User:PieCrate/Discrete Finite Phase Space

In mathematics and physics, discrete finite phase space is a phase space with a finite number of unique separate states. Discrete finite phase spaces are used in quantum state tomography, Grover's algorithm, quantum optics and quantum teleportation.

Discrete finite phase spaces
An alternate approach to defining discrete phase space is to first consider discrete space with finite $$N$$ isolated points. Discrete finite phase space can be defined as a discrete space where each possible state of a discrete system is represented. Each state of the discrete system correspond to an isolated point in the discrete finite phase space.

For $$N \times N$$ discrete phase space in quantum mechanics, position and momentum are both $$N \times N$$ hermitian operators, having $$N$$ orthogonal states that exist in an $$N$$ dimensional Hilbert space.

The Wigner function in phase space represents the states of a quantum system with continuous degrees of freedom. To represent the states of a discrete quantum system in discrete phase space, discrete Wigner functions must be defined. Discrete Wigner functions are of particular important for determining the state of discrete quantum systems.

Several different Wigner functions have been developed for different discrete finite phase spaces. One particular set of Wigner functions is defined on $$N \times N$$ discrete phase space in $$N = {p_{r}}^n$$ dimensional Hilbert space; $$n$$ positive integer power of a prime number, $$p_r$$ dimensional Hilbert space. Another set of Wigner functions is defined on $$N \times N$$ discrete phase space in composite $$N$$ dimensional Hilbert space.

$$N \times N$$ discrete phase space in $$N = {p_{r}}^n$$ dimensional Hilbert space
Discrete finite phase space is defined as a finite field having finite field elements p and q. The order of the finite field is given by the finite $$N$$ number of the elements of the field elements. The order $$N$$ of the finite field corresponds to the number $$N$$ of orthogonal states of the field elements which also correspond to the dimension $$N$$ of the Hilbert space.

First consider the case $$n = 1$$; prime number orthogonal states. The corresponding finite field is then given by $$\mathbb{F}_{p_{r}}$$; a finite field with prime number order. The discrete finite phase space would then be $${p_r} \times {p_r}$$. By the definition of finite fields, the states can then be labelled $$0, 1, 2, ..., p_r - 1$$, where arithmetic is performed modulo $$p_r$$.

Now consider $$n > 1$$ integer values; power of a prime number orthogonal states. The corresponding finite field is then given by $$\mathbb{F}_{{p_{r}}^n}$$; a finite field with power of a prime number order. The discrete finite phase space would then be $${p_r}^{n} \times {p_r}^{n}$$. The labels for the states for the case of $$n > 1$$ are found by explicitly constructing the finite field with order $${p_r}^{n}, n > 1$$.

The labelling and subsequent counting of the states allow for additional definitions in the discrete phase space to be established. Such definitions form the foundation for defining the set of Wigner functions for the $${p_r}^{n}$$ dimensional Hilbert space case.

Lines
For continuous phase space, a line is defined as aq + bp = c where a, b and c are aribitrary on the condition that a and b are both not zero with p and q as continuous variables. Parallel lines are then defined as lines with the same a and b but different c and by conseqence do not intersect. In $${p_r}^{n} \times {p_r}^{n}$$ discrete phase space, a line is similarly defined as a set of points satisfying an equation of the form aq + bp = c with conditions that a b and c are arbitrary elements of the finite field with a and b both not zero with p and q as field elements. Parallel lines in discrete phase space are defined as lines that do not intersect. Given that the discrete phase space has properties of a field, lines in discrete finite phase space have the following properties:.

The importance of defining lines, more specifically parallel lines, in discrete finite phase space is that a complete set of $$N$$ parallel lines, correspond to a set of $$N$$ orthogonal bases. More precisely, each complete set of parallel lines form a set of orthogonal bases for the state space which, for the case where N is an integer power of prime number, turn out to be a set of mutually unbiased bases. A more direct result of defining lines in discrete phase space is that sums over all lines in phase space allow the construction of the entire discrete Wigner function which is used to determine the state of the system.There are a total of $$N(N+1)$$ number of lines in $${p_r}^{n} \times {p_r}^{n}$$ discrete phase space. There are then $$N+1$$ sets of orthogonal bases with each set containing $$N$$ bases.
 * 1) For any two points, there is only one line which contain both points.
 * 2) For a point not on a particular line, there exists a line parallel to that particular line which the point is contained.
 * 3) For any two non-parallel lines, there is only one point which those lines intersect.

Sets of parallel lines form sets of orthogonal bases. It can then be said that each line within that set can be expressed as a corresponding projection operator of that state. The function $$M$$ is defined as as the transformation of an arbitrary line k to a corresponding projection operator. By applying the function $$M$$ to a set of parallel lines, pure state projections are obtained.

Phase point operators
For each point in $$N \times N$$ discrete phase space, there exists a phase point operator which is analogous to the phase point operator of the continuous Wigner function. In a discrete phase space of N dimensional Hilbert space, there are $$N^2$$ points in the phase space meaning there will be $$N^2$$ phase point operators. It is then required that each phase point operator $$ \hat{A}$$ can act on the state space of the system and by consequence the phase point operator is an $$ N \times N$$ matrix. In the $$N = {p_r}^{n}$$ case, the point operators are defined as:


 * 1) For each point $$ \alpha, tr(\hat{A}_{\alpha}) = 1$$ . Where $$tr$$ is defined as the trace.
 * 2) For any two points $$ \alpha $$ and $$ \beta, tr(\hat{A}_{\alpha} \hat{A}_{\beta}) = N \delta_{\alpha\beta}$$ where $$\delta_{\alpha\beta}$$ is the Kronecker delta.
 * 3) For any complete set of N parallel lines, for each line k, construct the operator $$\hat{P}_{k} = \frac{1}{N} \sum_{\alpha \in k}\hat{A}_{\alpha}$$ which is the average of all the phase point operators on that line. The $$N$$ operators $$\hat{P}_{k}$$ are a set of mutually orthogonal projection operators which sum to the identity.

Wigner function for $$N = {p_{r}}^n$$ dimensional Hilbert space
Consider a density matrix, $$\rho$$ in $$N = {p_{r}}^n$$ dimensional Hilbert space. Recalling the second definition of a phase point operator, $$\hat{A}_{\alpha}$$ are linearly independent and form a complete basis for the N dimensional Hilbert space. The density matrix is then defined by the Wigner function $$W$$, the set of values in the relation:
 * $$ \hat{\rho} = \sum_{\alpha} W_{\alpha} \hat{A}_{\alpha}$$

Further using the second definition, the Wigner function is defined as:
 * $$ W_{\alpha} = \frac{1}{N} tr(\hat{\rho} \hat{A}_{\alpha})$$

The properties of Wigner functions follow the properties of the phase point operator:


 * 1) $$ \sum W_{\alpha} = 1 $$
 * 2) Given density operators $$\hat{\rho} $$ and $$\hat{\rho}'$$ and corresponding Wigner functions $$W_{\alpha}$$ and $$W_{\alpha}'$$, $$ N \sum_{\alpha}W_{\alpha} W_{\alpha}'=tr(\hat{\rho}\hat{\rho}')$$
 * 3) Define a set of parallel lines to be the basis for the discrete phase space. The set of the sum of values along a line, k $$Pr_k = \sum_{\alpha \in k}W_{\alpha}, \forall k$$ is set of $$N$$ real number probabilities of the outcomes of specific measurements.

$$N \times N$$ discrete phase space in composite $$ N $$ dimensional Hilbert space
For composite $$N$$ dimensional Hilbert space, discrete phase space cannot be directly defined as a finite field since it is not integer power of a prime number. However, it is possible to factor N into prime numbers $$N = p_{r1}\times p_{r2}\times...\times p_{rn}$$. Using finite field theory, prime discrete sub phase spaces $$ \phi_n $$ are created from for each $$p_{rn}$$ factor. The total discrete phase space, $$\Phi$$ is defined to be the Cartesian product of the prime number discrete phase spaces, ie $$ p_{r1} \times p_{r1} \times p_{r2} \times p_{r1} \times ...p_{rn}\times p_{r1}$$. A point in the total phase space is then represented as an ordered n-tuple $$ (n_1, n_2,...n_n) $$ where $$ n_n $$ is a point in sub phase space $$ \phi_n$$. In short a point in total phase space for composite N dimensional Hilbert space is defined by points in the discrete phase spaces of the prime factors. It turns out that this method construction of phase space has similar results to the integer power of a prime number dimensional Hilbert space.

Slices
A likewise foundation must be established to express the Wigner function in N composite discrete phase space. A 'slice' which behaves analogous to the line of the integer power prime case is defined for the total discrete phase space. A slice is constructed by choosing lines in each prime dimensional phase space where the set of all points along the chosen lines is a slice. Slices are parallel if and only if for each line used to created the slice are parallel or identical. For N composites phase space, there are a total of $$p_{r1}(p_{r1}+1)p_{r2}(p_{r2}+1)...p_{rn}(p_{rn}+1)$$ different slices. These are grouped into $$(p_{r1}+1)(p_{r2}+1)...(p_{rn}+1)$$ sets of N parallel slices.

Phase point operators
Phase point operators for N composite discrete phase space are defined in such a way that they retain the same properties. Instead of lines, slices are used. The phase point operator $$ \hat{A}_{\alpha}$$ for a point $$ \alpha = \left( \alpha_1, \alpha_2,..., \alpha_n \right)$$:
 * $$ \hat{A}_{\alpha} = \hat{A}_{\alpha_1} \otimes \hat{A}_{\alpha_2} \otimes \hat{A}_{\alpha_3} \otimes ... \otimes \hat{A}_{\alpha_n}$$


 * 1) $$ tr(\hat{A}_{\alpha}) = tr(\hat{A}_{\alpha_1} \otimes \hat{A}_{\alpha_2} \otimes \hat{A}_{\alpha_3} \otimes ... \otimes \hat{A}_{\alpha_n}) = tr(\hat{A}_{\alpha_1})tr(\hat{A}_{\alpha_2}...tr(\hat{A}_{\alpha_n} = 1$$
 * 2) $$ tr(\hat{A}_{\alpha} \hat{A}_{\beta}) = tr(\hat{A}_{\alpha_1} \hat{A}_{\beta_1})...tr(\hat{A}_{\alpha_n} \hat{A}_{\beta_n}) = \delta_{11} \delta_{12}...\delta_{nn}$$
 * 3) For any complete set of N parallel slices, for each slice k, construct the operator $$\hat{P}_{k} = \frac{1}{N} \sum_{\alpha \in k}\hat{A}_{\alpha} = \frac{1}{N} \sum_{\alpha_1 \in k_1}\hat{A}_{\alpha_1} \otimes \frac{1}{N} \sum_{\alpha_2 \in k_2}\hat{A}_{\alpha_2}\otimes ... \otimes \frac{1}{N} \sum_{\alpha_n \in k_n}\hat{A}_{\alpha_n} = \hat{P}_{k_1} \otimes \hat{P}_{k_2} \otimes ... \otimes \hat{P}_{k_n}$$. The $$N$$ operators $$\hat{P}_{k}$$ are still a set of mutually orthogonal projection operators which still sum to the identity.

Wigner function for N composite dimensional Hilbert space
The definitions of the Wigner function for N composite dimensional Hilbert space are the with the only exception that for the third definition, the sum of values is along a slice:

The density matrix in N composite dimensional Hilbert space is defined by the Wigner function $$W$$, the set of values in the relation:
 * $$ \hat{\rho} = \sum_{\alpha} W_{\alpha} \hat{A}_{\alpha}$$

Further using the second definition, the Wigner function is defined as:
 * $$ W_{\alpha} = \frac{1}{N} tr(\hat{\rho} \hat{A}_{\alpha})$$


 * 1) $$\sum{\alpha} W_{\alpha} = 1 $$
 * 2) Given density operators $$\hat{\rho} $$ and $$\hat{\rho}'$$ and corresponding Wigner functions $$W_{\alpha}$$ and $$W_{\alpha}'$$, $$ N \sum_{\alpha}W_{\alpha} W_{\alpha}'=tr(\hat{\rho}\hat{\rho}')$$
 * 3) Define a set of parallel slices to be the discrete phase space. The set of the sum of values along a slice, k $$Pr_k = \sum_{\alpha \in k}W_{\alpha}, \forall k$$ is set of $$N$$ real number probabilities of the outcomes of specific measurements.