Symbolic Objects for Local and Remote entanglement generation processes

docs/src/references.bib

@@ -230,3 +230,26 @@ @article{naomi2013topological
   journal={Nature}
 }
 
+@article{prajit2023entangling,
+  title={Entangling quantum memories via heralded photonic Bell measurement},
+  author={Prajit Dhara, Dirk Englund, and Saikat Guha},
+  url={https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.033149},
+  year={2023},
+  journal={Phys. Rev. Research 5, 033149}
+}
+
+@article{prajit2022heralded,
+  title={Heralded Multiplexed High-Efficiency Cascaded Source of Dual-Rail Entangled Photon Pairs Using Spontaneous Parametric Down-Conversion},
+  author={Prajit Dhara, Spencer J. Johnson, Christos N. Gagatsos, Paul G. Kwiat, and Saikat Guha},
+  url={https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.17.034071},
+  year={2022},
+  journal={Phys. Rev. Applied 17, 034071}
+}
+
+@article{kevin2023zero,
+  title={Zero-Added-Loss Entangled-Photon Multiplexing for Ground- and Space-Based Quantum Networks},
+  author={Kevin C. Chen, Prajit Dhara, Mikkel Heuck, Yuan Lee, Wenhan Dai, Saikat Guha, and Dirk Englund},
+  url={https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.19.054029},
+  year={2023},
+  journal={Phys. Rev. Applied 19, 054029}
+}

src/QuantumSavory.jl

@@ -332,6 +332,8 @@ include("plots.jl")
 
 include("CircuitZoo/CircuitZoo.jl")
 
+include("StatesZoo/StatesZoo.jl")
+
 include("precompile.jl")
 
 end # module

src/StatesZoo/StatesZoo.jl

@@ -0,0 +1,56 @@
+module StatesZoo
+
+using QuantumSymbolics, QuantumOpticsBase
+using QuantumSymbolics: withmetadata, @withmetadata, Metadata
+import QuantumSymbolics: express_nolookup
+
+export SingleRailMidSwapBell, DualRailMidSwapBell, ZALMSpinPair
+
+abstract type AbstractTwoQubitState <: QuantumSymbolics.AbstractTwoQubitOp end #For representing density matrices
+Base.show(io::IO, x::AbstractTwoQubitState) = print(io, "$(symbollabel(x))")
+
+_bspin = SpinBasis(1//2)
+
+const cascaded_source_basis = [0 0 0 0;
+                               0 0 0 1;
+                               0 0 0 2;
+                               0 0 1 0;
+                               0 0 1 1;
+                               0 0 2 0;
+                               0 1 0 0;
+                               0 1 0 1;
+                               0 1 0 2;
+                               0 1 1 0;
+                               0 1 1 1;
+                               0 1 2 0;
+                               0 2 0 0;
+                               0 2 0 1;
+                               0 2 0 2;
+                               0 2 1 0;
+                               0 2 1 1;
+                               0 2 2 0;
+                               1 0 0 0;
+                               1 0 0 1;
+                               1 0 0 2;
+                               1 0 1 0;
+                               1 0 1 1;
+                               1 0 2 0;
+                               1 1 0 0;
+                               1 1 0 1;
+                               1 1 0 2;
+                               1 1 1 0;
+                               1 1 1 1;
+                               1 1 2 0;
+                               2 0 0 0;
+                               2 0 0 1;
+                               2 0 0 2;
+                               2 0 1 0;
+                               2 0 1 1;
+                               2 0 2 0]
+
+
+include("zalm_pair/zalm_pair.jl")
+include("zalm_pair/ret_cxy.jl")
+include("single_dual_rail_midswap/single_dual_rail_midswap.jl")
+
+end # module

src/StatesZoo/readme.md

@@ -0,0 +1,68 @@
+## Quick Start
+
+Shown below are the typical values for the parameters and how to call the functions:
+
+```
+using QuantumSavory.StatesZoo: cascaded_source_photonic, cascaded_source_spin, midswap_dual_rail, midswap_single_rail
+```
+
+#### `cascaded_source_photonic()`
+
+```
+Ns = 1e-3
+eAs = 1
+eBs = 1
+eD = 0.9
+Pd = 1e-8
+VisF = 0.99
+
+cascaded_source_photonic(Ns,eAs,eBs,eD,Pd,VisF)
+```
+
+#### `cascaded_source_spin()`
+
+```
+Ns = 1e-3
+gA = 0.5
+gB = 0.5
+eAm = 1
+eBm = 1
+eAs = 1
+eBs = 1
+eD = 0.9
+Pd = 1e-8
+Pdo1 = 1e-8
+Pdo2 = 1e-8
+VisF = 0.99
+
+cascaded_source_spin(Ns,gA,gB,eAm,eBm,eAs,eBs,eD,Pd,Pdo1,Pdo2,VisF)
+```
+
+#### `midswap_single_rail`
+
+```
+eA = 0.9
+eB = 0.9
+gA = 0.5
+gB = 0.5
+Pd = 1e-8
+Vis = 0.99
+
+midswap_single_rail(eA,eB,gA,gB,Pd,Vis)
+```
+
+#### `midswap_dual_rail`
+
+```
+eA = 0.9
+eB = 0.9
+gA = 0.5
+gB = 0.5
+Pd = 1e-8
+Vis = 0.99
+
+
+midswap_dual_rail(eA,eB,gA,gB,Pd,Vis)
+```
+
+Note: Details and references in the respective readme.md files in the matlab folder

src/StatesZoo/single_dual_rail_midswap/entanglement_swap_with_emissive_memories/midswap_dual_rail.m

@@ -0,0 +1,46 @@
+function Mv=midswap_dual_rail(eA,eB,gA,gB,Pd,Vis)
+% Author: Prajit Dhara
+% Function to calculate the spin-spin density matrix for midpoint swap
+% using memories emitting dual rail photonic qubits
+% Inputs:
+% eA, eB: Link efficiencies for memories A and B upto the swap (include link loss, detector efficiency, etc.)
+%         Range: [0,1]; Typical value: 1-1e-4
+% gA, gB: Memory initialization parameter for memories A and B 
+%         Range: [0,1]; Typical value: 0.5
+%         Memory emission model: \sqrt{1-g} |0>_M\otimes |0,1>_P + \sqrt{g} |1>_M\otimes |1,0>_P
+% Pd: Detector dark count probability per photonic mode (assumed to be the same for both detectors)
+%     Range: [0,1]; Typical value: 1e-8
+% Vis: Interferometer visibility for the midpoint swap ()
+%     Range:[0,1]; Typical value: 0.9-1
+% Output:
+% Mv: Spin-spin density matrix for the two memories after the midpoint swap
+%     Basis: |00>, |01>, |10>, |11>
+
+    m11=((1/2).*eA.*(1+(-1).*eB).*gA.*gB+(1/2).*(1+(-1).*eA).*eB.* ...
+        gA.*gB).*(1+(-1).*Pd).^3.*Pd+(1+(-1).*eA).*(1+(-1).*eB).* ...
+        gA.*gB.*(1+(-1).*Pd).^2.*Pd.^2;
+
+    m22=(1/4).*eA.*eB.*gA.*(1+(-1).*gB).*(1+(-1).*Pd).^4+(1/2).*eA.* ...
+        (1+(-1).*eB).*gA.*(1+(-1).*gB).*(1+(-1).*Pd).^3.*Pd+(1/2).*( ...
+        1+(-1).*eA).*eB.*gA.*(1+(-1).*gB).*(1+(-1).*Pd).^3.*Pd+(1+( ...
+        -1).*eA).*(1+(-1).*eB).*gA.*(1+(-1).*gB).*(1+(-1).*Pd).^2.* ...
+        Pd.^2;
+
+    m33=(1/4).*eA.*eB.*(1+(-1).*gA).*gB.*(1+(-1).*Pd).^4+(1/2).*eA.* ...
+        (1+(-1).*eB).*(1+(-1).*gA).*gB.*(1+(-1).*Pd).^3.*Pd+(1/2).*( ...
+        1+(-1).*eA).*eB.*(1+(-1).*gA).*gB.*(1+(-1).*Pd).^3.*Pd+(1+( ...
+        -1).*eA).*(1+(-1).*eB).*(1+(-1).*gA).*gB.*(1+(-1).*Pd).^2.* ...
+        Pd.^2;
+
+    m44=((1/2).*eA.*(1+(-1).*eB).*(1+(-1).*gA).*(1+(-1).*gB)+(1/2).* ...
+        (1+(-1).*eA).*eB.*(1+(-1).*gA).*(1+(-1).*gB)).*(1+(-1).*Pd) ...
+        .^3.*Pd+(1+(-1).*eA).*(1+(-1).*eB).*(1+(-1).*gA).*(1+(-1).* ...
+        gB).*(1+(-1).*Pd).^2.*Pd.^2;
+
+    m23=(Vis.^2).*(1/4).*eA.*eB.*(1+(-1).*gA).^(1/2).*gA.^(1/2).*(1+(-1).*gB) ...
+        .^(1/2).*gB.^(1/2).*(1+(-1).*Pd).^4;
+
+    m32=conj(m23);
+    Mv=[m11, 0, 0, 0 ; 0, m22, m23, 0 ; 0, m32, m33, 0 ; 0, 0, 0, m44];
+
+end

src/StatesZoo/single_dual_rail_midswap/entanglement_swap_with_emissive_memories/midswap_single_rail.m

@@ -0,0 +1,35 @@
+function Mv=midswap_single_rail(eA,eB,gA,gB,Pd,Vis)
+% Author: Prajit Dhara
+% Function to calculate the spin-spin density matrix for midpoint swap
+% using memories emitting single rail photonic qubits
+% Inputs:
+% eA, eB: Link efficiencies for memories A and B upto the swap (include link loss, detector efficiency, etc.)
+%         Range: [0,1]; Typical value: 1-1e-4
+% gA, gB: Memory initialization parameter for memories A and B 
+%         Range: [0,1]; Typical value: 0.5 (set to achieve maximum hashing bound or reach target fidelity)
+%         Memory emission model: \sqrt{1-g} |0>_M\otimes |1>_P + \sqrt{g} |1>_M\otimes |0>_P
+% Pd: Detector dark count probability per photonic mode (assumed to be the same for both detectors)
+%     Range: [0,1]; Typical value: 1e-8
+% Vis: Interferometer visibility for the midpoint swap; can be a complex number to account for phase mismatch 
+%     Range (absolute value):[0,1]; Typical value: 0.9-1
+% Output:
+% Mv: Spin-spin density matrix for the two memories after the midpoint swap
+%     Basis: |00>, |01>, |10>, |11>
+        m11=gA.*gB.*(1-Pd).*Pd;
+        m22=(1/2).*eB.*gA.*(1-gB).*(1-Pd).^2 ...
+            +(1-eB).*gA.*(1-gB).*(1-Pd).*Pd;
+
+        m33=(1/2).*eA.*(1-gA).*gB.*(1-Pd).^2 ...
+            +(1-eA).*(1-gA).*gB.*(1-Pd).*Pd;
+
+        m23=(Vis).*(1/2).*((eA.*eB.*(1-gA).*gA.*(1-gB).*gB).^(1/2)).*(1-Pd).^2;
+
+        m32=(Vis).*(1/2).*((eA.*eB.*(1-gA).*gA.*(1-gB).*gB).^(1/2)).*(1-Pd).^2;
+        %
+        m44=((1/2).*eB.*(1-eA).*(1-gA).*(1-gB)...
+            +(1/2).*eA.*(1-eB).*(1-gA).*(1-gB)).*(1-Pd).^2 ...
+            +(1-eA).*(1-eB).*(1-gA).*(1-gB).*(1-Pd).*Pd;
+        %
+        Mv=[m11, 0, 0, 0 ; 0, m22, m23, 0 ; 0, m32, m33, 0 ; 0, 0, 0, m44];
+
+end

src/StatesZoo/single_dual_rail_midswap/entanglement_swap_with_emissive_memories/readme.md

@@ -0,0 +1,78 @@
+# Entanglement Swap with Emissive Memories
+## Overview
+This folder contains the functions required to generate the spin-spin density matrix generated by linear photonic entanglement swap with emissive memories. The functions are written in MATLAB. The functions are:
+- midswap_dual_rail.m: Generates the spin-spin density matrix for linear photonic entanglement swap with emissive memories emitting dual rail photonic qubits from the papers Ref. 1.
+- midswap_single_rail.m: Generates the spin-spin density matrix for linear photonic entanglement swap with emissive memories emitting single rail photonic qubits from the papers Ref. 1.
+
+## Detailed Descriptions
+### midswap_dual_rail.m
+This function generates the spin-spin density matrix for linear photonic entanglement swap with emissive memories emitting dual rail photonic qubits 
+
+**Inputs:**
+- eA, eB: Link efficiencies for memories A and B upto the swap (include link loss, detector efficiency, etc.)
+    - Range: $[0,1]$ 
+    - Typical value: $1\leftrightarrow10^{-4}$
+- gA, gB: Memory initialization parameter for memories A and B 
+  -  Range: $[0,1]$
+  -  Typical value: $0.5$
+  - Memory emission model: $\sqrt{1-g_k} \ket{0} _M\otimes \ket{0,1}_P + \sqrt{g_k} \ket{1}_M\otimes \ket{1,0}_P$
+- Pd: Detector dark count probability per photonic mode (assumed to be the same for both detectors)
+    -  Range: $[0,1)$
+    -  Typical value: $10^{-8}$ 
+- Vis: Interferometer visibility for the midpoint swap ()
+    -  Range: $[0,1]$
+    -  Typical value: $0.99$ 
+
+**Output:**
+- Mv: Spin-spin density matrix for the two memories after the midpoint swap
+  - Basis order: $\ket{0,0}, \ket{0,1}, \ket{1,0}, \ket{1,1}$
+
+### midswap_single_rail.m
+This function generates the spin-spin density matrix for linear photonic entanglement swap with emissive memories emitting single rail photonic qubits
+
+**Inputs:**
+- eA, eB: Link efficiencies for memories A and B upto the swap (include link loss, detector efficiency, etc.)
+    - Range: $[0,1]$ 
+    - Typical value: $1\leftrightarrow10^{-4}$
+- gA, gB: Memory initialization parameter for memories A and B 
+  -  Range: $[0,1]$
+  -  Typical value: $0.5$
+  - Memory emission model: $\sqrt{1-g_k} \ket{0} _M\otimes \ket{1}_P + \sqrt{g_k} \ket{1}_M\otimes \ket{0}_P$
+- Pd: Detector dark count probability per photonic mode (assumed to be the same for both detectors)
+    -  Range: $[0,1)$
+    -  Typical value: $10^{-8}$ 
+- Vis: Interferometer visibility for the midpoint swap' can be complex to account for phase instability
+    -  Range (absolute value): $[0,1]$
+    -  Typical value: $0.99$ 
+
+**Output:**
+- Mv: Spin-spin density matrix for the two memories after the midpoint swap
+  - Basis order: $\ket{0,0}, \ket{0,1}, \ket{1,0}, \ket{1,1}$
+
+## Usage
+The functions can be used as follows:
+```matlab
+% Parameters
+eA = 0.9; % Link efficiency for memory A
+eB = 0.9; % Link efficiency for memory B
+gA = 0.5; % Initialization parameter for memory A
+gB = 0.5; % Initialization parameter for memory B
+Pd = 1e-8; % Dark count probability per photonic mode
+Vis = 0.99; % Interferometer visibility
+%% Generate the spin-spin density matrix
+M_dual= midswap_dual_rail(eA,eB,gA,gB,Pd,Vis); % For dual rail photonic qubits
+M_single = midswap_single_rail(eA,eB,gA,gB,Pd,Vis); % For single rail photonic qubits
+
+%% Calculate the probability of success
+P_succ_dual = trace(M_dual)*4; % Multiply by 4 to account for all click patterns
+P_succ_single = trace(M_single)*2; % Multiply by 2 to account for all click patterns
+
+%% Calculate the fidelity
+F_dual = 0.5*(M_dual(2,2)+M_dual(3,3)+M_dual(2,3)+M_dual(3,2));
+F_single = 0.5*(M_single(2,2)+M_single(3,3)+M_single(2,3)+M_single(3,2));
+
+```
+
+
+## Reference
+1. [P. Dhara, D. Englund, S. Guha, Entangling quantum memories via heralded photonic Bell measurement. Phys. Rev. Res. 5, 033149 (2023).](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.033149)

src/StatesZoo/single_dual_rail_midswap/readme.md

@@ -0,0 +1,6 @@
+The above code was translated from the code in the matlab files in the adjoining folder. The process for converting to julia involved:
+- Removing the broadcasting operator `.`, because the calculation is being performed on scalars.
+- Some spots needed the addition of brackets to clarify the ambiguity caused by the priority of operations.
+- Some spots needed `\` merging with previous line to avoid to ambiguity about end of expression.
+
+Then the output of the functions were linked to their corresponding symboic object which can be expressed as a density matrix in QuantumOptics representation.