# States

TODO: INTRO

## Basic State

Describes the Fock state of $$n$$ indistinguishable photons over $$m$$ modes.

See reference perceval.utils.BasicState for detailed information.

A Fock state, called BasicState in Perceval coding language, is represented by |n_1,n_2,...,n_m> notation where n_k is the number of photons in mode k.

Example code:

>>> bs = pcvl.BasicState("|0,1>")      # Creates a two-mode Fock state with 0 photons in first mode, and 1 photon in second mode.
>>> print(bs)                          # Prints out the created Fock state
|0,1>
>>> bs.n                               # Displays the number of photons of the created Fock state
1
>>> bs.m                               # Displays the number of modes of the created Fock state
2
>>> bs[0]                              # Displays the number of photons in the first mode of the created Fock state ( note that the counter of the number of modes    starts at 0 and ends at m-1 for an m-mode Fock state)
0
>>> print(pcvl.BasicState([0,1])*pcvl.BasicState([2,3]))  # Tensors the |0,1> and |2,3> Fock states, and prints out the result (the Fock state |0,1,2,3>)
|0,1,2,3>


## Annotated Basic State

AnnotatedBasicState extends BasicState and describes state of $$n$$ annotated photons over $$m$$ modes.

See reference perceval.utils.AnnotatedBasicState for detailed information.

### Annotation

Annotation distinguishes individual photons and is represented generically as a map of key $$\rightarrow$$ values - where key are user-defined labels, and values can be string or complex numbers.

Special predefined annotations are:

• P for polarization used in circuit with polarization operators - see Polarization

• t used in time circuit with an integer value is defining the period from where the photon is generated (default 0 meaning that the photon is coming from current period).

Photons with annotations are represented individually using python dictionary notation:

{key_1:value_1,...,key_k:value_k}

key_i will be referred to as an annotation key, and represents for example a degree of freedom labelled i ( time, polarization,…) that a photon can have, value_i is the value on this degree of freedom. Note that a photon can have a set of annotation keys, representing different degrees of freedom, each with its own value.

Note

Two photons are indistinguishable if they share the same values on all their common annotation keys, or if they have no common annotation keys. For instance, for the following three photons,

• $$p_1=$$ {a1:1}

• $$p_2=$$ {a2:1}

• $$p_3=$$ {a1:2,a2:2}

$$p_1$$ and $$p_3$$ are distinguishable because their a1 annotation keys have different values (1 for p_1 as opposed to 2 for p_3). $$p_2$$ and $$p_3$$ are also distinguishable because the values of their annotation key a2 do not agree. However, $$p_1$$ and $$p_2$$ are indistinguishable, because they share no common annotation keys.

### Use of Annotation in AnnotatedBasicState

A AnnotatedBasicState notation extends the BasicState notation as following:

|AP_(1:1)...AP_(1:n_1),...,AP_(m:1)...AP_(m:n_m)> where AP_(k:i) is the representation of the i-th photon in mode k, n_i is the number of photons in mode i.

To simplify the notation, for each mode, annotated photons with same annotations can be grouped and represented prefixed by their count: e.g. 2{...}. Absence of photons is represented by 0, and non annotated photons are just represented by their count as for BasicState.

For instance the following are representing different BasicStates with 2 photons having polarization annotations (here limited to H/V:

• |{P:H},{P:V}> corresponding to an annotated |1,1> BasicState.

• |2{P:H},0> corresponding to an annotated |2,0> BasicState where the 2 photons in the first mode are annotated with the same annotation.

• |{P:H}{P:V},0> corresponding also to an annotated|2,0> BasicState where the two photons in the first mode have different annotations.

• |1,{t:-1}> corresponding to an annotated |1,1> BasicState, the degree of freedom being time, with the photon in mode 2 coming from previous period, and the photon in mode 1 is not annotated.

Example code:

>>> print(pcvl.AnnotatedBasicState("|0,1>"))
|0,1>
>>> a_bs = pcvl.AnnotatedBasicState("|{P:H}{P:V},0>")   # Creates an annotated state |2,0> , with two photons in the first mode, one having a horizontal polarization, and the other a vertical polarization.
>>> print(a_bs)
|{P:H}{P:V},0>
>>> a_bs[0]                      # prints the photons in the first mode
({"P":"H"},{"P":"V"})
>>> print(a_bs.clear())     #prints the non-annotated Basic state corresponding to a_bs
|2,0>


## State Vector

StateVector extends AnnotatedBasicState to represents state superpositions.

See reference perceval.utils.StateVector for detailed information.

StateVector instances are constructed through addition and linear combination operations.

>>> st1 = pcvl.StateVector("|1,0>")   # write basic states or annotated basic states with the 'StateVector' command in order to enable creating a superposition using the '+' command
>>> st2 = pcvl.StateVector("|0,1>")
>>> st3= st1 + st2
>> print(len(st3))
2
>>> print(st3)
1/sqrt(2)*|1,0>+1/sqrt(2)*|0,1>
>>> st3[0]    # outputs the first state in the superposition state st3
|1,0>
>>> st3[1]     # outputs the second state in the superposition st3
|0,1>
>>> st4 = alpha*st1 + beta*st2


Warning

StateVector will normalize themselves so it will add normalization terms to any combination.

## State Vector Distribution

SVDistribution is a recipe for constructing a mixed state using BasicState and/or StateVector commands (which themselves produce pure states).

For example, The following SVDistribution

state

probability

|0,1>

1/2

1/sqrt(2)*|1,0>+1/sqrt(2)*|0,1>

1/4

|1,0>

1/4

results in the mixed state 1/2|0,1><0,1|+1/4(1/sqrt(2)*|1,0>+1/sqrt(2)*|0,1>)(1/sqrt(2)*<1,0|+1/sqrt(2)*<0,1|)+1/4|1,0><1,0|

## TimeSVDistribution

TimedSVDistribution is representing a time sequence distribution of StateVector.