Can u love if you cant trust?

So I have been with my lady for 9 months now. We are a lesbian couple and doing well. She hasnt come out to her parents yet and therefore lies to them about EVERYTHING from what she ate that day to what she is doing or have done. Lies about EVERYTHING. I also catch her lying to her sister and other random friends. I keep thinking to myself, if she can lie SO easily to her parents what makes me think that she wont lie to me like that? She is self employeed and makes her own hours. Sometimes she works late and i know the people she is with but that means nothing to me bc i know some of them would lie for her. Sometimes when she gets home at 1 or 2 in the morning its fusterating. Or when shes at a meeting till 11 or 12 sometimes even 1am that started at 7:30 that night. Or how she is SO private with her cell phone. Always locked. Always faced down. Im not the type to go through her sh!t but sometimes it just gets sooo obnoxious and my insecurities start to play a HUGE role. I get mad and start arguments. She hates it but can you blame me? I feel like anyone in my positions would be the same way. Even when she tells me over and over how she loves me and how she doesnt lie to me. I caught her on a lie. She doesnt know about it tho. I love her so much and a part of me wants to trust her so much but the other part of me doesnt trust her one bit and it makes my life so much harder. Its the hardest thing wanting to trust the one you love but not being able to. So I kinda figured i'd stick it out till sh!t hits the fan. A relationship without trust is not a relationship at all. But im not sure if sticking it out is really want it want. If you knew your spouse or partner was a liar to that extent, what would you do?