Alice is told two classical bits of information and sends a qubit to Bob, who can then recover the two classical bits. Again, it relies on Alice and Bob sharing a prepared pair beforehand. It’s the opposite of quantum teleportation, where Alice sends two classical bits and Bob can recover a qubit.
First let’s talk about bases. This is the usual basis: |00>, |01>, |10>, |11>. This is the Bell basis: (|00> + |11>)/√2, (|00> - |11>)/√2, (|10> + |01>)/√2, (|10> - |01>)/√2. Check for yourself that each two vectors are orthogonal to each other. To “measure a state in a different basis” means to apply a rotation from one basis into another, then measure. For example, if you have a state and you know that it’s one of the Bell basis states, you can figure out which one, by rotating into the usual basis and measuring.
One cool thing about the Bell basis is that you can change any basis vector into any other basis vector by operations on the first qubit only. For example, rotating the first qubit by 90 degrees turns (|00> + |11>)/√2 into (|10> - |01>)√2. Flipping the first qubit gives (|10> + |01>)√2, and so on.
Now superdense coding is easy. Alice and Bob start by sharing two halves of a Bell state. Then depending on which two classical bits Alice needs to send, she either leaves the state alone or rotates into one of the other three, by operating only on her qubit. Then she sends her qubit to Bob, who now has both qubits and can rotate them into the usual basis and measure them, recovering the two classical bits.
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Superdense coding.
Alice is told two classical bits of information and sends a qubit to Bob, who can then recover the two classical bits. Again, it relies on Alice and Bob sharing a prepared pair beforehand. It’s the opposite of quantum teleportation, where Alice sends two classical bits and Bob can recover a qubit.
First let’s talk about bases. This is the usual basis: |00>, |01>, |10>, |11>. This is the Bell basis: (|00> + |11>)/√2, (|00> - |11>)/√2, (|10> + |01>)/√2, (|10> - |01>)/√2. Check for yourself that each two vectors are orthogonal to each other. To “measure a state in a different basis” means to apply a rotation from one basis into another, then measure. For example, if you have a state and you know that it’s one of the Bell basis states, you can figure out which one, by rotating into the usual basis and measuring.
One cool thing about the Bell basis is that you can change any basis vector into any other basis vector by operations on the first qubit only. For example, rotating the first qubit by 90 degrees turns (|00> + |11>)/√2 into (|10> - |01>)√2. Flipping the first qubit gives (|10> + |01>)√2, and so on.
Now superdense coding is easy. Alice and Bob start by sharing two halves of a Bell state. Then depending on which two classical bits Alice needs to send, she either leaves the state alone or rotates into one of the other three, by operating only on her qubit. Then she sends her qubit to Bob, who now has both qubits and can rotate them into the usual basis and measure them, recovering the two classical bits.