the sum. It is exact for this model, not an upper bound.
• A cloud is a pure state, with zero von Neumann entropy. The bits measure how unpredictable looking is.

Measuring

• A possibility is a kind of piece on a square. For example, a knight on c6.
• Attacking a piece reveals it. It was that piece all along.
• Giving check is a partial measurement. It narrows down what a piece may be.
• A piece with one possibility left is known by deduction. It is revealed without being measured.
• Measuring one piece tells you about the others. The back rank ties them together.

Possibilities

• You are not shown the move of a hidden piece. It is drawn on every square it could have reached unmeasured. One possibility is real. The rest are phantoms.
• You measure a piece by attacking it. You measure it partly when it gives check, takes something or changes your legal moves. A possibility that would have been measured is gone.
• A new game asks how many extra possibilities a move may show. 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 an unmeasured move, every opponent piece is drawn on every square it could have reached, pawns and found pieces too. It has no cap and is much harder.

Playing

• The card of a piece has two columns. The first adds up to 100%, with the chance of a phantom: nothing there. The second assumes the piece is real. These are estimates; see The odds.
• A move made illegal by a hidden piece is refused. That possibility is ruled out.
• Undo takes back your move and the reply. What it measured is hidden again, but you remember it. That makes it a way to cheat.

The odds

• The odds are ordinary probabilities, not quantum ones. They come from a simple model of the opponent. Every legal move is equally likely, which is the least informative assumption.
• A line through a hidden piece counts the chance that it is clear. Pins and check count once the king is known.
• The odds never look at the real board. They cannot leak what the engine knows.
• Stockfish prefers strong moves, so these are estimates. They can be well off when one move is obvious.
• Every possibility starts with its own chance. All alike, or the odds you set in Double Chess 960. Ruling something out rescales the rest to 100%.
• A piece that moved unseen was more likely one with many quiet moves.
• The pieces share one back rank. Learning about one tells you about the others.
• Some shortcuts are taken. Hyper mode weighs each piece against an average picture of the others. Two hidden pieces cannot share a square. Those chances come from sampling many possible worlds (Monte Carlo). They can be off by about a percentage point.

Animation

Move animation

Once On repeat

A moved piece is shown travelling to each possibility.

Superposition

Growth

How far a cloud spreads in one turn, as the time of its quantum walk. Low, and pieces hardly leave their squares. High, and they spread over the board, and the opponent has fewer pieces on it to play with. The default is 0.5.

Settle

Settle nearly certain clouds

A cloud whose likeliest square has at least this chance is measured anyway, so that it is on the board and the engine can move it. Without it only the pieces you attack ever land on the board, and the opponent has little to move. The king is the exception: it is always measured before the engine plays, whatever this says. This is an extra rule, not quantum mechanics: it is the easiest one to play without. The default is on, at 0.5.

Defaults