"Perhaps you are still quite happy with mean-field theory, which is valid in most cases, where no entanglement needs to be considered. This does not mean that the system is not entangled, but just perhaps not strongly entangled." – P4

What does that mean? How do we bipartite a system after mean field approximation?

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1.1 Classical correlation

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1.1.1 Joint Probability without correlations

Theorem 1.1.1 The following statements are equivalent:

There is no correlation in the joint probability distribution
The probability of
vice versa of 2

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1.1.2 Correlation functions from classical probability theory

Definition 1.1.2

(1)

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1.1.2.1 Mutual Information

Definition 1.1.3

(2)

Cute thing of mutual information is that it doesn't calculate expectation values, it's independent of the value of the variables.

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1.1.2.1.1 Mutual Information and KL divergence

Definition 1.1.4

(3)

Thus,

(4)

1.1.2.1.1.1 Mutual Information and correlation

Definition 1.1.5 perfect correlation

(5)

In this case,

(6)

Definition 1.1.6 entropy for joint distribution

(7)

and conditional entropy

(8)

In this language

(9)

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1.2 Quantum entanglement

What's different from a quantum bit comparing to a classical bit?

Remark

The probability distribution of a pure quantum state must be associated with a chosen measurement basis.

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1.2.1 Reduced density matrix

(10)

For a whole pure state , different trace basis will give different reduced density matrix.

The proof in the book seems a bit ad hoc, I copy and paste my previous note here:

Definition 1.2.1

(11)

(12)

Proof. Write , define

Define

(13)

Thus,

and are orthogonal! Adding a normalization constant makes them orthonormal, thus proving the Schimidt decomposition. ⁠

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1.2.2 Pure bipartite state

We didn't constrain basis before, but in the context of schmidt decomposition, we can always find a basis , such that they are orthonormal

Theorem 1.2.2 A state has no correlation iff

, or

Schmidt decomposition gives a criterion of entanglement:

Theorem 1.2.3 The following statements are equivalent:

is entangled
Schmidt rank
Reduced density matrix (or ) is mixed state

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1.2.3 Seperable mixed state

Definition 1.2.4 A mixed state is seperable iff it can be written as

(14)

with ,

This is generally an NP-hard problem to determine whether a mixed state is seperable or not. We define

Definition 1.2.5

(15)

where where is the schmidt coefficients of

For seperable states, .

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1.2.4 Peres-Horodecki Criterion

also termed as PPT criterion, for positive partial transpose. It is a necessary condition for separability.

Theorem 1.2.6 If a mixed state is separable, then the partial transpose is positive semidefinite.

Take Bell state as an example,

(16)

Taking partial transpose on B, you would get negative eigenvalues, thus it's entangled.

Remark

This is called negativity in many body context. Check it out!

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2 Open Quantum system

Sorry for personal reason I'm too lazy to read this part. Would follow up maybe.

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2.1 The orthogonalization of Kraus operators

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3 Quantum Error-Correcting Codes

Remark

In some sub Hilbert space, the projection of noise channel can be unitary.

Example. Consider Kraus Operator , in the subspace spanned by , the projection is unitary.

Because:

(17)

(18)

which an unitary operation on the subspace.

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3.1 Bit Flip Code

Definition 3.1.1

(19)

i.e. the Kraus operators are ,

Thus, for

(20)

Definition 3.1.2 Probability of failure

(21)

In general this error is inevitable. But at best we can alleviate it.

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3.1.1 Repitition Code

By letting

(22)

By performing orthogonal measurement with projectors

(23)

we find

(24)

These s are called the error syndromes.

When getting these syndromes, we can perform correction operations:

(25)

After correction,

(26)

Thus the final error probability is

(27)

If , then .

The repetition of code can improve this further to .

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3.2 Shor's code

Shor's code can correct both bit flip and phase flip errors.

Starting from

(28)

let

(29)

For Depolarizing channel

(30)

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3.3 Error Probability Calculation (Order Estimation)

For a code that corrects any single-qubit error ( ), the logical error occurs only when at least 2 qubits fail.

Let the probability of a physical error on one qubit be . The probability of logical failure is the sum of probabilities of having errors:

(31)

Thus, the logical error probability is suppressed quadratically:

(32)

Remark

What's the quantum part of this?

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3.4 Quantum Code

Definition 3.4.1 A quantum code is a subspace of the total Hilbert space . The dimension of the code is , and the number of physical qubits is .

To distinguish different errors,

Theorem 3.4.2 Quantum error-correcting condition: A quantum code can correct a set of errors iff

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Definition 3.4.3 Quantum Code distance A quantum code has distance iff it can detect all errors affecting up to qubits, i.e. it can correct up to qubit errors.

(34)

for all operators acting on up to qubits.

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3.5 Stabilizer Codes

Definition 3.5.1 A stabilizer is an abelian subgroup of the Pauli group that does not contain .

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3.5.1 Shor's Code

The Stabilizer group of Shor's code is generated by

(35)

The Projector onto the code space is

(36)

Define which gives .

Thus the basis state is stabilized by

(37)

We can also define , as .

Hence

(38)

(39)

As we are dealing with states,

(40)

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3.5.2 Stabilizer Formalism

Definition 3.5.2 (Stabilizer Code) A Stabilizer Code is defined by a Stabilizer Group , such that

(41)

Then is the Stabilizer code and is its Stabilizer.

Definition 3.5.3 (Stabilizer State) If a Stabilizer code of qubit has independent generators, then the code space is one-dimensional, and the unique state in the code space is called a Stabilizer State.

Problem. How to exhaust all Stabilizer States of N qubits?