Typical subspace

Short description: Term in quantum information theory

In quantum information theory, the idea of a typical subspace plays an important role in the proofs of many coding theorems (the most prominent example being Schumacher compression). Its role is analogous to that of the typical set in classical information theory.

## Contents

* 1 Unconditional quantum typicality
* 2 Conditional quantum typicality
* 3 See also
* 4 References

## Unconditional quantum typicality

Consider a density operator ρ with the following spectral decomposition:

ρ=∑xpX(x)|x⟩⟨x|.

The weakly typical subspace is defined as the span of all vectors such that the sample entropy H‾(xn) of their classical label is close to the true entropy H(X) of the distribution pX(x):

TδXn≡span{|xn⟩:|H‾(xn)−H(X)|≤δ},

where

H‾(xn)≡−1nlog⁡(pXn(xn)),
H(X)≡−∑xpX(x)log⁡pX(x).

The projector Πρ,δn onto the typical subspace of ρ is defined as

Πρ,δn≡∑xn∈TδXn|xn⟩⟨xn|,

where we have "overloaded" the symbol TδXn to refer also to the set of δ-typical sequences:

TδXn≡{xn:|H‾(xn)−H(X)|≤δ}.

The three important properties of the typical projector are as follows:

Tr{Πρ,δnρ⊗n}≥1−ϵ,
Tr{Πρ,δn}≤2n[H(X)+δ],
2−n[H(X)+δ]Πρ,δn≤Πρ,δnρ⊗nΠρ,δn≤2−n[H(X)−δ]Πρ,δn,

where the first property holds for arbitrary ϵ,δ>0 and sufficiently large n.

## Conditional quantum typicality

Consider an ensemble {pX(x),ρx}x∈𝒳 of states. Suppose that each state ρx has the following spectral decomposition:

ρx=∑ypY|X(y|x)|yx⟩⟨yx|.

Consider a density operator ρxn which is conditional on a classical sequence xn≡x1⋯xn:

ρxn≡ρx1⊗⋯⊗ρxn.

We define the weak conditionally typical subspace as the span of vectors (conditional on the sequence xn) such that the sample conditional entropy H‾(yn|xn) of their classical labels is close to the true conditional entropy H(Y|X) of the distribution pY|X(y|x)pX(x):

TδYn|xn≡span{|yxnn⟩:|H‾(yn|xn)−H(Y|X)|≤δ},

where

H‾(yn|xn)≡−1nlog⁡(pYn|Xn(yn|xn)),
H(Y|X)≡−∑xpX(x)∑ypY|X(y|x)log⁡pY|X(y|x).

The projector Πρxn,δ onto the weak conditionally typical subspace of ρxn is as follows:

Πρxn,δ≡∑yn∈TδYn|xn|yxnn⟩⟨yxnn|,

where we have again overloaded the symbol TδYn|xn to refer to the set of weak conditionally typical sequences:

TδYn|xn≡{yn:|H‾(yn|xn)−H(Y|X)|≤δ}.

The three important properties of the weak conditionally typical projector are as follows:

𝔼Xn{Tr{ΠρXn,δρXn}}≥1−ϵ,
Tr{Πρxn,δ}≤2n[H(Y|X)+δ],
2−n[H(Y|X)+δ] Πρxn,δ≤Πρxn,δ ρxn Πρxn,δ≤2−n[H(Y|X)−δ] Πρxn,δ,

where the first property holds for arbitrary ϵ,δ>0 and sufficiently large n, and the expectation is with respect to the distribution pXn(xn).

## See also

* Classical capacity
* Quantum information theory

## References

* Wilde, Mark M., 2017, Quantum Information Theory, Cambridge University Press, Also available at eprint arXiv:1106.1145

* v
* t
* e

Quantum information science

General|

* Quantum computing
* Qubit
* physical vs. logical
* DiVincenzo's criteria
* Quantum information
* Quantum programming
* Quantum processors
* Cloud-based quantum computing
* Timeline of quantum computing

Theorems|

* Bell's theorem
* Gleason's theorem
* Gottesman–Knill theorem
* Holevo's theorem
* Margolus–Levitin theorem
* No-broadcast theorem
* No-cloning theorem
* No-communication theorem
* No-deleting theorem
* No-hiding theorem
* No-teleportation theorem
* PBR theorem
* Quantum threshold theorem

Quantum
communication|

* Quantum capacity
* Classical capacity
* Entanglement-assisted classical capacity
* Quantum channel
* Quantum network
* Quantum cryptography
* Quantum key distribution
* BB84
* SARG04
* Three-stage quantum cryptography protocol
* Quantum teleportation
* Superdense coding
* LOCC
* Entanglement distillation

Quantum algorithms|

* Universal quantum simulator
* Deutsch–Jozsa algorithm
* Grover's algorithm
* Quantum Fourier transform
* Shor's algorithm
* Simon's problem
* Quantum phase estimation algorithm
* Quantum counting algorithm
* Quantum annealing
* Quantum algorithm for linear systems of equations
* Amplitude amplification

Quantum
complexity theory|

* Quantum Turing machine
* EQP
* BQP
* QMA
* PostBQP
* QIP

Quantum
computing models|

* Quantum circuit
* Quantum logic gate
* One-way quantum computer
* cluster state
* Adiabatic quantum computation
* Topological quantum computer

Quantum
error correction|

* Stabilizer codes
* Entanglement-Assisted Quantum Error Correction
* Shor code
* Steane code
* CSS code
* Quantum convolutional codes
* Toric code

Physical
implementations|

| Quantum optics|

* Cavity QED
* Circuit QED
* Linear optical quantum computing
* KLM protocol
* Boson sampling

|

Ultracold atoms|

* Trapped ion quantum computer
* Optical lattice

Spin-based|

* Nuclear magnetic resonance QC
* Kane QC
* Loss–DiVincenzo QC
* Nitrogen-vacancy center

Superconducting
quantum computing|

* Charge qubit
* Flux qubit
* Phase qubit
* Transmon

Software|

* libquantum
* OpenQASM
* Q#
* Qiskit
* IBM Q Experience

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