Assumptions with complex variables (possible BUG)

[Janusz Przewocki]

I think I have found a bug.

In the code below I am trying to define a symbolic unitary matrix. In doing so, I define variables a, b, c, d - entries of the matrix. Unitary condition gives us assumptions on a, b, c, d, that I force using assume function. While trying to check if my matrix is unitary I have observed strange results.

In particular, Out[5] does not make sense, since c*conjugate(c) is always nonnegative.

+*In[1]:*+
[source, ipython2]
----
#We define variables
a, b, c, d = var('a, b, c, d')

#U is supposed to be an unitary matrix
U = matrix([[a, b], [c, d]])
M = U.H*U - identity_matrix(2)

#Assumptions on the entries of U
U_asm = [M[i][j] == 0 for i in range(2) for j in range(2)]
assume([U_asm[a] for a in [0, 1, 3]])
----


+*In[2]:*+
[source, ipython2]
----
assumptions()
----


+*Out[2]:*+
----[a*conjugate(a) + c*conjugate(c) - 1 == 0,
 b*conjugate(a) + d*conjugate(c) == 0,
 b*conjugate(b) + d*conjugate(d) - 1 == 0]----


+*In[3]:*+
[source, ipython2]
----
U.H*U == identity_matrix(2)
----


+*Out[3]:*+
----True----


+*In[4]:*+
[source, ipython2]
----
bool(a*conjugate(a) + c*conjugate(c) - 1 == 0)
----


+*Out[4]:*+
----False----


+*In[5]:*+
[source, ipython2]
----
a*conjugate(a) + c*conjugate(c) - 1 == 0
----


+*Out[5]:*+
----c*conjugate(c) + 8 == 0----


+*In[6]:*+
[source, ipython2]
----
version()
----


+*Out[6]:*+
----'SageMath version 9.0, Release Date: 2020-01-01'----

---------------------------------------

My system is: Arch Linux

My architecture is: 64 bit.

[Markus Wageringel]

You are probably using Sage with Python 2. In Python 2, a list expression involving an index variable `a` like

    assume([U_asm[a] for a in [0, 1, 3]])

overwrites the previous content of the variable `a`. At the end of the loop, `a` will have the value `3`, which leads to the output you obtained. If you rename the loop index to something other than a,b,c,d, the problem goes away.

With Python 3, this problem does not exist either, as loop variables are local as one would expect. Therefore, I suggest to switch to using Sage with Python 3.

[Janusz Przewocki]

Thank you, I did not notice it in my code.

Regards

  Janusz