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QuantumChannelTests.m
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(* ::Package:: *)
(* ::Title:: *)
(*QuantumUtils for Mathematica*)
(*Quantum Channel Unit Tests*)
(* ::Subsection::Closed:: *)
(*Copyright and License Information*)
(* ::Text:: *)
(*This package is part of QuantumUtils for Mathematica.*)
(**)
(*Copyright (c) 2015 and later, Christopher J. Wood, Christopher E. Granade, Ian N. Hincks*)
(**)
(*Redistribution and use in source and binary forms, with or withoutmodification, are permitted provided that the following conditions are met:*)
(*1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.*)
(*2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.*)
(*3. Neither the name of quantum-utils-mathematica nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.*)
(**)
(*THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THEIMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE AREDISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLEFOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIALDAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS ORSERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVERCAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USEOF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.*)
(* ::Subsection::Closed:: *)
(*Preamble*)
BeginPackage["QuantumChannelTests`"];
(* ::Text:: *)
(*The following packages are needed, but their contexts should not be loaded globally.*)
Needs["QUDevTools`"];
Needs["QuantumChannel`"];
(* ::Section:: *)
(*Results*)
Begin["`UnitTests`"];
$RegisteredTests={};
$TestResults := RunTest[$RegisteredTests];
End[];
(* ::Section:: *)
(*Unit Tests*)
Begin["`UnitTests`"];
(* ::Subsection::Closed:: *)
(*Channels*)
(* ::Subsubsection::Closed:: *)
(*Representations*)
TestCase[$RegisteredTests,"QuantumChannel:Unitary",
SameQ[
Unitary[PauliMatrix[1]],
QuantumChannel[PauliMatrix[1],{ChannelRep->Unitary,InputDim->2,OutputDim->2,Basis->"Col"}]
]];
TestCase[$RegisteredTests,"QuantumChannel:Super",
SameQ[
Super[IdentityMatrix[4]],
QuantumChannel[IdentityMatrix[4],{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]
]];
TestCase[$RegisteredTests,"QuantumChannel:Choi",
And[
SameQ[
Choi[IdentityMatrix[4]],
QuantumChannel[IdentityMatrix[4],
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}]],
SameQ[
Choi[IdentityMatrix[8],InputDim->4],
QuantumChannel[IdentityMatrix[8],
{ChannelRep->Choi,InputDim->4,OutputDim->2,Basis->"Col"}]]
]];
TestCase[$RegisteredTests,"QuantumChannel:Chi",
SameQ[
Chi[IdentityMatrix[4]],
QuantumChannel[IdentityMatrix[4],
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:Kraus",
SameQ[
Kraus[Table[PauliMatrix[j],{j,4}]],
QuantumChannel[Table[PauliMatrix[j],{j,4}],
{ChannelRep->Kraus,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:Stinespring",
Module[{a,b},
And[
SameQ[
Stinespring[Array[a,{8,2}],OutputDim->2],
QuantumChannel[Array[a,{8,2}],
{ChannelRep->Stinespring,InputDim->2,OutputDim->2,Basis->"Col"}]],
SameQ[
Stinespring[{Array[a,{8,2}],Array[b,{8,2}]},OutputDim->2],
QuantumChannel[{Array[a,{8,2}],Array[b,{8,2}]},
{ChannelRep->Stinespring,InputDim->2,OutputDim->2,Basis->"Col"}]]
]]];
TestCase[$RegisteredTests,"QuantumChannel:SysEnv",
Module[{a,b},
And[
SameQ[
SysEnv[{Array[a,{4,4}],{1,0}}],
QuantumChannel[{Array[a,{4,4}],{1,0}},
{ChannelRep->SysEnv,InputDim->2,OutputDim->2,Basis->"Col"}]],
SameQ[
SysEnv[{{Array[a,{4,4}],Array[b,{4,4}]},{1,0}}],
QuantumChannel[{{Array[a,{4,4}],Array[b,{4,4}]},{1,0}},
{ChannelRep->SysEnv,InputDim->2,OutputDim->2,Basis->"Col"}]]
]]];
TestCase[$RegisteredTests,"QuantumChannel:ChannelParameters",
SameQ[
ChannelParameters[QuantumChannel[IdentityMatrix[4],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}
]];
TestCase[$RegisteredTests,"QuantumChannel:ChannelRep",
SameQ[
ChannelRep[QuantumChannel[IdentityMatrix[4],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]],
Super
]];
TestCase[$RegisteredTests,"QuantumChannel:ChannelRep",
SameQ[
ChannelRep[QuantumChannel[IdentityMatrix[4],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]],
Super
]];
TestCase[$RegisteredTests,"QuantumChannel:InputDim",
SameQ[
InputDim[QuantumChannel[IdentityMatrix[4],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]],
2
]];
TestCase[$RegisteredTests,"QuantumChannel:OutputDim",
SameQ[
OutputDim[QuantumChannel[IdentityMatrix[4],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]],
2
]];
TestCase[$RegisteredTests,"QuantumChannel:Basis",
SameQ[
Basis[QuantumChannel[IdentityMatrix[4],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]],
"Col"
]];
(* ::Subsubsection::Closed:: *)
(*Evolution*)
TestCase[$RegisteredTests,"QuantumChannel:UnitaryEvolution",
With[{chan=QuantumChannel[
PauliMatrix[1],
{ChannelRep->Unitary,InputDim->2,OutputDim->2,Basis->"Col"}]},
And[
chan[{{1},{0}}]==={{0},{1}},
chan[{1,0}]==={0,1},
chan[{{1,0},{0,0}}]==={{0,0},{0,1}},
chan[{1,0,0,0}]==={0,0,0,1},
chan[{{1},{0},{0},{0}}]==={{0},{0},{0},{1}}
]]];
TestCase[$RegisteredTests,"QuantumChannel:SuperEvolution",
With[{
chan=QuantumChannel[
{{1,0,0,1},{0,0,0,0},{0,0,0,0},{1,0,0,1}}/2,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}],
out=IdentityMatrix[2]/2},
And[
chan[{{1},{0}}]===out,
chan[{1,0}]===out,
chan[{{1,0},{0,0}}]===out,
chan[{1,0,0,0}]==={1,0,0,1}/2,
chan[{{1},{0},{0},{0}}]==={{1},{0},{0},{1}}/2
]]];
TestCase[$RegisteredTests,"QuantumChannel:ChoiEvolution",
With[{
chan=QuantumChannel[
IdentityMatrix[4]/2,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}],
out=IdentityMatrix[2]/2},
And[
chan[{{1},{0}}]===out,
chan[{1,0}]===out,
chan[{{1,0},{0,0}}]===out,
chan[{1,0,0,0}]==={1,0,0,1}/2,
chan[{{1},{0},{0},{0}}]==={{1},{0},{0},{1}}/2
]]];
TestCase[$RegisteredTests,"QuantumChannel:KrausEvolution",
With[{
chan=QuantumChannel[
(PauliMatrix/@Range[4])/2,
{ChannelRep->Kraus,InputDim->2,OutputDim->2,Basis->"Col"}],
out=IdentityMatrix[2]/2},
And[
chan[{{1},{0}}]===out,
chan[{1,0}]===out,
chan[{{1,0},{0,0}}]===out,
chan[{1,0,0,0}]==={1,0,0,1}/2,
chan[{{1},{0},{0},{0}}]==={{1},{0},{0},{1}}/2
]]];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvEvolution",
With[{
chan=QuantumChannel[
{{{0,0,1,0},{0,1,0,0},{1,0,0,0},{0,0,1,0}},{1,0}},
{ChannelRep->SysEnv,InputDim->2,OutputDim->2,Basis->"Col"}],
out={{0,0},{0,1}}},
And[
chan[{{1},{0}}]===out,
chan[{1,0}]===out,
chan[{{1,0},{0,0}}]===out,
chan[{1,0,0,0}]==={0,0,0,1},
chan[{{1},{0},{0},{0}}]==={{0},{0},{0},{1}}
]]];
TestCase[$RegisteredTests,"QuantumChannel:StinespringEvolution",
With[{
chan=QuantumChannel[
{{0,0},{0,0},{0,1},{1,0},{0,1},{1,0},{0,0},{0,0}}/Sqrt[2],
{ChannelRep->Stinespring,InputDim->2,OutputDim->2,Basis->"Col"}],
out=IdentityMatrix[2]/2},
And[
chan[{{1},{0}}]===out,
chan[{1,0}]===out,
chan[{{1,0},{0,0}}]===out,
chan[{1,0,0,0}]==={1,0,0,1}/2,
chan[{{1},{0},{0},{0}}]==={{1},{0},{0},{1}}/2
]]];
(* ::Subsection::Closed:: *)
(*Channel Transformations*)
(* ::Subsubsection::Closed:: *)
(*To/From Unitary*)
With[{chan=QuantumChannel[PauliMatrix[1],
{ChannelRep->Unitary,InputDim->2,OutputDim->2,Basis->"Col"}]},
TestCase[$RegisteredTests,"QuantumChannel:UnitaryToUnitary",
SameQ[Unitary[chan],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:UnitaryToSuper",
SameQ[Super[chan],
QuantumChannel[
{{0,0,0,1},{0,0,1,0},{0,1,0,0},{1,0,0,0}},
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]]];
TestCase[$RegisteredTests,"QuantumChannel:UnitaryToChoi",
SameQ[Choi[chan],
QuantumChannel[
{{0,0,0,0},{0,1,1,0},{0,1,1,0},{0,0,0,0}},
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}]]];
TestCase[$RegisteredTests,"QuantumChannel:UnitaryToChi",
SameQ[Chi[chan],
QuantumChannel[
{{0,0,0,0},{0,2,0,0},{0,0,0,0},{0,0,0,0}},
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Pauli"}]]];
TestCase[$RegisteredTests,"QuantumChannel:UnitaryToKraus",
SameQ[Kraus[chan],
QuantumChannel[
{{{0,1},{1,0}}},
{ChannelRep->Kraus,InputDim->2,OutputDim->2,Basis->"Col"}]]];
TestCase[$RegisteredTests,"QuantumChannel:UnitaryToStinespring",
SameQ[Stinespring[chan],
QuantumChannel[
{{0,1},{1,0}},
{ChannelRep->Stinespring,InputDim->2,OutputDim->2,Basis->"Col"}]]];
TestCase[$RegisteredTests,"QuantumChannel:UnitaryToSysEnv",
SameQ[SysEnv[chan],
QuantumChannel[
{{{0,1},{1,0}},{1}},
{ChannelRep->SysEnv,InputDim->2,OutputDim->2,Basis->"Col"}]]];
];
With[{chan=QuantumChannel[PauliMatrix[3],
{ChannelRep->Unitary,InputDim->2,OutputDim->2,Basis->"Col"}]},
TestCase[$RegisteredTests,"QuantumChannel:SuperToUnitary",
SameQ[Unitary[Super[chan]],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:ChoiToUnitary",
SameQ[Unitary[Choi[chan]],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:KrausToUnitary",
SameQ[Unitary[Kraus[chan]],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToUnitary",
SameQ[Unitary[Stinespring[chan]],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToUnitary",
SameQ[Unitary[SysEnv[chan]],chan]
];
];
(* ::Subsubsection::Closed:: *)
(*From Super*)
With[{chan=QuantumChannel[
{{3,0,0,1},{0,3,1,0},{0,1,3,0},{1,0,0,3}}/4,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]},
TestCase[$RegisteredTests,"QuantumChannel:SuperToNotUnitary",
SameQ[Unitary[chan],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:SuperToSuper",
SameQ[Super[chan],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:SuperToPauliSuper",
SameQ[Super[chan,Basis->"Pauli"],
QuantumChannel[
{{2,0,0,0},{0,2,0,0},{0,0,1,0},{0,0,0,1}}/2,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:SuperToChoi",
SameQ[Choi[chan],
QuantumChannel[
{{3,0,0,3},{0,1,1,0},{0,1,1,0},{3,0,0,3}}/4,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:SuperToChi",
SameQ[Choi[chan,Basis->"Pauli"],
QuantumChannel[
{{3,0,0,0},{0,1,0,0},{0,0,0,0},{0,0,0,0}}/2,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:SuperToKraus",
SameQ[Kraus[chan],
QuantumChannel[
{{{Sqrt[3],0},{0,Sqrt[3]}}/2,{{0,1},{1,0}}/2},
{ChannelRep->Kraus,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:SuperToStinespring",
SameQ[Stinespring[chan],
QuantumChannel[
{{Sqrt[3],0},{0,1},{0,Sqrt[3]},{1,0}}/2,
{ChannelRep->Stinespring,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:SuperToSysEnv",
SameQ[SysEnv[chan],
QuantumChannel[
{{{Sqrt[3],0,0,0},{0,0,1,0},{0,0,Sqrt[3],0},{1,0,0,0}}/2,{1,0}},
{ChannelRep->SysEnv,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
];
(* ::Subsubsection::Closed:: *)
(*From Choi*)
With[{chan=QuantumChannel[
{{3,0,0,3},{0,1,1,0},{0,1,1,0},{3,0,0,3}}/4,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}]},
TestCase[$RegisteredTests,"QuantumChannel:ChoiToNotUnitary",
SameQ[Unitary[chan],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:ChoiToSuper",
SameQ[Super[chan],
QuantumChannel[
{{3,0,0,1},{0,3,1,0},{0,1,3,0},{1,0,0,3}}/4,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:ChoiToPauliSuper",
SameQ[Super[chan,Basis->"Pauli"],
QuantumChannel[
{{2,0,0,0},{0,2,0,0},{0,0,1,0},{0,0,0,1}}/2,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:ChoiToChoi",
SameQ[Choi[chan],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:ChoiToChi",
SameQ[Choi[chan,Basis->"Pauli"],
QuantumChannel[
{{3,0,0,0},{0,1,0,0},{0,0,0,0},{0,0,0,0}}/2,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:ChoiToKraus",
SameQ[Kraus[chan],
QuantumChannel[
{{{Sqrt[3],0},{0,Sqrt[3]}}/2,{{0,1},{1,0}}/2},
{ChannelRep->Kraus,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:ChoiToStinespring",
SameQ[Stinespring[chan],
QuantumChannel[
{{Sqrt[3],0},{0,1},{0,Sqrt[3]},{1,0}}/2,
{ChannelRep->Stinespring,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:ChoiToSysEnv",
SameQ[SysEnv[chan],
QuantumChannel[
{{{Sqrt[3],0,0,0},{0,0,1,0},{0,0,Sqrt[3],0},{1,0,0,0}}/2,{1,0}},
{ChannelRep->SysEnv,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
];
(* ::Subsubsection::Closed:: *)
(*From Kraus*)
With[{chan=QuantumChannel[
{Sqrt[3]*{{1,0},{0,1}}/2,{{0,1},{1,0}}/2},
{ChannelRep->Kraus,InputDim->2,OutputDim->2,Basis->"Col"}]},
TestCase[$RegisteredTests,"QuantumChannel:KrausToNotUnitary",
SameQ[Unitary[chan],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:KrausToSuper",
SameQ[Super[chan],
QuantumChannel[
{{3,0,0,1},{0,3,1,0},{0,1,3,0},{1,0,0,3}}/4,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:KrausToPauliSuper",
SameQ[Super[chan,Basis->"Pauli"],
QuantumChannel[
{{2,0,0,0},{0,2,0,0},{0,0,1,0},{0,0,0,1}}/2,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:KrausToChoi",
SameQ[Choi[chan],
QuantumChannel[
{{3,0,0,3},{0,1,1,0},{0,1,1,0},{3,0,0,3}}/4,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:KrausToChi",
SameQ[Choi[chan,Basis->"Pauli"],
QuantumChannel[
{{3,0,0,0},{0,1,0,0},{0,0,0,0},{0,0,0,0}}/2,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:KrausToKraus",
SameQ[Kraus[chan],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:KrausToStinespring",
SameQ[Stinespring[chan],
QuantumChannel[
{{Sqrt[3],0},{0,1},{0,Sqrt[3]},{1,0}}/2,
{ChannelRep->Stinespring,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:KrausToSysEnv",
SameQ[SysEnv[chan],
QuantumChannel[
{{{Sqrt[3],0,0,0},{0,0,1,0},{0,0,Sqrt[3],0},{1,0,0,0}}/2,{1,0}},
{ChannelRep->SysEnv,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
];
(* ::Subsubsection::Closed:: *)
(*From Stinespring*)
With[{chan=QuantumChannel[
{{Sqrt[3],0},{0,1},{0,Sqrt[3]},{1,0}}/2,
{ChannelRep->Stinespring,InputDim->2,OutputDim->2,Basis->"Col"}]},
TestCase[$RegisteredTests,"QuantumChannel:StinespringToNotUnitary",
SameQ[Unitary[chan],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToSuper",
SameQ[Super[chan],
QuantumChannel[
{{3,0,0,1},{0,3,1,0},{0,1,3,0},{1,0,0,3}}/4,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToPauliSuper",
SameQ[Super[chan,Basis->"Pauli"],
QuantumChannel[
{{2,0,0,0},{0,2,0,0},{0,0,1,0},{0,0,0,1}}/2,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToChoi",
SameQ[Choi[chan],
QuantumChannel[
{{3,0,0,3},{0,1,1,0},{0,1,1,0},{3,0,0,3}}/4,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToChi",
SameQ[Choi[chan,Basis->"Pauli"],
QuantumChannel[
{{3,0,0,0},{0,1,0,0},{0,0,0,0},{0,0,0,0}}/2,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToKraus",
SameQ[Kraus[chan],
QuantumChannel[
{Sqrt[3]*{{1,0},{0,1}}/2,{{0,1},{1,0}}/2},
{ChannelRep->Kraus,InputDim->2,OutputDim->2,Basis->"Col"}]
]];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToStinespring",
SameQ[Stinespring[chan],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToSysEnv",
SameQ[SysEnv[chan],
QuantumChannel[
{{{Sqrt[3],0,0,0},{0,0,1,0},{0,0,Sqrt[3],0},{1,0,0,0}}/2,{1,0}},
{ChannelRep->SysEnv,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
];
(* ::Subsubsection::Closed:: *)
(*From SysEnv*)
With[{chan=QuantumChannel[
{{{Sqrt[3],0,0,0},{0,0,1,0},{0,0,Sqrt[3],0},{1,0,0,0}}/2,{1,0}},
{ChannelRep->SysEnv,InputDim->2,OutputDim->2,Basis->"Col"}]},
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToNotUnitary",
SameQ[Unitary[chan],chan]
];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToSuper",
SameQ[Super[chan],
QuantumChannel[
{{3,0,0,1},{0,3,1,0},{0,1,3,0},{1,0,0,3}}/4,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToPauliSuper",
SameQ[Super[chan,Basis->"Pauli"],
QuantumChannel[
{{2,0,0,0},{0,2,0,0},{0,0,1,0},{0,0,0,1}}/2,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToChoi",
SameQ[Choi[chan],
QuantumChannel[
{{3,0,0,3},{0,1,1,0},{0,1,1,0},{3,0,0,3}}/4,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToChi",
SameQ[Choi[chan,Basis->"Pauli"],
QuantumChannel[
{{3,0,0,0},{0,1,0,0},{0,0,0,0},{0,0,0,0}}/2,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Pauli"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToKraus",
SameQ[Kraus[chan],
QuantumChannel[
{Sqrt[3]*{{1,0},{0,1}}/2,{{0,1},{1,0}}/2},
{ChannelRep->Kraus,InputDim->2,OutputDim->2,Basis->"Col"}]
]];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToStinespring",
SameQ[Stinespring[chan],
QuantumChannel[
{{Sqrt[3],0},{0,1},{0,Sqrt[3]},{1,0}}/2,
{ChannelRep->Stinespring,InputDim->2,OutputDim->2,Basis->"Col"}]]
];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToSysEnv",
SameQ[SysEnv[chan],chan]
];
];
(* ::Subsubsection::Closed:: *)
(*Non-CP Transformations*)
With[{
choi=QuantumChannel[
{{-1,0,0,-1},{0,1,1,0},{0,1,1,0},{-1,0,0,-1}},
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}],
super=QuantumChannel[
{{-1,0,0,1},{0,-1,1,0},{0,1,-1,0},{1,0,0,-1}},
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}],
kraus=QuantumChannel[
{{{{-1,0},{0,-1}},{{0,1},{1,0}}},
{{{1,0},{0,1}},{{0,1},{1,0}}}},
{ChannelRep->Kraus,InputDim->2,OutputDim->2,Basis->"Col"}],
stine=QuantumChannel[
{{{-1,0},{0,1},{0,-1},{1,0}},{{1,0},{0,1},{0,1},{1,0}}},
{ChannelRep->Stinespring,InputDim->2,OutputDim->2,Basis->"Col"}],
syse=QuantumChannel[
{{{{-1,0,0,0},{0,0,1,0},{0,0,-1,0},{1,0,0,0}},
{{1,0,0,0},{0,0,1,0},{0,0,1,0},{1,0,0,0}}},{1,0}},
{ChannelRep->SysEnv,InputDim->2,OutputDim->2,Basis->"Col"}],
rho=Array["a",{2,2}]
},
TestCase[$RegisteredTests,"QuantumChannel:ChoiToKrausNonCP",
Kraus[choi]===kraus];
TestCase[$RegisteredTests,"QuantumChannel:ChoiToStinespringNonCP",
Stinespring[choi]===stine];
TestCase[$RegisteredTests,"QuantumChannel:ChoiToSysEnvNonCP",
SysEnv[choi]===syse];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToChoiNonCP",
Choi[syse]===choi];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToSuperNonCP",
Super[syse]===super];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToKrausNonCP",
Kraus[syse]===kraus];
TestCase[$RegisteredTests,"QuantumChannel:SysEnvToStinespringNonCP",
Stinespring[syse]===stine];
TestCase[$RegisteredTests,"QuantumChannel:KrausToChoiNonCP",
Choi[kraus]===choi];
TestCase[$RegisteredTests,"QuantumChannel:KrausToSuperNonCP",
Super[kraus]===super];
TestCase[$RegisteredTests,"QuantumChannel:KrausToSysEnvNonCP",
SysEnv[kraus]===syse];
TestCase[$RegisteredTests,"QuantumChannel:KrausToStinespringNonCP",
Stinespring[kraus]===stine];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToChoiNonCP",
Choi[stine]===choi];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToSuperNonCP",
Super[stine]===super];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToSysEnvNonCP",
SysEnv[stine]===syse];
TestCase[$RegisteredTests,"QuantumChannel:StinespringToKrausNonCP",
Kraus[stine]===kraus];
]
(* ::Subsection::Closed:: *)
(*Channel Operations*)
(* ::Subsubsection::Closed:: *)
(*Linear Operations*)
TestCase[$RegisteredTests,"QuantumChannel:Times",
With[{op=Array["a",{4,4}]},
And[
10*Choi[op]===Choi[10*op],
10*Super[op]===Super[10*op],
10*Unitary[op]===Unitary[10*op]]
]];
TestCase[$RegisteredTests,"QuantumChannel:Plus",
With[{op=Array["a",{4,4}]},
And[
Super[op]+Super[op]===Super[op+op],
Choi[op]+Choi[op]===Choi[op+op]
]
]];
TestCase[$RegisteredTests,"QuantumChannel:Dot",
With[{
op1=Array["a",{4,4}],
op2=Array["b",{9,4}],
op3=Array["c",{9,9}]},
Super[op3].Super[op2].Super[op1]===Super[op3.op2.op1]
]];
TestCase[$RegisteredTests,"QuantumChannel:CircleTimes",
With[{X=PauliMatrix[1],Z=PauliMatrix[3],id=PauliMatrix[0]},
CircleTimes[Super[KroneckerProduct[X,X]],
Super[KroneckerProduct[Z,Z]],
Super[KroneckerProduct[id,id]]]===
Super[KroneckerProduct[X,Z,id,X,Z,id]]
]];
(* ::Subsubsection::Closed:: *)
(*Conjugate Transpose*)
Module[{chan, op=Array["a",{4,4}]},
chan=QuantumChannel[op,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}];
TestCase[$RegisteredTests,"QuantumChannel:Transpose",
Transpose[chan]===
QuantumChannel[Transpose@op,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]
];
TestCase[$RegisteredTests,"QuantumChannel:Conjugate",
Conjugate[chan]===
QuantumChannel[Conjugate@op,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]
];
TestCase[$RegisteredTests,"QuantumChannel:ConjugateTranspose",
ConjugateTranspose[chan]===
QuantumChannel[ConjugateTranspose@op,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]
];
TestCase[$RegisteredTests,"QuantumChannel:TrSuper",
Tr[chan]===Tr[op]
];
]
(* ::Subsubsection::Closed:: *)
(*Super ops*)
Module[{chan,op={{1,0,0,3},{0,1,3,0},{0,3,1,0},{3,0,0,1}}/4},
chan=QuantumChannel[op,
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}];
TestCase[$RegisteredTests,"QuantumChannel:MatrixPower",
MatrixPower[chan,2]===
QuantumChannel[MatrixPower[op,2],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]
];
TestCase[$RegisteredTests,"QuantumChannel:MatrixExp",
MatrixExp[chan]===
QuantumChannel[MatrixExp[op],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]
];
TestCase[$RegisteredTests,"QuantumChannel:MatrixLog",
MatrixLog[chan]===
QuantumChannel[MatrixLog[op],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}]
];
]
(* ::Subsubsection::Closed:: *)
(*Eigenvalues*)
Module[{chan,op={{1,0,0,1},{0,3,3,0},{0,3,3,0},{1,0,0,1}}/4},
chan=QuantumChannel[op,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}];
TestCase[$RegisteredTests,"QuantumChannel:Eigenvalues",
Eigenvalues[chan]==={3/2,1/2,0,0}
];
TestCase[$RegisteredTests,"QuantumChannel:Eigenvectors",
Eigenvectors[chan]==={{0,1,1,0},{1,0,0,1},{-1,0,0,1},{0,-1,1,0}}
];
TestCase[$RegisteredTests,"QuantumChannel:Eigensystem",
Eigensystem[chan]==={{3/2,1/2,0,0},{{0,1,1,0},{1,0,0,1},{-1,0,0,1},{0,-1,1,0}}}
];
TestCase[$RegisteredTests,"QuantumChannel:TrChoi",
Tr[chan]===Tr[op]
];
]
(* ::Subsubsection::Closed:: *)
(*Simplify Functions*)
TestCase[$RegisteredTests,"QuantumChannel:SimplifyFunctions",
Module[{a,b},
And[
SameQ[
FullSimplify[Super[{{Abs[a],0},{0,Abs[b]}}],a>=0&&b>=0],
Super[{{a,0},{0,b}}]],
SameQ[
ExpToTrig[Super[{{Exp[I*a],0},{0,1}}]],
Super[{{Cos[a]+I*Sin[a],0},{0,1}}]],
SameQ[
TrigToExp[Super[{{2*Cos[a],0},{0,1}}]],
Super[{{Exp[I*a]+Exp[-I*a],0},{0,1}}]]
]]];
(* ::Subsection::Closed:: *)
(*Channel Functions and Metrics*)
Module[{a,b,chan1,chan2},
chan1=QuantumChannel[DiagonalMatrix@Array[a,4],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}];
chan2=QuantumChannel[DiagonalMatrix@Array[b,4],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Col"}];
TestCase[$RegisteredTests,"QuantumChannel:ProcessFidelity",
And[
ProcessFidelity[chan1]===Total@Array[a,4]/4,
ProcessFidelity[chan1,chan2]===Array[a,4].Conjugate[Array[b,4]]/4
]];
TestCase[$RegisteredTests,"QuantumChannel:GateFidelity",
And[
AllMatchQ[Abs[a[1]b[1]],
Map[FullSimplify@GateFidelity[#,chan1,chan2]&,
{{1,0},{{1},{0}},{{1,0},{0,0}}}]],
{a[1],a[1],Abs[a[1]]}===
Map[FullSimplify@GateFidelity[#,chan1]&,
{{1,0},{{1},{0}},{{1,0},{0,0}}}]
]];
TestCase[$RegisteredTests,"QuantumChannel:AverageGateFidelity",
And[
AllMatchQ[0,
{FullSimplify[1/6 (2+Total@Array[a,4])-AverageGateFidelity[chan1]],
FullSimplify[1/6 (2+Array[a,4].Conjugate[Array[b,4]])
-AverageGateFidelity[chan1,chan2]]}]
]];
TestCase[$RegisteredTests,"QuantumChannel:EntanglementFidelity",
And[
AllMatchQ[a[1]Conjugate[b[1]],
Map[FullSimplify@EntanglementFidelity[#,chan1,chan2]&,
{{1,0},{{1},{0}},{{1,0},{0,0}}}]],
AllMatchQ[a[1],
Map[FullSimplify@EntanglementFidelity[#,chan1]&,
{{1,0},{{1},{0}},{{1,0},{0,0}}}]
]]];
];
TestCase[$RegisteredTests, "QuantumChannel:DiamondNormDistance",
And[
Module[{\[Theta]},FullSimplify[DiamondNormDistance[Unitary[MatrixExp[-I \[Theta] {{0,1},{1,0}}/2]]]-2*Sin[\[Theta]/2],Assumptions->0<=\[Theta]<2\[Pi]]==0]
]
]
TestCase[$RegisteredTests, "QuantumChannel:Unitarity",
And[
Unitarity[
QuantumChannel[{{0,1},{1,0}},
{ChannelRep->Unitary,InputDim->2,OutputDim->2,Basis->"Col"}]
] == 1,
Unitarity[
QuantumChannel[DiagonalMatrix[{1,0,0,0}],
{ChannelRep->Super, InputDim->2, OutputDim->2, Basis -> "Pauli"}]
] == 0,
Unitarity[
QuantumChannel[{
{1, 0, 0, 1},
{0, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 0, 0}},
{ChannelRep->Super, InputDim->2, OutputDim->2, Basis -> "Pauli"}]
] == 0,
Unitarity[
QuantumChannel[{Sqrt[1/2]{{1,0},{0,1}},Sqrt[1/2]{{0,1},{1,0}}},
{ChannelRep->Kraus, InputDim->2, OutputDim->2, Basis->"Col"}]
] == 1/3
]
]
TestCase[$RegisteredTests,"QuantumChannel:ChannelVolume",
Module[{a},
FullSimplify[
ChannelVolume[QuantumChannel[DiagonalMatrix@Array[a,4],
{ChannelRep->Super,InputDim->2,OutputDim->2,Basis->"Pauli"}]]-Sqrt[a[2]]Sqrt[a[3]]Sqrt[a[4]]]===0
]];
(* ::Subsection::Closed:: *)
(*Channel Predicates*)
TestCase[$RegisteredTests,"QuantumChannel:CompletelyPositiveQ",
And[
Not@CompletelyPositiveQ@QuantumChannel[
{{1,0,0,0},{0,0,1,0},{0,1,0,0},{0,0,0,1}},
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}],
CompletelyPositiveQ@QuantumChannel[
IdentityMatrix[4]/2,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}]
]];
TestCase[$RegisteredTests,"QuantumChannel:TracePreservingQ",
And[
TracePreservingQ@QuantumChannel[
IdentityMatrix[4]/2,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}],
Not@TracePreservingQ@QuantumChannel[
{{1,0,0,0},{0,1,0,0},{0,0,0,0},{0,0,0,0}},
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}]
]];
TestCase[$RegisteredTests,"QuantumChannel:HermitianPreservingQ",
And[
HermitianPreservingQ@QuantumChannel[
IdentityMatrix[4]/2,
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}],
Not@HermitianPreservingQ@QuantumChannel[
{{1,0,0,1},{0,0,0,0},{0,0,0,0},{0,0,0,1}},
{ChannelRep->Choi,InputDim->2,OutputDim->2,Basis->"Col"}]