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no deleting theorem wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 overview 2 formal statement toggle formal statement subsection 2 1 proof 3 consequence 4 see also 5 references toggle the table of contents no deleting theorem 5 languages 日本語 한국어 polski português 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from quantum no deleting theorem foundational theorem of quantum information processing in physics the no deleting theorem of quantum information theory is a no go theorem which states that in general given two copies of some arbitrary quantum state it is impossible to delete one of the copies it is a time reversed dual to the no cloning theorem 1 2 which states that arbitrary states cannot be copied it was proved by arun k pati and samuel l braunstein 3 deleting in this sense is not the same as irreversible erasure as described by landauer s principle an unknown state can be erased for example by swapping it with a standard state and discarding it into the environment deleting is closer to a reversible uncopying of one of two identical copies and the theorem shows that this cannot be done for an unknown quantum state either reversibly or irreversibly 4 intuitively it is because information is conserved under unitary evolution 5 this theorem seems remarkable because in many senses quantum states are fragile the theorem asserts that in a particular case they are also robust the no deleting theorem together with the no cloning theorem underpin the interpretation of quantum mechanics in terms of category theory and in particular as a dagger symmetric monoidal category 6 7 this formulation known as categorical quantum mechanics in turn allows a connection to be made from quantum mechanics to linear logic as the logic of quantum information theory in exact analogy to classical logic being founded on cartesian closed categories overview edit suppose that there are two copies of an unknown quantum state a pertinent question in this context is to ask if it is possible given two identical copies to delete one of them using quantum mechanical operations it turns out that one cannot the no deleting theorem is a consequence of linearity of quantum mechanics like the no cloning theorem this has important implications in quantum computing quantum information theory and quantum mechanics in general the process of quantum deleting takes two copies of an arbitrary unknown quantum state at the input port and outputs a blank state along with the original mathematically this can be described by u ψ a ψ b a c ψ a 0 b a c displaystyle u psi rangle _ a psi rangle _ b a rangle _ c psi rangle _ a 0 rangle _ b a rangle _ c where u displaystyle u is a unitary operator ψ a displaystyle psi rangle _ a is the unknown quantum state 0 b displaystyle 0 rangle _ b is the blank state a c displaystyle a rangle _ c is the initial state of the deleting machine and a c displaystyle a rangle _ c is the final state of the machine it may be noted that classical bits can be copied and deleted as can qubits in orthogonal states for example a controlled not gate transforms the two copy states 00 displaystyle 00 rangle and 11 displaystyle 11 rangle into 00 displaystyle 00 rangle and 10 displaystyle 10 rangle resetting the second copy to the standard state 0 displaystyle 0 rangle in this case we have deleted the second copy however it follows from linearity of quantum theory that there is no u displaystyle u that can perform the deleting operation for any arbitrary state ψ displaystyle psi rangle formal statement edit given three hilbert spaces for systems a b c displaystyle a b c such that the hilbert spaces for systems a b displaystyle a b are identical if u displaystyle u is a unitary transformation and a c displaystyle a rangle _ c is an ancilla state such that u ψ a ψ b a c ψ a 0 b a ψ c ψ displaystyle u psi rangle _ a psi rangle _ b a rangle _ c psi rangle _ a 0 rangle _ b a_ psi rangle _ c quad forall psi rangle where the final state of the ancilla a ψ c displaystyle a_ psi rangle _ c may depend on ψ displaystyle psi rangle then u displaystyle u is a swapping operation in the sense that the map ψ a a ψ c displaystyle psi rangle _ a mapsto a_ psi rangle _ c is an isometric embedding proof edit the theorem holds for quantum states in a hilbert space of any dimension for simplicity consider the deleting transformation for two identical qubits if two qubits are in orthogonal states then deletion requires that 0 a 0 b a c 0 a 0 b a 0 c displaystyle 0 rangle _ a 0 rangle _ b a rangle _ c rightarrow 0 rangle _ a 0 rangle _ b a_ 0 rangle _ c 1 a 1 b a c 1 a 0 b a 1 c displaystyle 1 rangle _ a 1 rangle _ b a rangle _ c rightarrow 1 rangle _ a 0 rangle _ b a_ 1 rangle _ c let ψ α 0 β 1 displaystyle psi rangle alpha 0 rangle beta 1 rangle be the state of an unknown qubit if we have two copies of an unknown qubit then by linearity of the deleting transformation we have ψ a ψ b a c α 2 0 a 0 b β 2 1 a 1 b α β 0 a 1 b 1 a 0 b a c displaystyle psi rangle _ a psi rangle _ b a rangle _ c alpha 2 0 rangle _ a 0 rangle _ b beta 2 1 rangle _ a 1 rangle _ b alpha beta 0 rangle _ a 1 rangle _ b 1 rangle _ a 0 rangle _ b a rangle _ c α 2 0 a 0 b a 0 c β 2 1 a 0 b a 1 c 2 α β φ a b c displaystyle qquad rightarrow alpha 2 0 rangle _ a 0 rangle _ b a_ 0 rangle _ c beta 2 1 rangle _ a 0 rangle _ b a_ 1 rangle _ c sqrt 2 alpha beta phi rangle _ abc in the above expression the following transformation has been used 1 2 0 a 1 b 1 a 0 b a c φ a b c displaystyle 1 sqrt 2 0 rangle _ a 1 rangle _ b 1 rangle _ a 0 rangle _ b a rangle _ c rightarrow phi rangle _ abc however if we are able to delete a copy then at the output port of the deleting machine the combined state should be ψ a 0 b a c α 0 a 0 b β 1 a 0 b a c displaystyle psi rangle _ a 0 rangle _ b a rangle _ c alpha 0 rangle _ a 0 rangle _ b beta 1 rangle _ a 0 rangle _ b a rangle _ c in general these states are not identical and hence we can say that the machine fails to delete a copy if we require that the final output states are same then we will see that there is only one option φ 1 2 0 a 0 b a 1 c 1 a 0 b a 0 c displaystyle phi rangle 1 sqrt 2 0 rangle _ a 0 rangle _ b a_ 1 rangle _ c 1 rangle _ a 0 rangle _ b a_ 0 rangle _ c and a c α a 0 c β a 1 c displaystyle a rangle _ c alpha a_ 0 rangle _ c beta a_ 1 rangle _ c since final state a displaystyle a rangle of the ancilla is normalized for all values of α β displaystyle alpha beta it must be true that a 0 c displaystyle a_ 0 rangle _ c and a 1 c displaystyle a_ 1 rangle _ c are orthogonal this means that the quantum information is simply in the final state of the ancilla one can always obtain the unknown state from the final state of the ancilla using local operation on the ancilla hilbert space thus linearity of quantum theory does not allow an unknown quantum state to be deleted perfectly consequence edit if it were possible to delete an unknown quantum state then using two pairs of epr states we could send signals faster than light thus violation of the no deleting theorem is inconsistent with the no signalling condition the no cloning and the no deleting theorems point to the conservation of quantum information stronger versions of the no cloning theorem and the no deleting theorem provide permanence to quantum information to create a copy one must import the information from some part of the universe and to delete a state one needs to export it to another part of the universe where it will continue to exist see also edit no broadcast theorem no cloning theorem no communication theorem no hiding theorem 8 quantum cloning quantum entanglement quantum information quantum teleportation uncertainty principle landauer s principle references edit w k wootters and w h zurek a single quantum cannot be cloned nature 299 1982 p 802 d dieks communication by epr devices physics letters a vol 92 6 1982 p 271 a k pati and s l braunstein impossibility of deleting an unknown quantum state nature 404 2000 p 164 pati arun k braunstein samuel l 2000 impossibility of deleting an unknown quantum state nature 404 164 165 arxiv quant ph 9911090 doi 10 1038 35005001 horodecki michał horodecki ryszard sen de aditi sen ujjwal 2005 12 01 common origin of no cloning and no deleting principles conservation of information foundations of physics 35 12 2041 2049 arxiv quant ph 0407038 bibcode 2005foph 35 2041h doi 10 1007 s10701 005 8661 4 issn 1572 9516 john baez physics topology logic and computation a rosetta stone 2009 bob coecke quantum picturalism 2009 arxiv 0908 1787 quantum no hiding theorem experimentally confirmed for first time archived from the original on 2025 11 22 retrieved 2026 02 02 v t e quantum information science general divincenzo s criteria nisq era quantum computing timeline quantum information quantum programming quantum simulation qubit physical vs logical quantum processors cloud based theorems bell s eastin knill gleason s gottesman knill holevo s no broadcasting no cloning no communication no deleting no hiding no teleportation pbr quantum speed limit threshold solovay kitaev schrödinger hjw quantum communication classical capacity entanglement assisted quantum capacity entanglement distillation entanglement swapping monogamy of entanglement locc quantum channel quantum network state purification quantum teleportation quantum energy teleportation quantum gate teleportation superdense coding quantum cryptography decoy state hidden matching post quantum cryptography quantum coin flipping quantum money quantum key distribution bb84 sarg04 other protocols quantum secret sharing quantum algorithms algorithmic cooling amplitude amplification bernstein vazirani bht boson sampling deutsch jozsa grover s hhl hidden subgroup magic state distillation quantum annealing quantum counting quantum fourier transform quantum optimization quantum phase estimation shor s simon s vqe quantum complexity theory bqp dqc1 eqp qip qma postbqp quantum processor benchmarks quantum supremacy quantum volume qc scaling laws randomized benchmarking xeb relaxation times t 1 t 2 quantum computing models adiabatic quantum computation continuous variable quantum information one way quantum computer cluster state quantum circuit quantum logic gate quantum machine learning quantum neural network quantum turing machine topological quantum computer hamiltonian quantum computation quantum error correction codes 5 qubit css gkp quantum convolutional stabilizer shor bacon shor steane toric gnu entanglement assisted physical implementations quantum optics cavity qed circuit qed linear optical qc klm protocol ultracold atoms neutral atom qc trapped ion qc spin based kane qc spin qubit qc nv center nmr qc superconducting charge qubit flux qubit phase qubit transmon quantum programming openqasm qiskit ibm qx quil forest rigetti qcs cirq q libquantum many others quantum information science quantum mechanics topics retrieved from https en wikipedia org w index php title no deleting_theorem oldid 1376476243 categories quantum information science theorems in quantum mechanics no go theorems hidden categories use american english from january 2019 all wikipedia articles written in american english articles with short description short description is different 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