You are presumed to be eligible for unemployment benefits, since EDD presumes that an employer was terminated through no fault of his or her own, and not for cause.
However, based on the statement that you received from your employer, it is likely that your employer will allege that you were terminated for cause due to misconduct since there were alleged problems with your work quality.
In order to demonstrate this , the employer would need to prove:
The claimant owes a material duty to the employer under the contract of employment.
There is a substantial breach of that duty.
The breach is a wilful or wanton disregard of that duty.
The breach disregards XXXXX XXXXX's interests and injures or tends to injure the employer's interests.
Based on the facts that you mentioned, unfortunately, your employer may be successful in proving misconduct, so it's possible you will not be eligible for unemployment benefits.
You can read more about misconduct here:http://www.edd.ca.gov/uibdg/Misconduct_MC_5.htm
I sincerely XXXXX XXXXX had better news to give you, but I hope you appreciate a direct and honest answer to your question.