Terra incognita of the quantum dimension – „Nonlocal grains of rice“

We epitomise the dimensions of a qubit using grains of rice. Thus, each grain of rice corresponds to a free parameter, that is, a single dimension within the quantum dimension. The first qubit has four dimensions. Based on the condition P■ + P□ =1, there are three freely selectable parameters. These three dimensions, two phases and one probability, are locally assigned to the one qubit.

Next, we assign a number to each chessboard field. We designate the first field qubit 1{3, 0}. Only two of these three dimensions can be observed in the experiment, as only phase differences lead to observable effect. The so-called “Bloch sphere” is often used to describe observable states of a single qubit.

The combination of two Q-bits has eight dimensions; only seven of them are independent of one another. Three dimensions are assigned locally per qubit. One red grain of rice, that corresponds to a non-local dimension, is left over. The state Omega |Ω> represents this dimension. All other quantum states in this seventh dimension can be derived from Omega by local, complex rotations. We can think of this as a family tree, with relatives connected with one another via certain routes. In particular, all other Bell states can be generated from Omega in this way.

|Ω>≡|Ф^+> =1/√2(|00> +|11>)

|Ф^-> =1/√2(|00> -|11>)

|Ψ^+> =1/√2(|01> +|10>)

|Ψ^-> =1/√2(|01> -|10>

It is therefore sufficient to associate Omega to this non-local dimension, as a representative of the family, so to speak.
Alice and Bob are always assigned locally to one chessboard field. The quantum state Omega is assigned non-locally to both chessboard fields. We describe the two first fields, as a unit, with the number qubit 2{6, 1}, where the first number in curly brackets describes the number of local dimensions, and the last number the number of non-local dimensions.

Now we proceed a step further, and look at the combination of three qubits with 16 dimensions, of which 15 are independent. Three dimensions are assigned locally per qubit. Six red grains of rice are left over, as representatives of the non-local dimensions.
Which quantum states arise in these six non-local dimensions? There are 3 possibilities for selecting 2 fields out of these 3 chessboard fields. For each of these 3 combinations, there is one respective Omega-like quantum state, which connects these two fields. These are the first three red, non-local dimensions. Three non-local dimensions are left over. Oscillations which connect not only two, but all three, fields non-locally can be found in these dimensions. It has been shown that there are exactly two types of entangled states that can connect all three fields with one another. The first type is represented by the so-called GHZ state. “GHZ” is short for the three physicists Greenberger, Horn and Zeilinger.

|GHZ> = 1/√2(|000> +|111>)

The GHZ state was realized experimentally for the first time in 1998. The second type is represented by the W state. “W” stands for the physicist Wolfgang Dür.

|W> = 1/√3(|100> +|010> + |001> )

These two states describe the entanglement of three particles. To observe them, therefore, not only the detectors Alice and Bob would be necessary, but also a third detector at a further location. The sixth non-local dimension, being an irrelevant phase, does not lead to any observable effect. Thus, we have completely described the entanglement of a system consisting of 3 qubits.

As a next step, we will look at 4 qubits. Here, something exciting happens: For the first time, the number of non-local dimensions prevails. Out of the 32 – 1 = 31 dimensions, 
4 • 3 = 12 are local, and 31 – 12 = 19 non-local. We designate the combinations of four chessboard fields with the qubit number 4{12, 19}.

The deeper we explore the quantum dimension, the more dimensions are non-local. Whereas the number of local dimensions only increased by three per qubit, the number of non-local dimensions increase exponentially.
The invisible dimensions are doubled with each further field, and point the way to a gigantic quantum world, many aspects of which are still unknown. Expressed in our metaphor of the red grains of rice which increase exponentially, the “volume” that the grains of rice absorb is doubled with each field. The comparison with the volume of billions and trillions of grains of rice can only show what exponential growth means. Each red grain of rice is, in itself, a small universe, a dimension that is interwoven with these billions of other dimensions in a complex and fascinating way.