diff --git a/ext/TensorOperationsChainRulesCoreExt.jl b/ext/TensorOperationsChainRulesCoreExt.jl
index 4ea1123f..1ca4c721 100644
--- a/ext/TensorOperationsChainRulesCoreExt.jl
+++ b/ext/TensorOperationsChainRulesCoreExt.jl
@@ -71,6 +71,8 @@ function ChainRulesCore.rrule(
 end
 function _rrule_tensoradd!(C, A, pA, conjA, α, β, 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)
@@ -81,16 +83,16 @@ function _rrule_tensoradd!(C, A, pA, conjA, α, β, ba)
         ΔC = unthunk(ΔC′)
 
         dC = β === Zero() ? ZeroTangent() : @thunk projectC(pullback_dC(ΔC, β))
-        dA = @thunk projectA(tensoradd_pullback_dA(ΔC, C, A, pA, conjA, α, ba...))
+        dA = @thunk projectA(tensoradd_pullback_dA(ΔC, C_β, A, pA, conjA, α, ba...))
         dα = if _needs_tangent(α)
-            @thunk projectα(tensoradd_pullback_dα(ΔC, C, A, pA, conjA, α, ba...))
+            @thunk projectα(tensoradd_pullback_dα(ΔC, C_β, A, pA, conjA, α, ba...))
         else
             ZeroTangent()
         end
         dβ = if _needs_tangent(β)
-            @thunk projectβ(pullback_dβ(ΔC, C, β))
+            @thunk projectβ(pullback_dβ(ΔC, C_β, β))
         else
-            ZeroTangent()
+            NoTangent()
         end
         dba = map(_ -> NoTangent(), ba)
         return NoTangent(), dC, dA, NoTangent(), NoTangent(), dα, dβ, dba...
@@ -112,6 +114,7 @@ function ChainRulesCore.rrule(
 end
 function _rrule_tensorcontract!(C, A, pA, conjA, B, pB, conjB, pAB, α, β, 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)
@@ -123,17 +126,17 @@ function _rrule_tensorcontract!(C, A, pA, conjA, B, pB, conjB, pAB, α, β, ba)
         ΔC = unthunk(ΔC′)
 
         dC = β === Zero() ? ZeroTangent() : @thunk projectC(pullback_dC(ΔC, β))
-        dA = @thunk projectA(tensorcontract_pullback_dA(ΔC, C, A, pA, conjA, B, pB, conjB, pAB, α, ba...))
-        dB = @thunk projectB(tensorcontract_pullback_dB(ΔC, C, A, pA, conjA, B, pB, conjB, pAB, α, ba...))
+        dA = @thunk projectA(tensorcontract_pullback_dA(ΔC, C_β, A, pA, conjA, B, pB, conjB, pAB, α, ba...))
+        dB = @thunk projectB(tensorcontract_pullback_dB(ΔC, C_β, A, pA, conjA, B, pB, conjB, pAB, α, ba...))
         dα = if _needs_tangent(α)
-            @thunk projectα(tensorcontract_pullback_dα(ΔC, C, A, pA, conjA, B, pB, conjB, pAB, α, ba...))
+            @thunk projectα(tensorcontract_pullback_dα(ΔC, C_β, A, pA, conjA, B, pB, conjB, pAB, α, ba...))
         else
             ZeroTangent()
         end
         dβ = if _needs_tangent(β)
-            @thunk projectβ(pullback_dβ(ΔC, C, β))
+            @thunk projectβ(pullback_dβ(ΔC, C_β, β))
         else
-            ZeroTangent()
+            NoTangent()
         end
         dba = map(_ -> NoTangent(), ba)
         return NoTangent(), dC,
@@ -156,6 +159,7 @@ function ChainRulesCore.rrule(
 end
 function _rrule_tensortrace!(C, A, p, q, conjA, α, β, 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)
@@ -166,16 +170,16 @@ function _rrule_tensortrace!(C, A, p, q, conjA, α, β, ba)
         ΔC = unthunk(ΔC′)
 
         dC = β === Zero() ? ZeroTangent() : @thunk projectC(pullback_dC(ΔC, β))
-        dA = @thunk projectA(tensortrace_pullback_dA(ΔC, C, A, p, q, conjA, α, ba...))
+        dA = @thunk projectA(tensortrace_pullback_dA(ΔC, C_β, A, p, q, conjA, α, ba...))
         dα = if _needs_tangent(α)
-            @thunk projectα(tensortrace_pullback_dα(ΔC, C, A, p, q, conjA, α, ba...))
+            @thunk projectα(tensortrace_pullback_dα(ΔC, C_β, A, p, q, conjA, α, ba...))
         else
             ZeroTangent()
         end
         dβ = if _needs_tangent(β)
-            @thunk projectβ(pullback_dβ(ΔC, C, β))
+            @thunk projectβ(pullback_dβ(ΔC, C_β, β))
         else
-            ZeroTangent()
+            NoTangent()
         end
         dba = map(_ -> NoTangent(), ba)
         return NoTangent(), dC, dA, NoTangent(), NoTangent(), NoTangent(), dα, dβ, dba...
diff --git a/test/ad.jl b/test/ad.jl
index 3d8e50e1..1be00fc8 100644
--- a/test/ad.jl
+++ b/test/ad.jl
@@ -2,6 +2,8 @@ using TensorOperations
 using TensorOperations: StridedBLAS, StridedNative
 using Test
 using ChainRulesTestUtils
+using ChainRulesCore: NoTangent
+using VectorInterface: Zero
 
 ChainRulesTestUtils.test_method_tables()
 
@@ -23,6 +25,7 @@ ChainRulesTestUtils.test_method_tables()
 
     test_rrule(tensortrace!, C, A, p, q, false, α, β; atol, rtol)
     test_rrule(tensortrace!, C, A, p, q, true, α, β; atol, rtol)
+    test_rrule(tensortrace!, C, A, p, q, false, α, Zero() ⊢ NoTangent(); atol, rtol)
 
     test_rrule(tensortrace!, C, A, p, q, true, α, β, StridedBLAS(); atol, rtol)
     test_rrule(tensortrace!, C, A, p, q, false, α, β, StridedNative(); atol, rtol)
@@ -52,6 +55,7 @@ end
     β = rand(T)
     test_rrule(tensoradd!, C, A, pA, false, α, β; atol, rtol)
     test_rrule(tensoradd!, C, A, pA, true, α, β; atol, rtol)
+    test_rrule(tensoradd!, C, A, pA, true, α, Zero() ⊢ NoTangent(); atol, rtol)
 
     test_rrule(tensoradd!, C, A, pA, false, α, β, StridedBLAS(); atol, rtol)
     test_rrule(tensoradd!, C, A, pA, true, α, β, StridedNative(); atol, rtol)
@@ -80,6 +84,7 @@ end
     test_rrule(tensorcontract!, C, A, pA, true, B, pB, false, pAB, α, β; atol, rtol)
     test_rrule(tensorcontract!, C, A, pA, false, B, pB, true, pAB, α, β; atol, rtol)
     test_rrule(tensorcontract!, C, A, pA, true, B, pB, true, pAB, α, β; atol, rtol)
+    test_rrule(tensorcontract!, C, A, pA, false, B, pB, true, pAB, α, Zero() ⊢ NoTangent(); atol, rtol)
 
     test_rrule(
         tensorcontract!, C, A, pA, false, B, pB, false, pAB, α, β, StridedBLAS();