The 34-year-old actress starred as the British monarch in the hit Netflix series, but Claire has revealed she's not given Olivia any specific advice about the challenge because although they're playing the same character, it's at two distinctly different stages in the Queen's life.
Speaking to Digital Spy, she shared: ''I managed to speak to her about it, but not in a character way.
''It's more about how she's doing, is she enjoying it, is she enjoying her days at work. It's not about the character, we don't need to talk about that.''
Olivia is set to assume the role for the next two seasons of the drama series.
However, Claire doesn't feel she's able to offer her any real insight into the character.
She said: ''We're both playing two very different women in a sense and she's an absolutely extraordinary actress, so she doesn't need me to give my two-pence about what I thought about Elizabeth, she doesn't need it.''
Meanwhile, Olivia previously admitted she was thrilled to have been offered the opportunity to star in 'The Crown'.
The British actress is a big fan of the show and she welcomed the chance to play the Queen.
She explained: ''I got a phone call from my agent when I was in a car. I think I had fairly recently finished watching the first series, which I think I had done in about three nights, so I was very excited.
''My agent was trying to be subtle, not knowing who was in the car with me and she went, 'It's something about a tiara' and I went, 'Oh it's 'The Crown!'''
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