chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
3,830 | lean_file_chunk | d076371ef841402eafcf82e47005a15c820a5b28b80c6dff14de46743eec17ed | lean_source_file | physlib_source_file_chunks | · simp
have hp0' : (0 : ℝ) < p := Prob.zero_lt_coe hp0
have hp1' : (p : ℝ) < 1 := lt_of_le_of_ne Prob.coe_le_one fun h => hp1 (Subtype.ext h)
-- If either relative entropy on the right is `⊤`, the bound is trivial.
by_cases hker₁ : σ₁.M.ker ≤ ρ₁.M.ker
swap
· have h_ne : ((p : NNReal) : ENNReal) ≠ 0 := by
... |
23,958 | lean_file_chunk | c4acda3bf24aab090cdb856024c2cf57120414ad6c2fa1abe8f4bf8f9b743eb0 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.Entropy.VonNeumann
public import Physlib.Meta.Sorry
@[expose] public section
noncomputable section
variable {d d₁ d₂ d₃ : Type*}
vari... |
23,862 | lean_file_chunk | 48f0a1651c7575384bada2369148fe060c4dd84253c871b93c3e95da021fb365 | lean_source_file | physlib_source_file_chunks | exact_mod_cast h_inner_def.trans h_trace;
have h_deriv : ∀ i, HasDerivAt (fun α : ℝ => (B.H.eigenvalues i) ^ α) (B.H.eigenvalues i * Real.log (B.H.eigenvalues i)) 1 := by
intro i
by_cases h_pos : 0 < B.H.eigenvalues i;
· convert! HasDerivAt.rpow ( hasDerivAt_const _ _ ) ( hasDerivAt_id 1 ) h_pos using 1
... |
23,899 | lean_file_chunk | 07e4b08b1d923ffa1000cbfbd67f78088862d9823a062c90d0f42bbf8ff3269d | lean_source_file | physlib_source_file_chunks | convert! Filter.Tendsto.comp ( show Filter.Tendsto ( fun M : { M : Matrix d d ℂ // M.IsHermitian } ↦ ( HermitianMat.conj M.val ) ρM ) ( nhds ( A.supportProj ) ) ( nhds ρM ) from ?_ ) h_conj using 2;
convert Continuous.tendsto _ _;
· convert! conj_supportProj_eq_of_ker_le A ρM hker |> Eq.symm;
· fun_prop;
... |
23,977 | lean_file_chunk | 139db09591a29e4033c2b1598da237719965fc955dd95cb73b70a02c2babfd10 | lean_source_file | physlib_source_file_chunks | have h_deriv : HasDerivAt (fun α : ℝ => ((M α) ^ α).trace - (M α).trace - ((ρ.M) ^ α).trace + ρ.M.trace) 0 1 := by
convert hasDerivAt_iff_tendsto_slope_zero.mpr _ using 1
convert cross_term_slope_tendsto_zero hM_nonneg hM_cont hM_one using 2 ; norm_num [ hM_one ] ; ring!
convert! h_deriv.add ( hasDerivAt_trac... |
23,902 | lean_file_chunk | ebab93b901d9860d278e9a4d49fe85fb7eb219a0bac62111f10b8f90d8540c7e | lean_source_file | physlib_source_file_chunks | simp_all [ ker_prod_le_iff ]
rw [not_and_or] at h_ker
rcases h_ker with h_ker | h_ker
· simp [SandwichedRelRentropy, h_ker_prod, h_ker, hα0]
· simp [SandwichedRelRentropy, h_ker_prod, h_ker, hα0]
end additivity
/-- The Sandwiched Renyi Relative entropy is additive when the inputs are product states -... |
23,551 | lean_file_chunk | cf52ceb2f577d24c33085793519b9d4c04223e320811759844fa6201b026a635 | lean_source_file | physlib_source_file_chunks | Filter.Tendsto (fun N : ℕ => ⟪ρ.M, x.M.cfc (approxLog N)⟫) Filter.atTop Filter.atBot := by
have h_split_sum : Filter.Tendsto (fun N : ℕ => ∑ i ∈ Finset.univ.filter (fun i => x.M.H.eigenvalues i = 0), approxLog N (x.M.H.eigenvalues i) * eigenWeight ρ x i) Filter.atTop Filter.atBot := by
have h_split_sum : Filter.T... |
23,975 | lean_file_chunk | a8817df00a6ff41f8d21d7c60a85d79c5e61807e99655a22f09e26e31cf289e2 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.Entropy.VonNeumann
public import QuantumInfo.ForMathlib.HermitianMat.Sqrt
/-!
Quantities of quantum _relative entropy_, namely the (sta... |
23,917 | lean_file_chunk | 0f4370a2aaf5078ecb194a2bba38bbf29f794a08244d3a93cc681ebdc97756bb | lean_source_file | physlib_source_file_chunks | private lemma W_mat_eq_factored
(ρAB : HermitianMat (dA × dB) ℂ) (σBC : HermitianMat (dB × dC) ℂ) :
W_mat ρAB σBC =
let F := ρAB.sqrt.mat * (ρAB.traceRight⁻¹.sqrt.mat ⊗ₖ (1 : Matrix dB dB ℂ))
let G := ((1 : Matrix dB dB ℂ) ⊗ₖ σBC.traceLeft⁻¹.sqrt.mat) * σBC.sqrt.mat
(F ⊗ₖ (1 : Matrix dC dC ℂ))... |
19,357 | lean_file_chunk | 2369c921e3dba3db64efb3e8dfcb3fd464f1e716540328fed6dba7006ad774dc | lean_source_file | physlib_source_file_chunks | simp [ inner_add_left, inner_one_kron_eq_inner_traceLeft, inner_kron_one_eq_inner_traceRight ];
congr! 2;
convert MState.traceLeft_assoc' ρ using 1;
unfold MState.assoc' MState.traceLeft; aesop;
/-- Weak monotonicity (form 2) for positive definite states:
S(ρ₁₂) + S(ρ₂₃) ≥ S(ρ₁) + S(ρ₃).
Proved by applyi... |
14,102 | lean_file_chunk | d258966494526829686cfd511ec4b4b123b6d1252f470e41e0ffc38054b40b88 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.States.Pure.Braket
public import QuantumInfo.Channels.Bundled
public import QuantumInfo.Channels.CPTP
public import QuantumInfo.Channels... |
17,398 | lean_file_chunk | 3477a25d46034df0b6313eaeea2b48629141bc2b75341912d5cd69cebaa8922c | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Dennj Osele. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dennj Osele
-/
module
public import Mathlib.Analysis.Complex.HasPrimitives
public import Mathlib.Algebra.Order.Star.Real
public import Mathlib.MeasureTheory.Constructions.BorelSpace.With... |
2,506 | lean_file_chunk | 491cef8c4259f6ed2fe4a782d8f0f6af6f1288575a777634bc4f2555504e825a | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Analysis.InnerProductSpace.Spectrum
public import Mathlib.Order.CompletePartialOrder
/-!
# Continuous linear maps
This file collects auxil... |
13,754 | lean_file_chunk | 61af1c4a19257c1fcd95f5fbdfc8579e5138440ad537faeb59f8a23570c01c5c | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Analysis.InnerProductSpace.Basic
public import Mathlib.Analysis.Normed.Module.FiniteDimension
public import Mathlib.Algebra.Order.Star.Real
... |
3,269 | lean_file_chunk | b4003f4a4a374da8a427ae98d1e6c7becc0fbebbbc649a116f46c4c4e67ffe5e | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib
public import Mathlib.Tactic.Bound
@[expose] public section
open Topology
--This is a stupid name for a stupid lemma
theorem Filter.Tend... |
10,253 | lean_file_chunk | 3a29d3aeebb196bf83cdc678e3cbadbebfb474fb1b76abf227d9a31d6a7d14a9 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Hayata Yamasaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kei Tsukamoto, Kento Mori, Hayata Yamasaki
-/
module
public import QuantumInfo.ForMathlib.HayataGroup.TraceInequality.LownerHeinzTheorem
/-!
# `2 × 2` block operators on a Hilbert s... |
23,731 | lean_file_chunk | 1a42bd72c834169a29626307af1924aa4a4bdcedb1f862997a54b6069a7b0912 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Hayata Yamasaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kei Tsukamoto, Kento Mori, Hayata Yamasaki
-/
module
public import QuantumInfo.ForMathlib.HayataGroup.TraceInequality.JensenOperatorInequality
public import QuantumInfo.ForMathlib.Ha... |
23,975 | lean_file_chunk | f1061bdbdf27950c3bf20d3bf82c464d11c6e1662f08a5580d3fc655cda4777a | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Hayata Yamasaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kei Tsukamoto, Kento Mori, Hayata Yamasaki
-/
module
public import QuantumInfo.ForMathlib.HayataGroup.TraceInequality.LownerHeinzTheorem
public import QuantumInfo.ForMathlib.HayataGr... |
3,021 | lean_file_chunk | c48b21960aad3b5c6a70b0c0aaa37036716c0d978be1e714af23b30bbfd04f86 | lean_source_file | physlib_source_file_chunks | lemma cfc_op_eq_op_cfc [ContinuousFunctionalCalculus ℝ (L ℋ) IsSelfAdjoint]
[ContinuousFunctionalCalculus ℝ ((L ℋ)ᵐᵒᵖ) IsSelfAdjoint] [Nontrivial (L ℋ)]
(f : ℝ → ℝ) (A : L ℋ) (hA : IsSelfAdjoint A)
(hf : ContinuousOn f (spectrum ℝ A)) :
cfc (R := ℝ) (A := (L ℋ)ᵐᵒᵖ) (p := IsSelfAdjoint) f (MulOpposite.op... |
18,663 | lean_file_chunk | d4c0d4a5f8c75aa02f845763c04ec226f48e3fec5c3b033c215535c876895719 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Hayata Yamasaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kei Tsukamoto, Kento Mori, Hayata Yamasaki
-/
module
public import QuantumInfo.ForMathlib.HayataGroup.TraceInequality.JensenOperatorInequalityIImpIV
@[expose] public section
namesp... |
23,976 | lean_file_chunk | fceb63fcd3cee6a8e0b36d77fce3543f807c3a1ee730c6f88ab8ec38eff3e955 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Hayata Yamasaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kei Tsukamoto, Kento Mori, Hayata Yamasaki
-/
module
public import QuantumInfo.ForMathlib.HayataGroup.TraceInequality.BlockDiagonal
public import QuantumInfo.ForMathlib.HayataGroup.T... |
23,970 | lean_file_chunk | ea00428b728e3c72a3796bff4397914294f75a753e6c16b76eb0b973b350b400 | lean_source_file | physlib_source_file_chunks | have hSnorm : ‖S‖ ≤ 1 := by
simpa [S] using blockSwap_norm_le_one (ℋ := ℋ) X hX
letI : Algebra ℝ (L (HSum ℋ)) := by
infer_instance
have hU_mem : S + Complex.I • CFC.sqrt (1 - S ^ 2) ∈ unitary (L (HSum ℋ)) := by
exact IsSelfAdjoint.self_add_I_smul_cfcSqrt_sub_sq_mem_unitary S hSsa hSnorm
let U : unitar... |
12,630 | lean_file_chunk | 44a183a56840a11d2ad822f397a9a0edffeebeff4698e376886173c9d1ab1fd2 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Hayata Yamasaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kei Tsukamoto, Kento Mori, Hayata Yamasaki
-/
module
public import QuantumInfo.ForMathlib.HayataGroup.TraceInequality.BlockDiagonal
public import QuantumInfo.ForMathlib.HayataGroup.T... |
23,980 | lean_file_chunk | b91f21cf997247a4c0fe9b66023df699b287988e2762c25365a9b85430e8e5fa | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Hayata Yamasaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kei Tsukamoto, Kento Mori, Hayata Yamasaki
-/
module
public import QuantumInfo.ForMathlib.HayataGroup.TraceInequality.OperatorGeometricMean
public import QuantumInfo.ForMathlib.Hayat... |
23,934 | lean_file_chunk | 9ac1cf327aadc31b86bd47196365a4d99f88b2148531b5cfdd1e66fd39b42dbc | lean_source_file | physlib_source_file_chunks | private lemma phiK_operatorPowerMean_eq_liebTraceMap
{s : ℝ} (K A B : L ℋ) (hA : A ∈ pdSet (ℋ := ℋ)) (hB : B ∈ pdSet (ℋ := ℋ)) :
phiK (ℋ := ℋ) K
(operatorPowerMean (ℋ := HSOp ℋ) s 1
(leftMulHS (ℋ := ℋ) A) (rightMulHS (ℋ := ℋ) B)) =
liebTraceMap (ℋ := ℋ) s K A B := by
rcases hA with ⟨hA... |
15,856 | lean_file_chunk | 9938cf2325d7bbae499f197c098dd8b54da612508fe93c3c3c727b3f52fc8130 | lean_source_file | physlib_source_file_chunks | (leftMulHS_pdSet (ℋ := ℋ) hA₁) (leftMulHS_pdSet (ℋ := ℋ) hA₂)
(rightMulHS_pdSet (ℋ := ℋ) hB₁) (rightMulHS_pdSet (ℋ := ℋ) hB₂)
hθ0 hθ1
have hconc :
(1 - θ) • operatorPowerMean (ℋ := HSOp ℋ) s 1
(leftMulHS (ℋ := ℋ) A₁) (rightMulHS (ℋ := ℋ) B₁) +
θ • operatorPowerMean (ℋ := HSOp ℋ) s ... |
23,980 | lean_file_chunk | 4aa5052f74529af090954bb38db7570df68e17ee1546fd6e0c237fe71cdc0fca | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Hayata Yamasaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kei Tsukamoto, Kento Mori, Hayata Yamasaki
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order
public import Mathlib.Analysis.SpecialFunctions.C... |
23,919 | lean_file_chunk | a74609e9876763c76c036b085167a4a392dee2a6613fb092c6036f29a578bb9a | lean_source_file | physlib_source_file_chunks | exact hA.add hT
have hB1_sa : IsSelfAdjoint B1 := by
subst B1
exact hB.add hT
have specA1 : spectrum ℝ A1 ⊆ Set.Ioi (0 : ℝ) := by
intro x hx
have hs : spectrum ℝ (cfc shift A) = shift '' spectrum ℝ A := by
simpa [shift] using
(cfc_map_spectrum (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (a... |
23,982 | lean_file_chunk | b49dc052ba56b06745466b188e07eef06ec82354f4d95210f9fff0cba2aad419 | lean_source_file | physlib_source_file_chunks | a • cfcₙ (Real.rpowIntegrand₀₁ (q : ℝ) t) A + b • cfcₙ (Real.rpowIntegrand₀₁ (q : ℝ) t) B =
r • (a • cfcR ratio A + b • cfcR ratio B) := by
simp [hrepr A hA0, hrepr B hB0, smul_add, smul_smul, mul_comm]
have hR :
cfcₙ (Real.rpowIntegrand₀₁ (q : ℝ) t) (a • A + b • B) = r • cfcR ratio (a • A + b • B) ... |
23,921 | lean_file_chunk | 3441dac36576b2aa101456efe5bccce3258ec804e42ac315dc0b08772c2d3948 | lean_source_file | physlib_source_file_chunks | (u * (1 - u)) • ((A - B) * (A - B)) = α • ((A - B) * (A - B)) := by simp [α]
_ = α • (A * A - A * B - B * A + B * B) := hsmul
_ = α • (A * A) - α • (A * B) - α • (B * A) + α • (B * B) :=
smul_sub_mul_sub (α := α) A B
_ = (u * (1 - u)) • (A * A) - (u * (1 - u)) • (A * B) - (u * (1 - u)) • (B * A)
... |
12,715 | lean_file_chunk | f29909deae42f21c05421c7f1bae131c816071ceb65b61e5c4bf7f874d4ae006 | lean_source_file | physlib_source_file_chunks | exact (Real.continuousAt_rpow_const x r (Or.inl (ne_of_gt hx0))).continuousWithinAt
have hspec :
spectrum ℝ Br = (fun x : ℝ ↦ x ^ r) '' spectrum ℝ B := by
dsimp [Br, cfcR]
simpa using
(cfc_map_spectrum (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x ^ r)
... |
11,775 | lean_file_chunk | 26931fde8d1bfffc055c15c6a09608fb5d10bc531e9194e5bf01b8ce5f275c6a | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Hayata Yamasaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kei Tsukamoto, Kento Mori, Hayata Yamasaki
-/
module
public import QuantumInfo.ForMathlib.HayataGroup.TraceInequality.LownerHeinzCore
public import Mathlib.Analysis.CStarAlgebra.Con... |
4,347 | lean_file_chunk | b7f2b361ff2661544e99c34a4351677de2d192f3aba78a0f0e8366de1938e929 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Hayata Yamasaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kei Tsukamoto, Kento Mori, Hayata Yamasaki
-/
module
public import QuantumInfo.ForMathlib.HayataGroup.TraceInequality.GeneralizedPerspectiveFunction
@[expose] public section
namesp... |
23,949 | lean_file_chunk | 07672f8ce21e61c3dab16ea4f2c971b27df99a844115ca702fc0bc6e49f5dd80 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.Matrix
public import QuantumInfo.ForMathlib.IsMaximalSelfAdjoint
public import QuantumInfo.ForMathlib.ContinuousLinearMap
pub... |
1,436 | lean_file_chunk | c48fc29ca7349b37d7758b77371f8512048b63dc20538a9cbe4f257aa0a39748 | lean_source_file | physlib_source_file_chunks | variable [Fintype n] [DecidableEq n] [Fintype m] [DecidableEq m]
theorem _root_.Matrix.IsHermitian.spectrum_rcLike {A : Matrix n n 𝕜} (hA : A.IsHermitian) :
RCLike.ofReal '' spectrum ℝ A = spectrum 𝕜 A := by
rw [hA.spectrum_eq_image_range, hA.spectrum_real_eq_range_eigenvalues]
/-- We fix a simp-normal form t... |
23,973 | lean_file_chunk | cf5d96a716338ce1d00ab3a6a3fe2ffdca48856de699880e3f6882cfc168721a | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Inner
public import QuantumInfo.ForMathlib.HermitianMat.NonSingular
public import QuantumInfo.ForMathlib.Isometr... |
23,925 | lean_file_chunk | cb7ba4c15a2c5625b0062b306abaf8b745a03a5ea2e8f0a89e8ca38dfb8d1a15 | lean_source_file | physlib_source_file_chunks | have h_seq_cont : ∀ n, ContinuousOn (fun A : HermitianMat d ℂ => A.cfc (fun x => (p_n n).eval x)) {A : HermitianMat d ℂ | spectrum ℝ A.mat ⊆ K} := by
fun_prop
refine' Metric.continuousOn_iff.mpr _;
intro A hA ε εpos
obtain ⟨N, hN⟩ := h_uniform (ε / 3) (by linarith)
obtain ⟨δ, δpos, hδ⟩ : ∃ δ > 0, ... |
23,937 | lean_file_chunk | fa2cfec89ba29500718b3f5a7e05a5814997609a6780f9090190516855c3d7ff | lean_source_file | physlib_source_file_chunks | -- The open set W := ⋃ᵢ V_i contains spectrum(A x₀) (since each λᵢ ∈ V_i and spectrum = range of eigenvalues). W is open as a union of open sets.
set W := ⋃ i, V_i i with hW_def
have hW_open : IsOpen W := by
exact isOpen_iUnion hV_i
have hW_spectrum : spectrum ℝ (A x₀).mat ⊆ W := by
in... |
7,984 | lean_file_chunk | e223fa2a4158128f3cada5db5a51318b74c37f2d334e6c511b1c50db43536d10 | lean_source_file | physlib_source_file_chunks | -- Since the eigenvectors form an orthonormal basis, we can express x as a linear combination of these eigenvectors.
obtain ⟨c, hc⟩ : ∃ c : d → ℂ, x = ∑ i, c i • A.H.eigenvectorBasis i := by
have := A.H.eigenvectorBasis.sum_repr x;
exact ⟨ _, this.symm ⟩;
-- By definition of $A$, we know that $A.mulVec (x.o... |
2,910 | lean_file_chunk | c3e1529ac8bb4c1caf7e92268540a9eae0d524f6256887d667a3cdf413f3d452 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Dennj Osele. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dennj Osele
-/
module
public import QuantumInfo.ForMathlib.Majorization
public import QuantumInfo.ForMathlib.HermitianMat.Order
/-! # Compound matrices on `HermitianMat`
This file lift... |
21,405 | lean_file_chunk | 2ba550525143e0012dd60ccef9a74ba26d36808b6a28d34fb5ed442d25378089 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Order
public import Mathlib.Analysis.Convex.Contractible
public import Mathlib.Topology.Instances.Real.Lemmas
/... |
3,940 | lean_file_chunk | 67b37938bd04f3ff4547b08f8d36fd3d05038b63f37345ebd6d3ab1b22e604fe | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Algebra.Jordan.Basic
public import QuantumInfo.ForMathlib.HermitianMat.CFC
public import QuantumInfo.ForMathlib.HermitianMat.Order
/-!
Her... |
23,986 | lean_file_chunk | 9f601d365c3fb381fab0be5c08db2935781ce00a5442b8ba026d9cbdfbb0dd4f | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Alex Meiburg. All rights reserved.
Released under MIT license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HayataGroup.TraceInequality.LiebAndoTrace
public import QuantumInfo.ForMathlib.HermitianMat.Schatten
@[expose] public section
/-! ... |
23,927 | lean_file_chunk | f368aa6c58b1df117d78542ad97ed883754db404b9d0a1f759b53ed7e9019df5 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Proj
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
public import Mathlib.Analysi... |
15,060 | lean_file_chunk | 20f00a8d7bd97a5f5ea54cb082b10469899228e72ff2748d2a4e5d1f7376b672 | lean_source_file | physlib_source_file_chunks | have h_block_pos : (Matrix.fromBlocks A (1 : Matrix d d 𝕜) (1 : Matrix d d 𝕜) (A⁻¹)) = (Matrix.fromBlocks 1 0 A⁻¹ 1) * (Matrix.fromBlocks A 0 0 (A⁻¹ - A⁻¹ * A * A⁻¹)) * (Matrix.fromBlocks 1 A⁻¹ 0 1) := by
simp [ Matrix.fromBlocks_multiply ];
have := hA.det_pos;
exact ⟨ by rw [ Matr... |
6,925 | lean_file_chunk | 67bebe7afc275077cb212a9e8904fdc964da59f19c896607b0334327a4cd0e2c | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Order
public import QuantumInfo.ForMathlib.Isometry
@[expose] public section
noncomputable section
namespace ... |
23,982 | lean_file_chunk | 797b65931084bc7c70a0df39fc2fbde32fb118999248e896f783be7bacfefcff | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Trace
public import Mathlib.Analysis.RCLike.Basic
@[expose] public section
namespace HermitianMat
open Comple... |
5,767 | lean_file_chunk | fe3fa3750f26783eb4d170b701a02834a0818fb52c66635bd495dd81733e5a66 | lean_source_file | physlib_source_file_chunks | theorem _root_.Matrix.nonneg_self_mul_conjTranspose {m : Type*} [Fintype m]
(M : Matrix n m 𝕜) : 0 ≤ M * M.conjTranspose :=
Matrix.nonneg_iff_posSemidef.mpr (Matrix.posSemidef_self_mul_conjTranspose M)
omit [Fintype m] in
open MatrixOrder in
theorem subtype_mk_nonneg {M : Matrix m m 𝕜} (h : 0 ≤ M) :
0 ≤ (⟨... |
12,107 | lean_file_chunk | 7cc9acf9f16796d32cc159663df0baa7b9eb6d33988b04a9f1557e25a541ca9b | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Sqrt
public import QuantumInfo.ForMathlib.HermitianMat.LiebConcavity
@[expose] public section
noncomputable se... |
21,414 | lean_file_chunk | 3bbe8760cc8d61bd726a589c26db07d6415393475a3d7766675fdeb2c54a4172 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Leonardo A Lessa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo A Lessa, Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.CFC
public import Mathlib.Analysis.CStarAlgebra.Classes
public import Mathlib.Analysis.In... |
3,700 | lean_file_chunk | 9480d01e9eec2dd074f567b0ccc5d19a4836bda1a532bbc7c423a46490d6b0cc | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Basic
public import QuantumInfo.ForMathlib.ContinuousLinearMap
public import QuantumInfo.ForMathlib.LinearEquiv
... |
23,969 | lean_file_chunk | 7fc5d1b0eef2397e7dd24fd71375eab9024cdcb913957a0302432094c9291133 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.CompoundMatrix
public import QuantumInfo.ForMathlib.HermitianMat.LogExp
public import QuantumInfo.ForMathlib.Her... |
23,937 | lean_file_chunk | fe75210b2882347a481cb05d5a8cbc98268415bfef2e31d3f1025a9b1c997f84 | lean_source_file | physlib_source_file_chunks | `scalarRpowApprox q T x → rpowConst q * (x^q - 1)` as `T → ∞`. -/
lemma scalarRpowApprox_tendsto {x : ℝ} (hx : 0 < x) (hq : 0 < q) (hq1 : q < 1) :
Filter.Tendsto (fun T => scalarRpowApprox q T x) atTop (nhds (rpowConst q * (x ^ q - 1))) := by
have h_def : ∀ T > 0, scalarRpowApprox q T x = x * (∫ t in (0)..T, t ^ ... |
18,898 | lean_file_chunk | 5a78724adcfbcd3b2591095bf833cf19a5c254030022f6310da26420be72e0aa | lean_source_file | physlib_source_file_chunks | rw [show H.H.eigenvalues i = Real.sqrt (H.H.eigenvalues i) ^ 2 from
(Real.sq_sqrt h_nn).symm, ← Real.rpow_natCast _ 2,
← Real.rpow_mul (Real.sqrt_nonneg _)]
push_cast
congr 1
simp [H, Matrix.conjTranspose_conjTranspose]
private lemma lieb_thirring_le_one_posDef
{A B : HermitianMat... |
16,223 | lean_file_chunk | 02a6522e1643e759ead7a327cbd007be187a3f27f7b7bab0a031d38a7332bacc | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Rpow
public import QuantumInfo.ForMathlib.Majorization
@[expose] public section
variable {d d₂ 𝕜 : Type*} [Fi... |
2,994 | lean_file_chunk | 6b086f72f2f21c925d48b91bf729bf324715ace802f05b735ed3feda94ceca12 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Proj
@[expose] public section
variable {d 𝕜 : Type*} [Fintype d] [DecidableEq d] [RCLike 𝕜]
variable {A B : ... |
8,016 | lean_file_chunk | b74226b4ab4d07bcfddbf010f85c171571c3b76034587e3f9ec6d5ec51970f3e | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Reindex
/-! # Trace of Hermitian Matrices
While the trace of a Hermitian matrix is, in informal math, typicall... |
3,794 | lean_file_chunk | be11e6c9b08af936cf4ac9e0f13e14e1cc05641c5608446621a5cc5a5c0be0bf | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Inner
public import QuantumInfo.ForMathlib.HermitianMat.NonSingular
public import QuantumInfo.ForMathlib.Isometr... |
954 | lean_file_chunk | 58a26fc7793e546022c66d8266c583017f3ec1a32a95f7590ad7d289df4603e1 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.HermitianMat.Basic
public import QuantumInfo.ForMathlib.HermitianMat.CFC
public import QuantumInfo.ForMathlib.HermitianMat.In... |
2,748 | lean_file_chunk | 625d24596c9ae93161682f8a68fc82371f758809e811827d14cc39e57c10aa05 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Analysis.Matrix.Normed
/-!
# Maximal self-adjoint subrings
This file introduces the `IsMaximalSelfAdjoint R α` typeclass, which records th... |
23,989 | lean_file_chunk | f9a70e45ffde2634a1fab1c0c23a9f1ae99d61f7d80a7651ca041c4aef8dbda8 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Analysis.InnerProductSpace.JointEigenspace
public import Mathlib.LinearAlgebra.Matrix.Permutation
public import QuantumInfo.ForMathlib.Matri... |
9,387 | lean_file_chunk | 2f7721bf503cd67e90f4b50849f1c3e7fb39232763b8d0f2ffe1d6a04b3e3a6d | lean_source_file | physlib_source_file_chunks | have h₂ := Module.End.mem_eigenspace_iff.mp h.2
rw [h₂]
congr; symm
simp only [sharedEigenvaluesB, h₂, inner_smul_right, OrthonormalBasis.inner_eq_one,
mul_one, ← RCLike.conj_eq_iff_re, ← RCLike.star_def]
have h₃ : (sharedEigenbasis hA hB hAB) i ≠ 0 := by
have := (sharedEigenbasis hA hB hAB).orthonormal... |
23,964 | lean_file_chunk | d1d4ef8db57a344abba97a5afe36614ed7a195fc2c440c31d771cc343fb9baf9 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Algebra.Order.Ring.Star
public import Mathlib.Analysis.Normed.Ring.Lemmas
public import Mathlib.Data.Finset.Attr
public import Mathlib.Data.... |
6,608 | lean_file_chunk | d7a7f3fb2d5e383c549b509266bbdb2f903b834447c6ea361de21fbdf9d51fbd | lean_source_file | physlib_source_file_chunks | -- Define the sequence $T_k$ such that $T_k \geq N_0(k)$ and each interval $[T_k, T_{k+1})$ contains some $n_k$ with $P(k, n_k)$.
obtain ⟨T, hT_mono, hT_bound, hT_wit⟩ : ∃ T : ℕ → ℕ, StrictMono T ∧ (∀ k, T k ≥ N0 k) ∧ (∀ k, ∃ n ∈ Set.Ico (T k) (T (k + 1)), f₁ (x k) n ≤ y₁ + (k + 1 : ℝ≥0)⁻¹) := by
have hP : ∀ k L,... |
3,972 | lean_file_chunk | 776276ddefad675fe176166de8402d41a70a604fad58d30abde0a13b389773e1 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Relabelling linear equivalences
## i. Overview
This module provides linear equivalences obtained by... |
23,904 | lean_file_chunk | 5bc73d64405d91a43c1d94a79eda91da78616de38b28f63638cdabce682d9a03 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib
public import QuantumInfo.ForMathlib.Matrix
/-! # Majorization and weak log-majorization
This file develops the theory of majorization for... |
23,991 | lean_file_chunk | 8151103e9ef2bf7ee4f3fe8da53d256899d03edcaf8ab46c2184a02fb8378918 | lean_source_file | physlib_source_file_chunks | convert congr_arg (fun x => compoundMatrix x k) this using 1 <;> simp [h_compoundMatrix_mul, h_compoundMatrix_conjTranspose]
rw [compoundMatrix_mul, compoundMatrix_mul]
rw [← compoundMatrix_conjTranspose]
rw [compoundMatrix_diagonal]; simp [Matrix.mul_assoc]
congr! 3
obtain ⟨σ, hσ⟩ := h_eigenv... |
12,634 | lean_file_chunk | b012f2a815b6d1e9c88e817903c14d108f7e67ace3c73fa77e72e13ee690d67d | lean_source_file | physlib_source_file_chunks | intro k hk
convert Real.rpow_le_rpow _ (h_log_maj k hk) hr.le using 1 <;>
norm_num [Real.finsetProd_rpow _ _ fun i _ => hx_nn _,
Real.finsetProd_rpow _ _ fun i _ => hy_nn _]
exact Finset.prod_nonneg fun _ _ => hx_nn _
/-
The Abel summation identity (a rewriting of the sum) gives:
∑_{i=0}^{n-1} x_... |
23,987 | lean_file_chunk | b9d343927e77c731c1a6dc73e3ae4f517713af38dc1bed604621e752bc9ef4d0 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
public import Mathlib.Algebra.Order.Group.Pointwise.CompleteLattice
public import Mathlib.Analysis.CS... |
23,958 | lean_file_chunk | e4cbdefe03f7f5847821bfadef0ed5f2b8149b8bee168d53b98fcc0c3d7b1ff7 | lean_source_file | physlib_source_file_chunks | (M : Matrix (dA × dB) (dA × dB) R) (B : Matrix dB dB R) :
(M * ((1 : Matrix dA dA R) ⊗ₖ B)).trace = (M.traceLeft * B).trace := by
simp [trace, mul_apply, kroneckerMap_apply, traceLeft, one_apply,
Fintype.sum_prod_type, Finset.sum_mul]
rw [Finset.sum_comm]
exact Finset.sum_congr rfl fun _ _ => Finset.sum_c... |
23,965 | lean_file_chunk | 7e73d64307667c2d664c7f7b74785b629410bd2e88e1567e114f92c33583678b | lean_source_file | physlib_source_file_chunks | -- Since $v \otimes w$ is non-zero, we have $(A \otimes B - yzI)(v \otimes w) = 0$ implies that $A \otimes B - yzI$ is not invertible.
have h_not_invertible : ¬IsUnit (A ⊗ₖ B - (y * z) • 1) := by
simp only [ne_eq, isUnit_iff_isUnit_det, isUnit_iff_ne_zero, Decidable.not_not, v_tensor_w] at *
rw [ ← Matr... |
1,439 | lean_file_chunk | afa34b1fdaaf1c646ab64b72fd9b0fd8a914966a5635493199799af435f468e3 | lean_source_file | physlib_source_file_chunks | mul_left_comm, mulVec, dotProduct, mul_comm, map_sum, map_mul, map_prod,
RingHomCompTriple.comp_apply, RingHom.id_apply, mul_assoc];
exact Finset.sum_congr rfl fun _ _ => Finset.sum_comm.trans ( Finset.sum_congr rfl fun _ _ => Finset.sum_congr rfl fun _ _ => by ring )
end finprod
--TODO: Can this be used fo... |
22,839 | lean_file_chunk | 2663e6a6df4950a878135b842bb6f5cbb8a06a8ec51cf8e8f9e330ce64f61f2c | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.Matrix
public import QuantumInfo.ForMathlib.Majorization
public import QuantumInfo.ForMathlib.HermitianMat.Unitary
public imp... |
8,891 | lean_file_chunk | fdd6744763b1eacdaf1d155588b41ffad32f0b6737c434c6851d6e00ae8f0552 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Topology.Algebra.Module.FiniteDimension
public import Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap
public import QuantumInfo.F... |
8,633 | lean_file_chunk | 31ac937dbf35bad64051e8657cee7b61ba1751434a3c57569034cd4964fb86d7 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Analysis.SpecialFunctions.Log.Basic
public import Mathlib.Order.CompletePartialOrder
@[expose] public section
--Can this be rewritten more... |
23,989 | lean_file_chunk | 30905f1c02d90b4adef7376ca36cd7f2f4589d849fbdca788a24e4bb6bde38d9 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Algebra.Order.Module.Field
public import Mathlib.Analysis.Convex.PathConnected
public import Mathlib.Analysis.Convex.Quasiconvex
public impo... |
10,190 | lean_file_chunk | 6b31aff25425036b8d434a9e9cf951f1d692f53ab89a1b85e912f44fb60ed526 | lean_source_file | physlib_source_file_chunks | replace hz_mem := hT₂.segment_subset hy₁ hy₂ hz_mem
exact (hfc₁.frequently_lt_of_tendsto hzs hx₂ hzI hz_mem).exists
have hJ : IsClosed J := by
apply IsSeqClosed.isClosed
intro zs z hzI hzs
simp only [Set.mem_setOf_eq, J, C] at hzI
replace ⟨hzI, hzI2⟩ := And.intro (hzI · |>.left) (hzI · |>.right)
... |
1,346 | lean_file_chunk | 0aaaca49bd0b0739e3fdae8ef82fcc156f30ae7880c78c033a8597465606fa20 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Analysis.Subadditive
@[expose] public section
def Superadditive (u : ℕ → ℝ) : Prop :=
∀ m n, u (m + n) ≥ u m + u n
namespace Superaddit... |
561 | lean_file_chunk | 56da05ab04a72db6f842d0dc305f4f786527b8ba08ba98bf46f497605e7dd68f | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.Init
public import Aesop.Frontend.Command
/-!
# Commutes Rule Set
This module defines the `Commutes` Aesop rule set which is used by the
`... |
8,061 | lean_file_chunk | 4cf4199cbebc8f3f049af4c2c32342e0f8fda89f5b8b1604fe9059abc75210c1 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2026 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.Tactic.Commutes.Attribute
public import Mathlib.Algebra.BigOperators.Group.List.Basic
public import Mathlib.Algebra.BigOperat... |
2,779 | lean_file_chunk | 8ca12f993cbea9968e0f0b5e20335536f5fda563979b9a14b1ab8d4da230a78f | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib
/-!
# Typeclass instances for `ULift`
This file equips `ULift` with algebraic and analytic typeclass instances lifted from the
underlying ... |
3,345 | lean_file_chunk | ae49edecc89e8a3a52fa58e45f5da957aba9906542ffc1ee4786176eb50f2f17 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import Mathlib.LinearAlgebra.Matrix.Kronecker
public import Mathlib.LinearAlgebra.Matrix.PosDef
public import QuantumInfo.ForMathlib.HermitianMat.Unitary
... |
8,242 | lean_file_chunk | 37a7b7016f15ef429edd67e70de5aa180f9025881eaa3b55ba88be2db229acc0 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.Channels.Bundled
public import QuantumInfo.Channels.CPTP
public import QuantumInfo.Channels.Dual
public import QuantumInfo.Channels.Matr... |
3,251 | lean_file_chunk | 71ae9085f4bc0ab8ae0ab1e2714a5f32eb22f18188f9e774cdea22b3aac1f6ba | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg, Rodolfo Soldati
-/
module
public import QuantumInfo.States.Mixed.MState
/-! # Unitary operators on quantum state
This file is intended for lemmas about unitary matrices ... |
8,333 | lean_file_chunk | 183f4fc97fee3161a874d4c3b75998ae898e572ed430be842a88405a9712a669 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.ForMathlib.Superadditive
public import Mathlib.Order.LiminfLimsup
public import Mathlib.Topology.Order.LiminfLimsup
public import Mathli... |
20,062 | lean_file_chunk | 404cee088135da67db048092a648a35f88d8f0534ee282bb559f94e8935e30e8 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg, Leonardo A. Lessa, Rodolfo R. Soldati
-/
module
public import Mathlib.Algebra.Module.Submodule.Lattice
public import Mathlib.Analysis.Subadditive
public import Mathlib.Cat... |
23,965 | lean_file_chunk | 1c7be97eafdd68c1a8bdb6e9a5209834ac961ff91eeec29d0fbea9b2de2e9a9f | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg, Leonardo A. Lessa, Rodolfo R. Soldati
-/
module
public import Mathlib.Algebra.Module.Submodule.Lattice
public import Mathlib.Analysis.Subadditive
public import Mathlib.Cat... |
1,041 | lean_file_chunk | cc5202496ff447252da5d3379a4f78ce0467f45f6588ab08f0705d2cf2a08bf9 | lean_source_file | physlib_source_file_chunks | conv => enter [1, σ]; rw [subtype_val_iInf']
exact Continuous.subtype_mk (h.comp MState.Continuous_HermitianMat) _
set_option backward.isDefEq.respectTransparency false in
/-- On the 1D Hilbert space, the optimal hypothesis testing rate is simply 1 - ε,
since there's nothing to learn. (More generally this would hold... |
8,299 | lean_file_chunk | c3864f71892caab1cf9e5b3d6ffad19bd41dbff76b2f3bfa09765dfca53db540 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg, Leonardo A. Lessa, Rodolfo R. Soldati
-/
module
public import QuantumInfo.ResourceTheory.FreeState
@[expose] public section
/-- A quantum resource theory extends a `Free... |
23,998 | lean_file_chunk | b53404d3ef862316f75d79f10241b5d8167b5c1affed93592216e39e6bf0c3e9 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg, Leonardo A. Lessa, Rodolfo R. Soldati
-/
module
public import QuantumInfo.Channels.Pinching
public import QuantumInfo.ForMathlib.HermitianMat.Jordan
public import QuantumI... |
23,985 | lean_file_chunk | 3f1bd17cd8b645b3755122b6080dba338e25748fa88b57bb22804a839589869a | lean_source_file | physlib_source_file_chunks | change 0 ≤ HermitianMat.mat (_ - _)
rw [← HermitianMat.cfc_sub]
change (0 : HermitianMat d ℂ) ≤ _
rw [HermitianMat.cfc_nonneg_iff]
intro i
simp only [Pi.sub_apply, Function.comp_apply, one_div, Pi.mul_apply]
set fi := f (C.H.eigenvalues i) with hfi
set gi := g (C.H.eigenvalues i) with hgi
have hfi₀ : 0 ... |
23,903 | lean_file_chunk | 6845b37046dc65b09ba3cc464639f11258d9218403c2acbc7890d93486c10897 | lean_source_file | physlib_source_file_chunks | (-((R2 ρ σ).toReal * ⟪(ℰ n (ρ ⊗ᵣ^[n])).M, P2 ε2 n⟫) -
ε₀ * ⟪(ℰ n (ρ ⊗ᵣ^[n])).M, P2 ε2 n⟫) +
⟪(ℰ n (ρ ⊗ᵣ^[n])).M, P2 ε2 n⟫ * c' ε2 n) =
R1 ρ ε + ENNReal.ofReal ε2 +
ENNReal.ofReal ⟪P1 ε2 n, ℰ n (ρ ⊗ᵣ^[n])⟫ *
(ENNReal.ofReal ε2 + R2 ρ σ + ENNReal.ofReal ε₀ -
... |
23,935 | lean_file_chunk | 7b941a56c8e05cb640c14903e20e644873e33aa7181794cee0a7927b4209725b | lean_source_file | physlib_source_file_chunks | infinity" eigenvalue, intuitively. This isn't something very well defined, certainly not supported
in our definitions. This only becomes mathematically meaningful when we see how it's used later, in
(S88): both sides are traced against `ℰ n (ρ ⊗ᵣ^[n])`, so that the 0 eigenvalues becomes irreleva... |
4,305 | lean_file_chunk | 204cacbc610d85d630c9a7a095ed99014dedaca047d48c89e54aea292d04aa17 | lean_source_file | physlib_source_file_chunks | suffices h : ∀ (m k : ℕ), R2 ρ (σₖ (m + k)) - R1 ρ ε ≤ (1 - ε')^k * (R2 ρ (σₖ m) - R1 ρ ε) by
convert! h 0; simp
intro m k; induction k generalizing m
· simp [σₖ]
rename_i k ih
have σₖ_succ (n) : σₖ (n + 1) = Lemma7_improver ρ hε hε' (σₖ n) :=
Function.iterate_succ_apply' ..
... |
16,681 | lean_file_chunk | b779edf8a824a19c988d3494ee227ff76d58dde321af566ce75ada2d83fcaca5 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Leonardo A Lessa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo A Lessa
-/
module
public import QuantumInfo.States.Mixed.MState
public import Physlib.Meta.Sorry
@[expose] public section
open MState
open BigOperators
open scoped RealIn... |
15,165 | lean_file_chunk | 8188adedd7c2fa0673a27654046a4bbd5be375a06c4e5e0e2bad6c1dd3dc32ef | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Leonardo A Lessa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo A Lessa
-/
module
public import QuantumInfo.States.Pure.Braket
public import QuantumInfo.States.Ensemble
public import QuantumInfo.Entropy.VonNeumann
public import QuantumI... |
23,938 | lean_file_chunk | de51168d521586ce7985ac194193a311bb6a92b0ffa1cfc56dadc8f9cf21283b | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg, Leonardo A. Lessa
-/
module
public import QuantumInfo.ForMathlib.ContinuousLinearMap
public import QuantumInfo.ForMathlib.ComplexLaplaceTransform
public import QuantumInfo... |
23,997 | lean_file_chunk | ec1e244925558f556ed7c5747527eaac24008b6f9d5d839712fb803f6375ea4a | lean_source_file | physlib_source_file_chunks | -- Apply `Matrix.IsHermitian.eigenvalues_eq_of_unitary_similarity_diagonal` to $A \otimes B$ and $U_A \otimes U_B$ and the diagonal entries.
obtain ⟨σ, hσ⟩ : ∃ σ : d₁ × d₂ ≃ d₁ × d₂, (ρ₁.prod ρ₂).M.H.eigenvalues ∘ σ = fun (i, j) => ((ρ₁.spectrum i) * (ρ₂.spectrum j)) := by
have h_unitary : ∃ U : Matrix (d₁ × d₂) ... |
12,983 | lean_file_chunk | e2894a62d9b5e5f666dcdddbd7ea46355ebbd92bd9a1cdf4264b342725060c24 | lean_source_file | physlib_source_file_chunks | simp_all only [mat_apply, Matrix.IsHermitian.eigenvectorUnitary_apply, Matrix.of_apply]
congr! 2;
norm_num [ Complex.ext_iff, sq ];
exact Or.inl (Real.mul_self_sqrt (ρ.psd.eigenvalues_nonneg _))
simp_all [ Complex.normSq, sq ];
have h1 := h_eigenvectorUnitary i j
convert! h1 using 1;
simp [mul_assoc... |
2,241 | lean_file_chunk | cd9f9325aad5862afa135bd30a573615f79b44e3620d9ac3c3555114296a6e68 | lean_source_file | physlib_source_file_chunks | /-
Copyright (c) 2025 Alex Meiburg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
module
public import QuantumInfo.States.Mixed.MState
public import QuantumInfo.ForMathlib.ContinuousLinearMap
public import QuantumInfo.ForMathlib.ComplexLaplaceTransfo... |
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