LiveGraphics3D

What you see on the right is a LiveGraphics3D Java applet. You can rotate it by dragging the mouse with the left button pressed. You can find more details about the user interface here.

Note: If you do not see anything on the right, you must install Java.

Notation

You see a toy example of a 1 ↦ 1 quantum random access code. The blue dot with label 1 at the North Pole indicates the direction of the first measurement (not the outcome "1"). The blue circle at the equator shows all points with equiprobable outcomes. The red dots with labels 0 and 1 show where the strings "0" and "1" are encoded.

Let us agree that the blue dot indicates the direction corresponding to measurement outcome zero (that is why 0 and 1 is on the Northern and Southern Hemisphere, respectively, but not vice versa).

Success probability

The probability to recover the bit correctly when its value is "0" depends on the location of 0. It is

  • 1, when 0 is at the North Pole,
  • 1/2, when 0 is on the equator,
  • 0, when 0 is at the South Pole.
In general, the closer 0 is to the North pole, the larger the probability. For 1 it is exactly opposite. We are interested in the average success probability over all strings of 0 and 1 (of length n) and all choices of the bit to be recovered. In this simple example it is just the average of two probabilities. It is equal to 0.6112605 and is shown in orange.

Optimal 1 ↦ 1 QRAC

Clearly this 1 ↦ 1 QRAC is not very good, since both 0 and 1 are close to equator (thus success probability is slightly above 1/2). The optimal encoding would be to place 0 and 1 at the North and South Pole, respectively. Then success probability would be equal to 1. Try dragging the red points with mouse and see if you can improve it! To reset everything back to the original positions, press Home key on your keyboard.

 This is simple. Here we have 3 measurements that are orthogonal to each other. They cut the sphere into four equal parts. There are two axis on poles and two on equator. They are either orthogonal or opposite. You can see only 12 red points, because the other 4 coincide with those who are on equator. The points on equator form a square (like in 2 -> 1 encoding), but the other 8 points lie on the vertices of cube (like in 3 -> 1 case) which is streched in vertical direction. Try to rotate the picture! This configuration was obtained by forcing some symmetry - using symmetric cutting of sphere into 14 parts with measurements placed on the vertices of regular tetrahedron. In this setting numerical optimization ended up with 14 points placed on the vertices of rhombic dodecahedron and 2 points placed at seemingly random places. Great circles correspond to measurements of each bit. They are placed symmetrically. Circles draw a cuboctahedron on the sphere which is dual to rhombic dodecahedron. It consists of 6 squares and 8 triangles and they all have equal side lengths. The 14 points are placed in the middle of these squares and triangles. Interactive 2 ↦ 1 QRAC

Drag the blue dots to make your own 2 ↦ 1 quantum random access code!

Blue dots determine the directions of measurements used to recover the two encoded bits. As you move them, the red dots indicating the optimal encoding will move accordingly.

Success probability of the resulting code is shown in orange. It cannot get above 0.8535533906. Maximum is reached when the measurements are performed along orthogonal directions.

To reset everything back to the original positions, press Home key on your keyboard.

 Interactive 3 ↦ 1 QRAC

Make your own 3 ↦ 1 code!

This time the success probability cannot get above 0.7886751346. Again, the maximum is reached when the measurements are performed along orthogonal directions.

Drag the blue dots to change the code. To reset everything back to the original positions, press Home key on your keyboard.

 Interactive 4 ↦ 1 QRAC

Can you make a 4 ↦ 1 quantum random access code with shared randomness?

We know that the success probability in this case can be as high as 0.7414814566. This happens when the measurements are performed along orthogonal directions, but two of them coincide.

Can you do better?

Drag the blue dots to change the code. To reset everything back to the original positions, press Home key on your keyboard.

 Interactive symmetric 4 ↦ 1 QRAC

This is an illustration of how the symmetric 4 ↦ 1 code came about. It consists of four symmetrically chosen measurements (indicated by four blue dots), that are chosen along the directions given by the vertices of a regular tetrahedron centered at the origin. The four great circles orthogonal to the measurement directions are parallel to the faces of the tetrahedron.

Drag the orange dot to move the circles up and down. Verify that they are parallel to the faces of the tetrahedron and orthogonal to the vectors pointing from the origin to the vertices of the tetrahedron.