The Nebraskan State Legislature in the United States is a unicameral house. There is no upper house such as the Senate in the Australian system or the Legislative Councils in the Australian states. The Nebraskan Senators at the federal level recently proposed that the state adopt a bicameral legislative structure.
From the article, Senators question Unicameral set-up ;
Friend argued that a two-house system could provide more checks and balances of laws, prevent what he called legislative tyranny and infuse the lawmaking process with more information and precision.In the Nebraskan Constitution the Legislature must be at least thirty members but no larger than fifty. Which is a relatively small body of legislators in what is a small state. Nebraska did not adopt a unicameral legislative structure until 1937 when the motion passed by referendum. Half the house is up for election every four years, so it kind of operates in a staggered format such as the Australian Senate does. Unlike the Australian parliamentary systems, Nebraska has a separate executive from the legislative so issues of separation of powers are not the same as in a unicameral parliamentary system. Additionally the Nebraskan House does not recognize parties with the intent of creating more independent legislators. The Nebraskan Constitution recognizes several formal executive positions which are all elected;
- Governor
- Lieutenant-Governor
- Secretary of State
- Auditor of Public Accounts
- Treasurer
- Attorney-General









Comments
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The States of the US actually give the executive mandate from the voters, which allows for freedom of conscience from the Parliament. If the Australian Parliament voted as freely as the US Congress, there would be changes in Government every two months.
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The separation of powers simply means that as you say, bicameralism is not needed to check the executive or legislature. Thus, a directly elected Governor with line-item veto and a House elected by Hare-Clark proportional representation (using multi member electorates) [with the two-thirds vote to overrule veto] would satisfy all necessary checks and balances whilst being highly representative and more efficient.
I would prefer we had the same elected executive and PR unicameral legislatures here.