diff --git a/ext/TensorKitChainRulesCoreExt/tensoroperations.jl b/ext/TensorKitChainRulesCoreExt/tensoroperations.jl
index 82bd9b578..af6f35125 100644
--- a/ext/TensorKitChainRulesCoreExt/tensoroperations.jl
+++ b/ext/TensorKitChainRulesCoreExt/tensoroperations.jl
@@ -12,6 +12,8 @@ function ChainRulesCore.rrule(
         α::Number, β::Number, ba...
     )
     C′ = tensoradd!(copy(C), A, pA, conjA, α, β, ba...)
+    # only keep `C` alive on the tape if `dβ` needs it
+    C_β = _needs_tangent(β) ? C : nothing
 
     projectA = ProjectTo(A)
     projectC = ProjectTo(C)
@@ -49,7 +51,7 @@ function ChainRulesCore.rrule(
         else
             ZeroTangent()
         end
-        dβ = _needs_tangent(β) ? @thunk(projectβ(inner(C, ΔC))) : ZeroTangent()
+        dβ = _needs_tangent(β) ? @thunk(projectβ(inner(C_β, ΔC))) : NoTangent()
         dba = map(_ -> NoTangent(), ba)
         return NoTangent(), dC, dA, NoTangent(), NoTangent(), dα, dβ, dba...
     end
@@ -66,6 +68,7 @@ function ChainRulesCore.rrule(
         α::Number, β::Number, ba...
     )
     C′ = tensorcontract!(copy(C), A, pA, conjA, B, pB, conjB, pAB, α, β, ba...)
+    C_β = _needs_tangent(β) ? C : nothing
 
     projectA = ProjectTo(A)
     projectB = ProjectTo(B)
@@ -138,7 +141,7 @@ function ChainRulesCore.rrule(
         else
             ZeroTangent()
         end
-        dβ = _needs_tangent(β) ? @thunk(projectβ(inner(C, ΔC))) : ZeroTangent()
+        dβ = _needs_tangent(β) ? @thunk(projectβ(inner(C_β, ΔC))) : NoTangent()
         dba = map(_ -> NoTangent(), ba)
         return NoTangent(), dC,
             dA, NoTangent(), NoTangent(),
@@ -156,6 +159,7 @@ function ChainRulesCore.rrule(
         α::Number, β::Number, ba...
     )
     C′ = tensortrace!(copy(C), A, p, q, conjA, α, β, ba...)
+    C_β = _needs_tangent(β) ? C : nothing
 
     projectA = ProjectTo(A)
     projectC = ProjectTo(C)
@@ -190,7 +194,7 @@ function ChainRulesCore.rrule(
         else
             ZeroTangent()
         end
-        dβ = _needs_tangent(β) ? @thunk(projectβ(inner(C, ΔC))) : ZeroTangent()
+        dβ = _needs_tangent(β) ? @thunk(projectβ(inner(C_β, ΔC))) : NoTangent()
         dba = map(_ -> NoTangent(), ba)
         return NoTangent(), dC, dA, NoTangent(), NoTangent(), NoTangent(), dα, dβ, dba...
     end
diff --git a/test/chainrules/tensoroperations.jl b/test/chainrules/tensoroperations.jl
index 13052bca6..c3fd557b8 100644
--- a/test/chainrules/tensoroperations.jl
+++ b/test/chainrules/tensoroperations.jl
@@ -3,6 +3,7 @@ using TensorKit
 using TensorKit: type_repr, SectorDict
 using TensorOperations
 using ChainRulesCore
+using VectorInterface: Zero
 using ChainRulesTestUtils
 using Random
 using LinearAlgebra
@@ -47,6 +48,7 @@ for V in spacelist
                     for conjA in (false, true)
                         C = randn!(TensorOperations.tensoralloc_add(T, A, p, conjA, Val(false)))
                         test_rrule(tensortrace!, C, A, p, q, conjA, α, β; atol, rtol)
+                        test_rrule(tensortrace!, C, A, p, q, conjA, α, Zero() ⊢ NoTangent(); atol, rtol)
                     end
                 end
             end
@@ -65,6 +67,7 @@ for V in spacelist
 
                     C2 = randn!(TensorOperations.tensoralloc_add(T, A, p, true, Val(false)))
                     test_rrule(tensoradd!, C2, A, p, true, α, β; atol, rtol)
+                    test_rrule(tensoradd!, C1, A, p, false, α, Zero() ⊢ NoTangent(); atol, rtol)
 
                     A = rand(Bool) ? C1 : C2
                 end
@@ -107,6 +110,10 @@ for V in spacelist
                             tensorcontract!, C, A, pA, conjA, B, pB, conjB, pAB, α, β;
                             atol, rtol
                         )
+                        test_rrule(
+                            tensorcontract!, C, A, pA, conjA, B, pB, conjB, pAB, α,
+                            Zero() ⊢ NoTangent(); atol, rtol
+                        )
                     end
                 end
             end